The lander structure was four triangles, folded into a tetrahedron until after the airbags deflated.
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Currently under construction, the tetrahedron appears to be swept away from the West Side Highway as if by the Hudson River's tides.
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"Between a semi-lifelike thing and a tetrahedron with eyeballs, there's no real difference in terms of building a relationship," he said.
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He achieved this by dividing the globe into 96 triangles and projecting them onto a tetrahedron, preserving the proportions of water and land.
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Then he unfolded the tetrahedron into a rectangle, where the 96 sections created a map resembling the surface of the original globe, only flat.
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When Mr. Rausing became chief executive, the company was selling only tetrahedron-shaped cartons, and annual production of Tetra Pak cartons had not yet reached one billion.
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He found that a four-satellite system has a very high dynamic symmetry — in fact, it is a cousin to the tetrahedron, the first of the Platonic solids.
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BIG's greatest gift is its ability to contort and adapt building envelopes to their sites, and its Via West 57th, a tetrahedron-shaped Manhattan apartment tower, is no exception.
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That's why he likes symbols so much, he said, and has them tattooed on his fingers, arms and neck: the sign of Saturn, a fire tetrahedron, a dagger-slash-cross.
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The new metropolis would consist of a giant tetrahedron—a pyramid made of equilateral triangles—a shape that Fuller, the popularizer of the geodesic dome, admired for its stability and symmetry.
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Tetra Pak has been in the business of packaging for more than 60 years, with its name a nod to the tetrahedron-shaped package that founder Ruben Rausing developed all those years ago.
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Two new Manhattan projects — the tetrahedron luxury rental at 625 West 23th Street (currently under construction) and the apparently non-Euclidean stack that will be 2 WTC (in concept stage) — have been designed by BIG, the firm led by Dane Bjarke Ingels.
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Animation of a Reuleaux tetrahedron, showing also the tetrahedron from which it is formed. Four balls intersect to form a Reuleaux tetrahedron. The Reuleaux tetrahedron is the intersection of four balls of radius s centered at the vertices of a regular tetrahedron with side length s. The spherical surface of the ball centered on each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron.
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Shockley partials and Frank partials can combine to form a Thompson tetrahedron, or a stacking fault tetrahedron.
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A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles. Every Heronian tetrahedron can be arranged in Euclidean space so that its vertex coordinates are also integers.
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The trigonometry of a tetrahedron explains the relationships between the lengths and various types of angles of a general tetrahedron.
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Chérest, M.; Felkin, H. Tetrahedron Lett., 1971, 383.Huet, J.; Maroni- Barnaud, Y.; Anh, N. T.; Seyden-Penne, J. Tetrahedron Lett., 1976, 159.
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A central cross section of a regular tetrahedron is a square. The two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a rectangle.Sections of a Tetrahedron When the intersecting plane is near one of the edges the rectangle is long and skinny.
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Each vertex of the Reeve tetrahedron lies on a fundamental lattice point (a point in ). No other fundamental lattice points lie on the surface or in the interior of the tetrahedron. The volume of the Reeve tetrahedron is . In 1957 Reeve used this tetrahedron to show that there exist tetrahedra with four lattice points as vertices, and containing no other lattice points, but with arbitrarily large volume.
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J. Urban et al., Tetrahedron Lett., 31, 4421–4424 (1990) and in the presence of phase transfer catalystsV. J. Patil, Tetrahedron Lett., 37, 1481–1484 (1996).
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This yields a tetrahedron with edge-length 2, centered at the origin. For the other tetrahedron (which is dual to the first), reverse all the signs. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube.
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PropargylMüller, T. J. J.; Netz, A. Tetrahedron Lett. 1999, 40, 3145. and oxoniumDavies, S. G.; Newton, R. F.; Williams, J. M. J. Tetrahedron Lett. 1989, 30, 2967.
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H. Nakamura, H. Wu, Y. Ohizumi and Y. Hirata, Tetrahedron Lett., 1984, 25, 2989H. Wu, H. Nakamura, J. Kobayashi and Y. Ohizumi, Tetrahedron Lett., 1984, 25, 3719R.
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In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Lhuilier in 1782, and got the name orthocentric tetrahedron by G. de Longchamps in 1890.. In an orthocentric tetrahedron the four altitudes are concurrent. This common point is called the orthocenter, and it has the property that it is the symmetric point of the center of the circumscribed sphere with respect to the centroid.
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Mount Steele Cabin in Tetrahedron Provincial Park Tetrahedron Provincial Park is a provincial park in British Columbia, Canada, located northeast of Sechelt in the Sunshine Coast area of the province. The park was created in 1995.Wilderness Committee - Sunshine Coast Its highest point is Tetrahedron Peak, at 1727 metres. It also contains Panther Peak (1681 metres), and Mt. Steele (1651 metres) BC Provincial Park Planning Document: Tetrahedron Provincial Park.
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In geometry, a disphenoid (from Greek sphenoeides, "wedgelike") is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are sphenoid,. bisphenoid, isosceles tetrahedron,.
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Tetrahedron Lett. 2001, 42, 285-287.Okada, M.; Nakamura, Y. Sago, A.; Hirokawa, H.; Taguchi, T. Stereoselective cosnstruction of functionalized (Z)- fluoroalkenes directed to o- depsipeptide isosteres. Tetrahedron Lett.
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The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres. It has respective centers such as incenter, circumcenter, excenters, Spieker center and points such as a centroid. However, there is generally no orthocenter in the sense of intersecting altitudes. Gaspard Monge found a center that exists in every tetrahedron, now known as the Monge point: the point where the six midplanes of a tetrahedron intersect.
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Erik Wallenberg with the tetrahedron package Erik Wallenberg (25 December 1915 – 18 October 1999) was a Swedish engineer. He is credited for inventing the Tetra Pak tetrahedron packaging in 1944.
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In geometry, the augmented truncated tetrahedron is one of the Johnson solids (J65). It is created by attaching a triangular cupola (J3) to one hexagonal face of a truncated tetrahedron.
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The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.
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Moreover, the angle made by the faces is \pi/3. The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.
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The tetrahedron is a 2-complex. The link of a vertex of a tetrahedron is the triangle. Let be a simplicial complex. The link of a vertex is the graph constructed as follows.
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Matemateca IME-USP) 3D model of regular tetrahedron. In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex.
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A tetrahedron is a 3-simplex. Unlike the case of the other Platonic solids, all the vertices of a regular tetrahedron are equidistant from each other (they are the only possible arrangement of four equidistant points in 3-dimensional space). A tetrahedron is a triangular pyramid, and the regular tetrahedron is self-dual. A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. For one such embedding, the Cartesian coordinates of the vertices are :(+1, +1, +1); :(−1, −1, +1); :(−1, +1, −1); :(+1, −1, −1).
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100px The Icosahedron colored as a snub tetrahedron has chiral symmetry.
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Hence the orthocenter coincides with the Monge point of the tetrahedron.
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In three dimensions, an orthoscheme is also called a birectangular tetrahedron.
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If the four faces of a tetrahedron have the same perimeter, then the tetrahedron is a disphenoid. If the four faces of a tetrahedron have the same area, then it is a disphenoid. The centers in the circumscribed and inscribed spheres coincide with the centroid of the disphenoid. The bimedians are perpendicular to the edges they connect and to each other.
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Vatèle, J.-M.; Dumas, D.; Goré, J. Tetrahedron Lett. 1990, 31, 2277.
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The Kelvin coupling is named after William Thompson (Lord Kelvin) who published the design in 1868–71. It consists of three spherical surfaces that rest on a concave tetrahedron, a V-groove pointing towards the tetrahedron and a flat plate. The tetrahedron provides three contact points, while the V-groove provides two and the flat provides one for a total required six contact points. The benefits of this design is that the center of rotation is located at the tetrahedron, however it suffers from contact stress problems in high load applications.
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Four balls intersect to form a Reuleaux tetrahedron. The intersection of four balls of radius s centered at the vertices of a regular tetrahedron with side length s is called the Reuleaux tetrahedron, but its surface is not a surface of constant width. Weber also has films of both types of Meissner body rotating as well as interactive images. It can, however, be made into a surface of constant width, called Meissner's tetrahedron, by replacing three of its edge arcs by curved surfaces, the surfaces of rotation of a circular arc.
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The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the Euler line of the tetrahedron that is analogous to the Euler line of a triangle. The nine-point circle of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is the twelve-point sphere and besides the centroids of the four faces of the reference tetrahedron, it passes through four substitute Euler points, one third of the way from the Monge point toward each of the four vertices.
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Silicate tetrahedron. A silicon–oxygen tetrahedron is the SiO4 anionic group, or a silicon atom with four surrounding oxygen atoms arranged to define the corners of a tetrahedron. This is a fundamental component of most silicates in the Earth's crust. A variety of silicate minerals can be identified by the way that the tetrahedra links differ, also by the cations present in the mineral.
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A regular tetrahedron is a tetrahedron in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity. In a regular tetrahedron, all faces are the same size and shape (congruent) and all edges are the same length. a thin volume of empty space is left, where the five edge angles do not quite meet.
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Banwell currently serves on the editorial boards for several journals such as Tetrahedron.
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Other enzymes that may be used for asymmetric ester hydrolysis include electric eel acetylcholinesterase,Deardorff, D. R.; Matthews, A. J.; McMeekin, D. S.; Craney, C. L. Tetrahedron Lett. 1986, 27, 1255. chymotrypsin, and Baker's yeast.Kerscher, V.; Kreiser, W. Tetrahedron Lett.
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For space-filling, the triakis truncated tetrahedron can be constructed as follows: # Truncate a regular tetrahedron such that the big faces are regular hexagons. # Add an extra vertex at the center of each of the four smaller tetrahedra that were removed.
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The Prada Transformer is an unusual building which has one of four different apparent shapes, depending on the function for which the building is needed at the moment. The building is roughly in the shape of a tetrahedron. Cranes rotate the building so that different surfaces of the tetrahedron face downward, thereby changing the building's form and function. The different faces of the "tetrahedron" are actually shapes other than triangles.
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A Goursat tetrahedron can be represented graphically by a tetrahedral graph, which is in a dual configuration of the fundamental domain tetrahedron. In the graph, each node represents a face (mirror) of the Goursat tetrahedron. Each edge is labeled by a rational value corresponding to the reflection order, being π/dihedral angle. :100px A 4-node Coxeter-Dynkin diagram represents this tetrahedral graphs with order-2 edges hidden.
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The octahedron represents the central intersection of two tetrahedra The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids.
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Tetrahedron Peak is a mountain in British Columbia, Canada. The highest peak on the Sunshine Coast, it rises to an elevation of 1739 metres (5705 ft), with a prominence of 1164 metres (3819 ft). It is located in the Tetrahedron Provincial Park.BC Parks.
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1970, 35, 10. Electrophiles that react with the benzylic anions formed by these methods include aldehydes and ketones, activated (primary, allylic, or benzylic) halides,Flippin, L. A. Tetrahedron Lett. 1991, 32, 6857. molecular oxygen,Wasserman, H. H.; Lipshutz, B. H. Tetrahedron Lett.
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The shape is distorted from the tetrahedron due to the non-symmetry of the tripod.
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All tetrahedra can be inscribed in a parallelepiped. A tetrahedron is orthocentric if and only if its circumscribed parallelepiped is a rhombohedron. Indeed, in any tetrahedron, a pair of opposite edges is perpendicular if and only if the corresponding faces of the circumscribed parallelepiped are rhombi. If four faces of a parallelepiped are rhombi, then all edges have equal lengths and all six faces are rhombi; it follows that if two pairs of opposite edges in a tetrahedron are perpendicular, then so is the third pair, and the tetrahedron is orthocentric. A tetrahedron ABCD is orthocentric if and only if the sum of the squares of opposite edges is the same for the three pairs of opposite edges:Reiman, István, "International Mathematical Olympiad: 1976-1990", Anthem Press, 2005, pp. 175-176.
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The ammonium ion is tetrahedral The tetrahedron shape is seen in nature in covalently bonded molecules. All sp3-hybridized atoms are surrounded by atoms (or lone electron pairs) at the four corners of a tetrahedron. For instance in a methane molecule () or an ammonium ion (), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called Tetrahedron.
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Reeve tetrahedron In geometry, the Reeve tetrahedron is a polyhedron, in three-dimensional space with vertices at , , and where is a positive integer. It is named after John Reeve, who used it to show that higher-dimensional generalizations of Pick's theorem do not exist.
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The three-dimensional case: the four spheres intercepts other spheres on black circles. There is also a three-dimensional analog, in which the four spheres passing through a point of a tetrahedron and points on the edges of the tetrahedron intersect in a common point.
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The four blocks can be rearranged such that they fit, equidistantly spaced, inside a regular tetrahedron.
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Tetrahedron 68, 8545 (2012). Many of the bioactive compounds found on Suberites are microbial in nature.
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A solid angle of π sr is one quarter of that subtended by all of space. When all the solid angles at the vertices of a tetrahedron are smaller than π sr, O lies inside the tetrahedron, and because the sum of distances from O to the vertices is a minimum, O coincides with the geometric median, M, of the vertices. In the event that the solid angle at one of the vertices, v, measures exactly π sr, then O and M coincide with v. If however, a tetrahedron has a vertex, v, with solid angle greater than π sr, M still corresponds to v, but O lies outside the tetrahedron.
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A tetrahedron is a three-dimensional object bounded by four triangular faces. Seven lines associated with a tetrahedron are concurrent at its centroid; its six midplanes intersect at its Monge point; and there is a circumsphere passing through all of the vertices, whose center is the circumcenter. These points define the "Euler line" of a tetrahedron analogous to that of a triangle. The centroid is the midpoint between its Monge point and circumcenter along this line.
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The rhetorical tetrahedron, on the other hand, carries those 3 points, along with adding context into the picture, as shown in the top image to the right. Simply put, context can help explain the "why" and "how" something is written by introducing the setting in which it was created. As seen in the more complex image, a tetrahedron is a triangular pyramid, and the rhetorical tetrahedron adds another dimension of detail to the situation or work.
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Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, which is a tessellation. Some tetrahedra that are not regular, including the Schläfli orthoscheme and the Hill tetrahedron, can tessellate. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.
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Xia, S.-L. Zhu, M. Zeng, Z. Gu, H. Wang, W. Li, Tetrahedron 2015, 71, 6099–6103, .
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Tetrahedron is a weekly peer-reviewed scientific journal covering the field of organic chemistry. According to the Journal Citation Reports, the journal has a 2014 impact factor of 2.641. Tetrahedron and Elsevier, its publisher, support an annual symposium. In 2010, complaints were raised over its high subscription cost.
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The compound of five tetrahedra is a geometric illustration of the notion of orbits and stabilizers, as follows. The symmetry group of the compound is the (rotational) icosahedral group I of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) tetrahedral group T of order 12, and the orbit space I/T (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset gT corresponds to which tetrahedron g sends the chosen tetrahedron to.
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Shigehisa, H.; Mizutani, T.; Tosaki, S. Y.; Ohshima, T.; Shibasaki, M, Tetrahedron 2005, 61, 5057-5065. (Figure 2).
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Finan, M.; Kishi, Y. Tetrahedron Lett. 1982, 23, 2719. The configuration at the untouched epoxide carbon is preserved.
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A variety of silica groups have been discovered, and are controlled by their bonding arrangement, and silica tetrahedron.
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"A one-pot reduction- transimination-reduction synthesis of N-substituted β-ethanolamines from cyanohydrins." Tetrahedron 48 3977-3982.
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1979, 43, 877. of α-thioalkoxynitriles, and transmetallationPereyre, M.; Odic, Y. Tetrahedron Lett., 1969, 505. from α-stannylnitriles.
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Goodyear, M. D.; Hill, M. L.; West, J. P.; Whitehead, A. J. Tetrahedron Lett. 2005, 46, 8535–8538.
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The cross product is used in calculating the volume of a polyhedron such as a tetrahedron or parallelepiped.
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A Sierpinski square-based pyramid and its 'inverse' Sierpinski pyramid recursion progression (7 steps) A Sierpiński triangle-based pyramid as seen from above (4 main sections highlighted). Note the self-similarity in this 2-dimensional projected view, so that the resulting triangle could be a 2D fractal in itself. The Sierpinski tetrahedron or tetrix is the three- dimensional analogue of the Sierpinski triangle, formed by repeatedly shrinking a regular tetrahedron to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process. A tetrix constructed from an initial tetrahedron of side-length L has the property that the total surface area remains constant with each iteration.
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In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron, tetrakis cube, and kiscubeConway, Symmetries of Things, p.284) is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It also can be called a disdyakis hexahedron or hexakis tetrahedron as the dual of an omnitruncated tetrahedron.
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Lee conformal tetrahedric projection of the world centered on the south pole. Tissot's indicatrix of deformation. Lee conformal tetrahedric projection tessellated several times in the plane. The Lee conformal world in a tetrahedron is a polyhedral, conformal map projection that projects the globe onto a tetrahedron using Dixon's elliptic functions.
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Figure 1: Closed triangulated polyhedra. (a) Tetrahedron (Td), (b) Trigonal bipyramid (D3h). (c) Octahedron (Oh). (d) Pentagonal bipyramid (D5d).
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The other four vertices exist at alternate corners of a central cube (a demicube, in this case a tetrahedron).
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In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}.
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Doxorubicin (DOX) was conjugated with the tetrahedron and was loaded into MCF-7 breast cancer cells that contained the P-glycoprotein drug efflux pump. The results of the experiment showed the DOX was not being pumped out and apoptosis of the cancer cells was achieved. The tetrahedron without DOX was loaded into cells to test its biocompatibility, and the structure showed no cytotoxicity itself. The DNA tetrahedron was also used as barcode for profiling the subcellular expression and distribution of proteins in cells for diagnostic purposes.
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Epidote often has a distinctive pistachio-green colour. Sorosilicates, also termed disilicates, have tetrahedron-tetrahedron bonding at one oxygen, which results in a 2:7 ratio of silicon to oxygen. The resultant common structural element is the [Si2O7]6− group. The most common disilicates by far are members of the epidote group.
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The methane molecule (CH4) is tetrahedral because there are four pairs of electrons. The four hydrogen atoms are positioned at the vertices of a tetrahedron, and the bond angle is cos−1(−) ≈ 109° 28′."Angle Between 2 Legs of a Tetrahedron" – Maze5.net This is referred to as an AX4 type of molecule.
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When halfway between the two edges the intersection is a square. The aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly. If the tetrahedron is bisected on this plane, both halves become wedges.
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In the churchyard, close to the porch, is a gravestone in the form of a three-sided pyramid; a tetrahedron.
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It is well known that the numbers along the three outside edges of the nth Layer of the tetrahedron are the same numbers as the nth Line of Pascal's triangle. However, the connection is actually much more extensive than just one row of numbers. This relationship is best illustrated by comparing Pascal's triangle down to Line 4 with Layer 4 of the tetrahedron. Multiplying the numbers of each line of Pascal's triangle down to the nth Line by the numbers of the nth Line generates the nth Layer of the Tetrahedron.
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Three-dimensional simple polyhedra include the prisms (including the cube), the regular tetrahedron and dodecahedron, and, among the Archimedean solids, the truncated tetrahedron, truncated cube, truncated octahedron, truncated cuboctahedron, truncated dodecahedron, truncated icosahedron, and truncated icosidodecahedron. They also include the Goldberg polyhedron and Fullerenes, including the chamfered tetrahedron, chamfered cube, and chamfered dodecahedron. In general, any polyhedron can be made into a simple one by truncating its vertices of valence four or higher. For instance, truncated trapezohedrons are formed by truncating only the high-degree vertices of a trapezohedron; they are also simple.
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A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4.
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E. P. Eckenberg et al., A useful application of benzyl trichloroacetimidate for the benzylation of alcohols, Tetrahedron, 49, 1619–1624 (1993).
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1966, 31, 1363. Organosilanes,J. Am. Chem. Soc. 1980, 102, 1577; Tetrahedron Lett. 1990, 31, 743 Photochemical protection,Synlett 2007, 1071.
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Graphical representation of a rotaxane, useful as a molecular switch. This DNA tetrahedron is an artificially designed nanostructure of the type made in the field of DNA nanotechnology. Each edge of the tetrahedron is a 20 base pair DNA double helix, and each vertex is a three-arm junction. Rotating view of C60, one kind of fullerene.
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For tetrahedra in hyperbolic space or in three- dimensional elliptic geometry, the dihedral angles of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by the Murakami–Yano formula. However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist.
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Asymmetric cyclization reactions by Rh(I) with chiral ligands Yukari Tauraa, Masakazu Tanakaa, Kazuhisa Funakoshia and Kiyoshi Sakai. Tetrahedron Letters. Volume 30, Issue 46, 1989, Pages 6349-6352 Asymmetric cyclization reactions. Cyclization of substituted 4-pentenals into cyclopentanone derivatives by rhodium(I) with chiral ligands Yukari Taura, Masakazu Tanaka, Xiao-Ming Wu, Kazuhisa Funakoshi and Kiyoshi Sakai. Tetrahedron.
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Other useful reducing agents for this transformation include lithium dialkylcupratesBull, J. R.; Tuinman, A. Tetrahedron Lett. 1973, 4349. and molybdenum hexacarbonyl-alumina.
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Y. Blum, D. Reshef, and Y. Shvo. H-transfer catalysis with Ru3(CO)12. Tetrahedron Lett. 22(16) 1981, pp. 1541-1544.
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A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as spaceframes.
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In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad (see figure). A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices. A normal surface may have many components of intersection, called normal disks, with one tetrahedron, but no two normal disks can be quads that separate different pairs of vertices since that would lead to the surface self-intersecting. A normal surface intersects a tetrahedron in (possibly many) triangles (see above left) and quads (see above right) Dually, a normal surface can be considered to be a surface that intersects each handle of a given handle structure on the 3-manifold in a prescribed manner similar to the above.
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Stork, G.; Gregson, M. J. Am. Chem. Soc. 1969, 91, 2372, footnote 6.Ledon, H.; Linstrumelle, G.; Julia, S. Tetrahedron Lett. 1973, 25.
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A regular tetrahedron or tetragonal disphenoid can be dissected into two halves with a central square. Each half is a topological triangular prism.
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These orbitals, which have identical energies, are arranged spatially in the form of a tetrahedron, the angle of cos−1(−) ≈ 109.47° between them.
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This same cyclization can also be achieved via a Friedel-Crafts alkylation and cyclization.Das, S.; Bhattacharyya, S.; Mukherjee, D. Tetrahedron. 1992, 48, 9101-9110.
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Sandford, G. (2003). " Perfluoroalkanes." Tetrahedron 59: 437–454. . This carbocation intermediate can readily undergo rearrangements, which can lead to a complex mixture of products.
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2001, 216, 1; Pure Appl. Chem., 2000, 72, 1671. and of 2+2 ketene olefin cycloadditions;Tetrahedron Lett. 1968, 3921; J. Am. Chem. Soc.
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Chen, Z.L., Wang, B.D., Chen, M.Q. Steroidal bitter principles from Tacca plantaginea. Structures of taccalonolide A and B. Tetrahedron Lett., 28 (1987), 1673-1678.
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In the 2013 film Oblivion the large structure in orbit above the Earth is of a tetrahedron design and referred to as the Tet.
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Sadly, in giving an example of a Heronian tetrahedron found by E. P. Starke, the book repeats a mistake of Starke in calculating its volume.
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This is tetrahedron-free, and the edge density is 5/9. However, the best known upper bound is 0.562, using the technique of flag algebras.
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A tetrahedron has 4 vertices, 6 edges, and is bounded by 4 triangular faces. In most cases a tetrahedral volume mesh can be generated automatically.
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The first human carrying pure solar balloon flight was made on 1 May 1973 by Tracy Barnes in his balloon 'Barnes Solar Firefly Tetrahedron'. This balloon was made from a spiral tube of fabric that was formed into a tetrahedron. Dominic Michaelis is recorded as having owned the first pure solar balloon in Europe. This balloon was flown by Julian Nott across the English Channel.
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The Tetrahedron Prize for Creativity in Organic Chemistry or Bioorganic and Medicinal Chemistry is awarded annually by Elsevier, the publisher of Tetrahedron Publications. It was established in 1980 and named in honour of the founding co-chairmen of these publications, Professor Sir Robert Robinson and Professor Robert Burns Woodward. The prize consists of a gold medal, a certificate, and a monetary award of US $15,000.
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Several shape modifications of the Dino Cube exist. These include the aforementioned BrainTwist, in the shape of a tetrahedron that can flip inside out, the Platypus, whose shape is also based on a tetrahedron, the Redi Cube, a cubic version with shallower cuts, and the Rainbow Cube, in the shape of a cuboctahedron. The latter three puzzles, unlike the Dino Cube, also have their "core" pieces visible.
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The elongated square gyrobicupola can form a space-filling honeycomb with the regular tetrahedron, cube, and cuboctahedron. It can also form another honeycomb with the tetrahedron, square pyramid and various combinations of cubes, elongated square pyramids, and elongated square bipyramids. The pseudo great rhombicuboctahedron The pseudo great rhombicuboctahedron is a nonconvex analog of the pseudo-rhombicuboctahedron, constructed in a similar way from the nonconvex great rhombicuboctahedron.
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Because of this property of rotating within a square, the Reuleaux triangle is also sometimes known as the Reuleaux rotor. The Reuleaux triangle is the first of a sequence of Reuleaux polygons, whose boundaries are curves of constant width formed from regular polygons with an odd number of sides. Some of these curves have been used as the shapes of coins. The Reuleaux triangle can also be generalized into three dimensions in multiple ways: the Reuleaux tetrahedron (the intersection of four balls whose centers lie on a regular tetrahedron) does not have constant width, but can be modified by rounding its edges to form the Meissner tetrahedron, which does.
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Tetrahedron Lett. 2008, 49(39), 5668–5671. # Prabhakar Peddikotla, Amar G. Chittiboyina, Ikhlas A. Khan, (2014) ChemInform Abstract: Synthesis of Pterostilbene by Julia Olefination. ChemInform 45, .
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Fourteen of the pieces would have an external surface, with one tetrahedron cut out of the middle. The only cake number which is prime is 2.
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In a framework silicate, each tetrahedron shares all 4 oxygen atoms with its neighbours, forming a 3D-4D structure. Quartz and feldspars are in this group.
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Additionally, radical probe experiments with 5-hexenyl and cyclopropyl enol ethers did not give any rearranged products.Differding, E.; Rüegg, G. M. Tetrahedron Lett. 1991, 32, 3815.
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For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces.
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As an example, consider the case of building a tetrahedron from a triangle, the latter of whose elements are enumerated by row 3 of Pascal's triangle: 1 face, 3 edges, and 3 vertices (the meaning of the final 1 will be explained shortly). To build a tetrahedron from a triangle, we position a new vertex above the plane of the triangle and connect this vertex to all three vertices of the original triangle. The number of a given dimensional element in the tetrahedron is now the sum of two numbers: first the number of that element found in the original triangle, plus the number of new elements, each of which is built upon elements of one fewer dimension from the original triangle. Thus, in the tetrahedron, the number of cells (polyhedral elements) is ; the number of faces is the number of edges is the number of new vertices is .
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In geometry, the Murakami–Yano formula, introduced by , is a formula for the volume of a hyperbolic or spherical tetrahedron given in terms of its dihedral angles.
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The more substituted double bond of the diene reacts preferentially.Doyle, M.; Dorow, R.; Tamblyn, W.; Buhro, W. Tetrahedron Lett. 1982, 23, 2261. (8)File:IntraCPScope4.png (9)File:MCPScopeSubbed.
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Yamamoto, K.; Biswas, K.; Gaul, C.; Danishefsky, S. J. (2003). “Effects of temperature and concentration in some ring closing metathesis reactions”. Tetrahedron Lett. 44 (16): 3297–3299. .
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Moreover, the angle made by the faces is \pi/3. The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together. The Gieseking manifold has a double cover homeomorphic to the figure-eight knot complement. The underlying compact manifold has a Klein bottle boundary, and the first homology group of the Gieseking manifold is the integers.
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1968, 90, 7139. (13)File:AlaneScope9.png Alkenyl- and alkynylalanes undergo transmetalation to a variety of metals, including boron,Negishi, E.-i.; Boardman, L. D. Tetrahedron Lett. 1982, 3327.
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In geometry, the augmented tridiminished icosahedron is one of the Johnson solids (J64). It can be obtained by joining a tetrahedron to another Johnson solid, the tridiminished icosahedron.
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Piers and J. Y. Roberge, Tetrahedron Lett., 1992, 33, 6923; (b) H. Iio, K. Asao and T. Tokoroyama, J. Chem. Soc., Chem. Commun., 1985, 774 (−)-agelasine E,A.
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Nearby Tetrahedron Provincial Park is a popular location for day hikes and mountain biking. It was named for Captain Henry Augustus Mellon who came to Vancouver in 1886.
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An example known to Leonhard Euler is a Heronian birectangular tetrahedron, a tetrahedron with a path of three edges parallel to the three coordinate axes and with all faces being right triangles. The lengths of the edges on the path of axis-parallel edges are 153, 104, and 672, and the other three edge lengths are 185, 680, and 697, forming four right triangle faces described by the Pythagorean triples (153,104,185), (104,672,680), (153,680,697), and (185,672,697). Eight examples of Heronian tetrahedra were discovered in 1877 by Reinhold Hoppe. 117 is the smallest possible length of the longest edge of a perfect tetrahedron with integral edge lengths. Its other edge lengths are 51, 52, 53, 80 and 84.
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Physiological Plant Pathology 19, 217–226.Crombie L and Mistry J. (1990) "The phytoalexins of oat leaves: 4-3,1-benzoazin-4-onmes or amides?" Tetrahedron Lettres 31, 2647–2648.
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Stangl and R. Jira, Tetrahedron Lett., 1970, 11, 3589–3592 found chlorohydrin formation was dependent on copper chloride concentrations. Work by Hosokawa and coworkersT. Hosokawa, T. Nomura, S.-I.
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Braun, R. A. J. Org. Chem. 1963, 28, 1383. Conjugated dienyl epoxides form similar products, lending support to the existence of an ylide intermediate.Eberbach, W.; Roser, J. Tetrahedron Lett.
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In 1979 he published the first light driven molecular machine in Tetrahedron Letters. To date (July 2019) he has published over 1024 original paper and 219 reviews and books.
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K. Bakkestuen, L.-L. Gundersen, D. Petersen, B. Utenova and A. Vik, Org. Biomol. Chem., 2005, 3, 1025 (−)-agelasine F,K. Asao, H. Iio and T. Tokoroyama, Tetrahedron Lett.
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Wacław Sierpiński, Pythagorean Triangles, Dover Publications, 2003 (orig. ed. 1962), p. 107. Note however that Sierpiński repeats an erroneous calculation of the volume of the Heronian tetrahedron example above.
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2.3 which is reduced to 3.3.3, being the tetrahedron, and all the 6 edges of the tetrahedra can also be seen as the degenerate faces of the original cube.
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In graph theory, the hypergraph removal lemma states that when a hypergraph contains few copies of a given sub-hypergraph, then all of the copies can be eliminated by removing a small number of hyperedges. It is a generalization of the graph removal lemma. The special case in which the graph is a tetrahedron is known as the tetrahedron removal lemma. It was first proved by Gowers and, independently, by Nagle, Rödl, Schacht and Skokan.
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The link of a vertex of a tetrahedron is a triangle – the three vertices of the link corresponds to the three edges incident to the vertex, and the three edges of the link correspond to the faces incident to the vertex. In this example, the link can be visualized by cutting off the vertex with a plane; formally, intersecting the tetrahedron with a plane near the vertex – the resulting cross-section is the link.
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The characterization regarding the edges implies that if only four of the six edges of an orthocentric tetrahedron are known, the remaining two can be calculated as long as they are not opposite to each other. Therefore the volume of an orthocentric tetrahedron can be expressed in terms of four edges a, b, c, d. The formula isAndreescu, Titu and Gelca, Razvan, "Mathematical Olympiad Challenges", Birkhäuser, second edition, 2009, pp. 30-31, 159.
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There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same rotations as the tetrahedron, and are somewhat analogous to the snub cube and snub dodecahedron, including some forms which are chiral and some with Th-symmetry, i.e. have different planes of symmetry from the tetrahedron. The icosahedron is unique among the Platonic solids in possessing a dihedral angle not less than 120°.
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The set of polytopes formed by alternating the hypercubes are known as demicubes. In three dimensions, this produces a tetrahedron; in four dimensions, this produces a 16-cell, or demitesseract.
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The crystal structure is a distorted wurtzite lattice. At the nitrogen atom, the coordination is a flattened tetrahedron. The N−C−C angle is slightly larger than the tetrahedral angle.
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Allylchloride via Allylalkohole With carboxylic acids, acyl chlorides are obtained.D. O. Jang et al., A mild and efficient procedure for the preparation of acid chlorides from carboxylic acids, Tetrahedron Lett.
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Tanaka, H.; Ogasawara, K. Tetrahedron Lett. 2002, 43, 4417. It has also shown to reduce aromatic esters to the corresponding alcohols as shown in eq 6 and 7. :Image:Ester to alcohol.
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1968, 45, 581. Experimental results agree with the theory—softer hydride reagents afford higher yields of the conjugate reduction product.Bottin, J.; Eisenstein, O.; Minot, C.; Anh, T. Tetrahedron Lett., 1972, 3015.
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Applied to the geometry of the Tetrahedron of edge length , we have an incircle radius (derived by dividing twice the face area through the perimeter ), a dihedral angle , and in consequence .
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Moessner, C.; Bolm, C. Angew. Chem. Int. Ed. 2005, 44, 7564.Zhu, G.; Zhang, X. Tetrahedron: Asymmetry 1998, 9, 2415..Ahn, K. H.; Ham, C.; Kim, S.-K.; Cho, C.-W.
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Pseudoprolines are obtained by reacting the free amino acids with aldehydes or ketone.Fülop F., Mattinen J., Pihlaja K., Tetrahedron (1990) 46, p.6545Kallen RG., J. Am. Chem. Soc., (1971) 93, p.
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This formula is a special case of the multinomial formula for . The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron..
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Stanley Kubrick originally intended the monolith in 2001: A Space Odyssey to be a tetrahedron, according to Marvin Minsky, a cognitive scientist and expert on artificial intelligence who advised Kubrick on the HAL 9000 computer and other aspects of the movie. Kubrick scrapped the idea of using the tetrahedron as a visitor who saw footage of it did not recognize what it was and he did not want anything in the movie regular people did not understand. In Season 6, Episode 15 of Futurama, named "Möbius Dick", the Planet Express crew pass through an area in space known as the Bermuda Tetrahedron. Many other ships passing through the area have mysteriously disappeared, including that of the first Planet Express crew.
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Boron nitride exists as several forms (polymorphs). Nitrides of silicon and phosphorus are also known, but only the former is commercially important. The nitrides of aluminium, gallium, and indium adopt diamond-like wurtzite structure in which each atom occupies tetrahedral sites. For example, in aluminium nitride, each aluminium atom has four neighboring nitrogen atoms at the corners of a tetrahedron and similarly each nitrogen atom has four neighboring aluminium atoms at the corners of a tetrahedron.
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The tetrahedron is further equipped with targeting protein, three folate molecules, which lead the DNA nanoparticles to the abundant folate receptors found on some tumors. The result showed that the gene expression targeted by the RNAi, luciferase, dropped by more than half. This study shows promise in using DNA nanotechnology as an effective tool to deliver treatment using the emerging RNA Interference technology. The DNA tetrahedron was also used in an effort to overcome the phenomena multidrug resistance.
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They coordinate tetrahedrally around T(1) forming a giant tetrahedron. The supertetrahedra T(2) are located at the symmetry-related positions (0.25, 0.25, 0.75); they also form a giant tetrahedron surrounding T(1). Edges of both giant tetrahedra orthogonally cross each other at their centers; at those edge centers, each B10 polyhedron bridges all the super-structure clusters T(1), T(2) and O(1). The superoctahedron built of B10 polyhedra is located at each cubic face center.
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The complete list remains an open problem. If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in many different ways. For example, one can divide an octahedron into four identical tetrahedra and combine them again with two regular ones. (As a side- note: these two kinds of tetrahedron have the same volume.) The tetrahedron is unique among the uniform polyhedra in possessing no parallel faces.
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A sphere, a surface of constant radius and thus diameter, is a surface of constant width. Contrary to common belief the Reuleaux tetrahedron is not a surface of constant width. However, there are two different ways of smoothing subsets of the edges of the Reuleaux tetrahedron to form Meissner tetrahedra, surfaces of constant width. These shapes were conjectured by to have the minimum volume among all shapes with the same constant width, but this conjecture remains unsolved.
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8064 is the smallest possible volume (and 6384 is the smallest possible surface area) of a perfect tetrahedron. The integral edge lengths of a Heronian tetrahedron with this volume and surface area are 25, 39, 56, 120, 153 and 160. In 1943, E. P. Starke published another example, in which two faces are isosceles triangles with base 896 and sides 1073, and the other two faces are also isosceles with base 990 and the same sides.
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The Logos Tetrahedron is a concert hall in Ghent, Belgium adjacent to the foundation's recording studio and offices. It has seating for 150 people and is equipped with sound and light infrastructure. Since the concert hall is in the shape of a tetrahedron, it has no straight angles and, as a result, no standing waves can occur. Acoustic waves can never amplify one another in phase and are reflected by the walls under ever changing angles.
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Fire is one of the four classical elements in ancient Greek philosophy and science. Fire is considered to be both hot and dry, and according to Plato is associated with the tetrahedron.
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One of Man Ray's paintings, Hamlet, was based on a photograph he took of a Meissner tetrahedron,. which he thought of as resembling both Yorick's skull and Ophelia's breast from Shakespeare's Hamlet.
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The Rainy River is a short river that enters the Thornbrough Channel at Port Mellon, British Columbia, Canada. While it does come close to doing so, it never enters Tetrahedron Provincial Park.
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Further reaction of these compounds with alkali metal fluorides leads to nitrile sulfides, which undergo [3+2] cycloaddition in the presence of dipolarophiles.Grunwell, R.; Dye, L. Tetrahedron Lett., 1975, 1739. (10)File:SF4Scope5.
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Scheme 4. Intramolecular hydrogen transfer- mediated synthesis of carbonyl ylides from 5-hydroxy-4-pyrones. Modified from Garst, M. E.; McBride, B. J.; Douglass III, J. G. Tetrahedron Lett. 1983, 24, 1675.
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There are generic geometric names for the most common polyhedra. The 5 regular polyhedra are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively.
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A tetrahedron is a disphenoid if and only if its circumscribed parallelepiped is right-angled. We also have that a tetrahedron is a disphenoid if and only if the center in the circumscribed sphere and the inscribed sphere coincide.. Another characterization states that if d1, d2 and d3 are the common perpendiculars of AB and CD; AC and BD; and AD and BC respectively in a tetrahedron ABCD, then the tetrahedron is a disphenoid if and only if d1, d2 and d3 are pairwise perpendicular.. The disphenoids are the only polyhedra having infinitely many non-self-intersecting closed geodesics. On a disphenoid, all closed geodesics are non-self-intersecting.. The disphenoids are the tetrahedra in which all four faces have the same perimeter, the tetrahedra in which all four faces have the same area, and the tetrahedra in which the angular defects of all four vertices equal . They are the polyhedra having a net in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints..
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Tetrahedron Lett. 49 4903-4904. frameless 1,3-Dienes were found to be formed upon the treatment of an allylic alcohol with an aryl sulfide in the presence of triethylamine.Wollowitz, S.; Reich, H.J. (1982).
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Hong, C. Y.; Overman, L. E. Tetrahedron Lett. 1994, 35, 3453. (9)File:HeckScope4.png Asymmetric IMHR may establish tertiary or quaternary stereocenters. BINAP is the most commonly chiral ligand used in this context.
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Scheme 6. The chemoenzymatic synthesis of (+)-totarol. Chemoenzymatic synthesis of (+)-totarol has also been achieved with high yield (41.8%) (Scheme 6).Miyake, T.; Kigoshi, H.; Akita, H. Tetrahedron: Asymmetry. 2007, 18, 2915-2922.
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E Malan - Phytochemistry, 1995A novel doubly- linked proteracacinidin analogue from Acacia caffra. E Malan, A Sireeparsad, JFW Burger and D Ferreira, Tetrahedron letters, 1994 The oxydative depolymerisation of proteracacinidins yields the anthocyanidin teracacinidin.
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The 5-cell is self-dual, and its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism. Its dihedral angle is cos−1(), or approximately 75.52°.
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Pascal's triangle can be used as a lookup table for the number of elements (such as edges and corners) within a polytope (such as a triangle, a tetrahedron, a square and a cube).
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Tetrahedron and octahedron metal cores formed by aggregation of the Li3 triangle - carbanion coordinate complex Like other species consisting of polar subunits, organolithium species aggregate. Formation of aggregates is influenced by electrostatic interactions, the coordination between lithium and surrounding solvent molecules or polar additives, and steric effects. A basic building block toward constructing more complex structures is a carbanionic center interacting with a Li3 triangle in an η- 3 fashion. In simple alkyllithium reagents, these triangles aggregate to form tetrahedron or octahedron structures.
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If the Roman surface were to be inscribed inside the tetrahedron with least possible volume, one would find that each edge of the tetrahedron is tangent to the Roman surface at a point, and that each of these six points happens to be a Whitney singularity. These singularities, or pinching points, all lie at the edges of the three lines of double points, and they are defined by this property: that there is no plane tangent to any surface at the singularity.
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If six equal resistors are soldered together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.Záležák, Tomáš (18 October 2007); "Resistance of a regular tetrahedron" (PDF), retrieved 25 January 2011 Since silicon is the most common semiconductor used in solid- state electronics, and silicon has a valence of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how crystals of silicon form and what shapes they assume.
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The tetrahedron-first orthographic projection of the tetrahedral prism into 3D space has a tetrahedral projection envelope. Both tetrahedral cells project onto this tetrahedron, while the triangular prisms project to its faces. The triangular-prism-first orthographic projection of the tetrahedral prism into 3D space has a projection envelope in the shape of a triangular prism. The two tetrahedral cells are projected onto the triangular ends of the prism, each with a vertex that projects to the center of the respective triangular face.
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He has also published many critical essays and articles in specialized publications. In 1982 he received the Louis Paul Boon Award for the social engagement in his artistic work. In 1990 he designed and constructed the Logos Tetrahedron (a tetrahedron-shaped concert- hall) for the Logos Foundation in Ghent, a project for which it received the Tech-Art prize 1990. Next to his reputation as a composer, he is also a well known expert in computer technology and electronic art.
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The triakis tetrahedron, a polyhedral realization of an 8-vertex Apollonian network Apollonian networks are planar 3-connected graphs and therefore, by Steinitz's theorem, can always be represented as the graphs of convex polyhedra. The convex polyhedron representing an Apollonian network is a 3-dimensional stacked polytope. Such a polytope can be obtained from a tetrahedron by repeatedly gluing additional tetrahedra one at a time onto its triangular faces. Therefore, Apollonian networks may also be defined as the graphs of stacked 3d polytopes.
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A midplane is defined as a plane that is orthogonal to an edge joining any two vertices that also contains the centroid of an opposite edge formed by joining the other two vertices. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class of orthocentric tetrahedron. An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and the foot of the altitude dropped from the opposite vertex. A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a median and a line segment joining the midpoints of two opposite edges is called a bimedian of the tetrahedron.
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In particular, the regular tetrahedron has four equilateral triangles for faces and can be considered the three- dimensional analogue of the shape. The plane can be tiled using equilateral triangles giving the triangular tiling.
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The origin of diastereoselectivity in reductions of chiral ketones has been extensively analyzed and modeled.Cram, J.; Abd Elhafez, A. J. Am. Chem. Soc. 1952, 74, 5828.Chérest, M.; Prudent, N. Tetrahedron 1980, 36, 1599.
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Pharmacal Res. 2008, 31, 866-871. (+)-Totarol's anti-cancer activity is hypothesized to be due to the natural product's ability to form an o-quinone methide in vivo.Water, R. W.; Pettus, T. R. Tetrahedron.
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High yields and Z stereoselectivity occurred at low temperature and at the presence of HMPA.Stork, Gilbert, and Kang Zhao. "A stereoselective synthesis of (Z)-1-iodo-1-alkenes." Tetrahedron Letters 30.17 (1989): 2173-2174.
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2002, 35, 867. center Transition-metal catalyzed reactions have the potential to quickly generate an interesting functionality. Propellanes have been generated from intramolecular cyclization under palladium catalysis.Yamago, S.; Nakamura, E. Tetrahedron 1989, 45, 3081.
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Guanti, G.; Banfi, L.; Narisano, E.; Riva, R.; Thea, S. Tetrahedron Lett. 1986, 27, 4639. These substrates have been used for the enantioselective construction of biologically relevant sugars (see Synthetic Applications below). (7)File:EsteraseScope5.
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Trinitrotriazine, or 2,4,6-trinitro-1,3,5-triazine, is a theoretical explosive compound. Synthesis of this compound has been elusive despite its simple structure,G. D. Hartman, R. D. Hartman and J. E. Schwering. Tetrahedron Letters.
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Removing a third end node defines 2 types of 4-faces: rectified 5-cell, 021, and 24-cell, 0111. Removing a fourth end node defines 2 types of cells: octahedron, 011, and tetrahedron, 020.
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1985, 107, 1768.Greico, P. A.; Bahsas, A. Tetrahedron Lett. 1988, 29, 5855. The reaction process involves the formation of an aryl immonium ion intermediate followed by an aza Diels-Alder reaction with an alkene.
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This paradigm shift using monodentate phosphine ligands challenged the belief that a tight metal-ligand complex is necessary for high stereocontrol. Then in 2000, Reetz,M. T. Reetz, T. Sell, Tetrahedron Lett. 2000, 41, 6333.
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A DNA tetrahedron was used to deliver RNA Interference (RNAi) in a mouse model, reported a team of researchers in MIT. Delivery of the interfering RNA for treatment has showed some success using polymer or lipid, but there are limits of safety and imprecise targeting, in addition to short shelf life in the blood stream. The DNA nanostructure created by the team consists of six strands of DNA to form a tetrahedron, with one strand of RNA affixed to each of the six edges.
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The structure is a unit cube with the Te2− ions at the corners and face centers. The four mercury atoms are coordinated so that each mercury atom lies at the center of a regular tetrahedron of tellurium atoms and each tellurium lies at the center of a regular tetrahedron of mercury atoms. Its crystal point group of 3m and space group is F3m. It is a covalent compound with a high proportion of metallic bonding, due to its low valencies and even lower interatomic distances .
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A hexameric compound [Ni(SR)2]6 is produced in the reaction of nickel carbonyl with dialkyl sulfides (RSR). Nickel can be part of a cubane-type cluster with iron and chalcogens. The metal atoms are arranged in a tetrahedron shape, with the sulfur or selenium making up another tetrahedron that combines to make a cube. For example, the [NiFe3S4(PPh3)(SEt)3]2− is a dianion that has a tetraethyl ammonium salt. Similar ion clusters are [NiFe3Se4(PPh3)(SEt)3]2− and [NiFe3Se4(SEt)4]3−.
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Prepared Carbocyclic Nucleoside Analogs by Using Vince Lactam Vince lactam is also an intermediate in the synthesis of various nucleoside analogs such as difluoro guanosine derivatives (V),Toyota, A.; Habutani, C.; Katagiri, N.; Kaneko, C. Tetrahedron Lett. 1994, 35, 5665-5668.Toyota, A.; Aizawa, M.; Habutani, C.; Katagiri, N.; Kaneko, C. Tetrahedron 1995, 36, 8783-8798. carbocyclic oxanosine and related derivatives (VI),Saito, Y.; Nakamura, M.; Ohno, T.; Chaicharoenpong, C.; Ichikawa, E.; Yamamura, S.; Kato, K.; Umezawa, K. J. Antibiotics 2000, 53, 309-313.
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A tetrahedron can also be distorted by increasing the angle between two of the bonds. In the extreme case, flattening results. For carbon this phenomenon can be observed in a class of compounds called the fenestranes.
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The selenoxide elimination has been used in converting ketones, esters, and aldehydes to their α,β-unsaturated derivatives.Lauer, R.F; Young, M.W.; Sharpless, K.B. (1973). "Reactions of Selenoxides: Thermal Syn elimination and H2O-18 Exchange". Tetrahedron Lett.
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Pteridium esculentum contains the known bracken carcinogen ptaquiloside.Fletcher M.T., Hayes P.Y., Somerville M.J., De Voss J.J."Ptesculentoside, a novel norsesquiterpene glucoside from the Australian bracken fern Pteridium esculentum". Tetrahedron Letters. 51 (15) (pp 1997-1999), 2010.
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Cook, G.R; Kargbo, R; Maity, B. Org. Lett. 2005, 7, 2767–2770. Similarly, a chiral amino alcohol allows for extremely high enantioselectivity in the allylation of ketones.Haddad, T.D; Hirayama, L.C; Taynton, P; Singaram, B. Tetrahedron Lett.
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Chiral auxiliaries on the alkene partner have been used for stereoselective transformations. In the reaction of camphorsultam-derived unsaturated amides, lower temperatures were needed to achieve high selectivities.Binger, P.; Schäfer, B. Tetrahedron Lett. 1988, 29, 529.
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This recurrence for the binomial coefficients is known as Pascal's rule. Pascal's triangle has higher dimensional generalizations. The three- dimensional version is called Pascal's pyramid or Pascal's tetrahedron, while the general versions are called Pascal's simplices.
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Tetrahedron Letters 1984, 25 (16), 1661–1664. DOI: 10.1016/S0040-4039(01)81138-2 and even benzene have been described. While the existence of these reactions has been confirmed, the isomerization and automerization mechanisms remain unknown.
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Substituting dimethyl sulfide with something less noxious has been the goal of several research projects. Ohsugia et al.Ohsugia, S.-I.; Nishidea, K.; Oonob, K.; Okuyamab, K.; Fudesakaa, M.; Kodamaa, S.; Node, M. Tetrahedron 2003, 59, 8393–8398.
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Another, S. japonicas, is native to the waters around Japan.C. Tanaka, J. Tanaka, R. F. Bolland, G. Marriott, T. Higa, Seragamides A–F, new actin-targeting depsipeptides from the sponge Suberites japonicus Thiele. Tetrahedron 62, 3536 (2006).
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The Strother Field Tetrahedron Wind Indicator at Strother Field in Winfield, Kansas was listed on the National Register of Historic Places in 2015. With photos. It played a role in the World War II Battle of Kansas.
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Projection envelopes of the 16-cell. (Each cell is drawn with different color faces, inverted cells are undrawn) The cell-first parallel projection of the 16-cell into 3-space has a cubical envelope. The closest and farthest cells are projected to inscribed tetrahedra within the cube, corresponding with the two possible ways to inscribe a regular tetrahedron in a cube. Surrounding each of these tetrahedra are 4 other (non-regular) tetrahedral volumes that are the images of the 4 surrounding tetrahedral cells, filling up the space between the inscribed tetrahedron and the cube.
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Here is the expansion of (A + B + C)4: Writing the expansion in this non-linear way shows the expansion in a more understandable way. It also makes the connection with the tetrahedron obvious−the coefficients here match those of layer 4. All the implicit coefficients, variables, and exponents, which are normally not written, are also shown to illustrate another relationship with the tetrahedron. (Usually, "1A" is "A"; "B1" is "B"; and "C0" is "1"; etc.) The exponents of each term sum to the layer number (n), or 4, in this case.
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Art historian and critic T.K. Sabapathy has commented that Han is concerned with the uneasy relation between men and nature in the modern era, and has expressed a desire for people to realize that "[we] are part of nature and not apart from nature".: see . In addition, he has said: 20 Tonnes (2002), photographed on 14 June 2009 Beyond Han's sculptures which embrace organic forms, she has also designed geometric pieces. Four Dimensions (1993) was made up of ten geometrical structures, including single objects (such as Tetrahedron–Tetrahedron Interpenetration)See .
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The cubical cells to be meshed can also be sliced into 5 tetrahedra, using a (Diamond cubic) lattice as a basis. Cubes are mated on each side with another that has an opposite alignment of the tetrahedron around the centroid of the cube. Alternating vertices have a different number of tetrahedra intersecting on it, resulting in a slightly different mesh depending on position. When sliced this way, additional planes of symmetry are provided; having a tetrahedron around the centroid of the cube also generates very open spaces around points that are outside of the surface.
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13, September 1971, pp 472, 473 and 475Michael Gerzon, Experimental Tetrahedral Recording: part three, Studio Sound, Vol. 13, October 1971, pp 510, 511, 513 and 515 In this setup, the tetrahedron was inscribed into a cuboid, using every other corner. Despite Gerzon's somewhat over-enthusiastic description (which pre-dates the introduction of Ambisonics and the proper formulation of its psychoacoustic criteria), the tetrahedron exhibits the same stability problems in 3D that plague the triangle for horizontal-only reproduction. It is a viable option for adequate full-sphere reproduction only at low frequencies.
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II. A Simple Pathway to a "Cisnucleoside," 9-β-D- Arabinofuranosyladenine (Spongoadenosine), J. Org. Chem., 1963, 28, 3004–3006; Reist, E. J.; Goodman, L. Synthesis of 9-β-D-Arabinofuranosylguanine, Biochemistry, 1964, 3, 15–18; Ikehara, M.; Ogiso, Y. Studies of Nucleosides and Nucleotides. LIV. Purine Cyclonucleosides - 19. Further Investigation on the Cleavage of the 8,2'-O-Anhydro Linkage. A New Synthesis of 9-β-D- Arabinofuranosyl-adenine, Tetrahedron, 1972, 28, 3695–3704; Ranganathan, R. A Novel Purine Nucleoside Synthesis of 9-β-D-arabinofuranosyl-adenine, Tetrahedron Lett.
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A novel synthetic pathway to vitisin B compounds. Joana Oliveira, Victor de Freitas and Nuno Mateus, Tetrahedron Letters, Volume 50, Issue 27, 8 July 2009, Pages 3933-3935, Charge equilibria and pK values of 5-carboxypyranomalvidin-3-glucoside (vitisin A) by electrophoresis and absorption spectroscopy. Robert E. Asenstorfer and Graham P. Jones, Tetrahedron, Volume 63, Issue 22, 28 May 2007, Pages 4788-4792, Effect of acetaldehyde and several acids on the formation of vitisin A in model wine anthocyanin and colour evolution. Romero C. and Bakker J., International journal of food science & technology, 2000, vol.
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Heubach is the co-author of the book Combinatorics of Compositions and Words (with Toufik Mansour, CRC Press, 2009). She is a contributor to a text in bioinformatics, "Concepts in Bioinformatics and Genomics" by Jamil Momand and Alison McCurdy, Oxford University Press, 2016. Her research in combinatorial game theory has also included analysis of a variant of nim in which piles of pebbles are placed on the edges of a tetrahedron and each move removes at least one pebble from the set of edges incident to a single triangle of the tetrahedron.
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Chemical structure of castavinol C3. Castavinols are natural phenolic compounds found in red wines.Castavinol, a new series of polyphenols from Bordeaux red wines. Chantal Castagnino and Joseph Vercauteren, Tetrahedron Letters, Volume 37, Issue 43, 21 October 1996, pp.
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Grigg, R.; Sridharan, V. Tetrahedron Lett. 1992, 33, 7965. When dienes are involved in the intramolecular Heck reaction, insertion affords π-allylpalldium intermediates, which may be intercepted by nucleophiles. This idea was applied to a synthesis of (–)-morphine.
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Conjugative hydrocyanation has been heavily used for steroid synthesis. Originally, it was used to prepare the steroidal D ring, but other approaches have also been employed.Nagata, W. ; Terasawa, T. ; Hirai, S. ; Takeda, K. Tetrahedron Lett., 1960, 17, 27.
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1973, 47, 417. Alternative methods to produce nitrile anions include conjugate addition to α,β-unsaturated nitriles,Barrett, G.; Grattan, T. Tetrahedron Lett. 1970, 4237. reductionT. Saegusa, Y. Ito, H. Kinoshita, and S. Tomita, Bull. Chem. Soc. Jpn.
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Corey, E. J.; Boaz, N. W. Tetrahedron Lett. 1984, 25, 3059. Both of these mechanisms predict inversion at the electrophilic carbon, which is observed in a number of cases.Johnson, C. R.; Dutra, G. A. J. Am. Chem. Soc.
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2005, 340, 2360–2368. 600px Thioglycosides used in glycosylation reactions as donors can be activated under a wide range of conditions, most notably using NIS/AgOTf.Veeneman, G. H., van Leeuwen, S. H., van Boom, J. H. Tetrahedron Lett.
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The Corey–Winter olefin synthesis is a series of chemical reactions for converting 1,2-diols into olefins.Corey, E. J.; Winter, R. A. E. J. Am. Chem. Soc. 1963, 85, 2677. ()Corey, E. J.; Hopkiss, P. B. Tetrahedron Lett.
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The New Age group Share International, founded by Benjamin Creme, practices a form of meditation called 'Transmission Meditation' using an open metal-poled tetrahedron in order to tune into the cosmic energy of Maitreya and other spiritual masters.
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Methyl vinyl ketone and various substituted benzaldehydes were found to undergo asymmetric MBH reaction. The chiral pyrrolidine catalyst was effective for ortho- and para-substituted electron-deficient benzaldehydes (75-99% yield, 8-73% ee).J. Tetrahedron: Asymmetry, 2010, 1511.
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"New cytochalasins from the fungus Xylaria hypoxylon." Tetrahedron 53(18): 6485–92. X. hypoxylon also contains a carbohydrate-binding protein, a lectin, with a unique sugar specificity, and which has potent anti-tumor effects in various tumor cell lines.
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A general scheme involving ionic intermediates is shown below. (2)440px Lithium organocuprates undergo oxidative addition to enones to give, after reductive elimination of an organocopper(III) species, β-substituted lithium enolates.Hannah, J.; Smith, J. Tetrahedron Lett., 1975, 187.
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P4O10 can be seen as the end product of condensation reactions, where each tetrahedron shares three corners with the others. Conversely, a complex mix of polymers is produced when a small amount of water is added to phosphorus pentoxide.
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Conformational analysis of odd-membered rings suggests they tend to reside in less symmetrical forms with smaller energy differences between stable conformations.Anet, F.A.L.; St. Jacques, M.; Henrichs, P.M.; Cheng, A.K.; Krane, J.; Wong, L. Tetrahedron 1974, 30, 1629-1637.
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These determine a plane. It will be called the plane abc. Thus the planes a, b, c, abc, form a tetrahedron with vertices bc, ac, ab, 0. (iv) Four planes a, b, c, d, give four planes abc, abd, acd, bcd.
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In addition to the polyhedral boranes, the program explored pi-allyl, fluoroalkyl, and boron hydride complexes of the transition metals. Research also extended to stereochemically-non-rigid complexes.Muetterties, E. L.. Polytopal form and isomerism. Tetrahedron (1974), 30(12), 1595-604.
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Keto-ACE was used as the basis for the design of ketomethylene derivates. Its analogues contain a ketomethylene isostere replacement at the scissile bond that is believed to mimic the tetrahedron transition state of the proteolytic reaction at the active site.
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In this and many other modes of intramolecular Heck cyclization, annulations typically produce a cis ring juncture.Larock, R. C.; Song, H.; Baker, B. E.; Gong, W. H. Tetrahedron Lett. 1988, 29, 2919. (6)File:HeckScope1.png 6-Exo cyclization is also common.
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Unlike the IMHR, radical cyclization does not require the coupling of two sp2-hybridized carbons. In some cases, the results of radical cyclization and IMHR are complementary.Ishibashi, H.; Ito, K.; Hirano, T.; Tabuchi, M.; Ikeda, M. Tetrahedron 1993, 49, 4173.
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Tetrahedron, 2008, 64(20), 4511-4574. Using an allene instead of a simple alkene as the precursor for the gives an intermediate that can react at the gamma carbon rather than at the alpha.J. Am. Chem. Soc. 2009, 131, 4196.
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Ali et al. (1995). p. 12. In 2002, a research article was published in the journal Tetrahedron in which, authors Faizi and Naz dedicated their break-through research to the memory of Siddiqui, their mentor.Faizi and Naz (2002). p. 6185.
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The Å distances between iron and oxygen are between 2.33-2.93. The phosphate tetrahedron is almost regular, with 2 short bonds and 2 longer bonds (Bridson, et al., 1997). The iron atom has four surrounding oxygen atoms giving it tetrahedral coordination.
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Kündig, E. P.; Xu, L. H.; Romanens, P.; Bernardinelli, G. Tetrahedron Lett. 1993, 34, 7049. (8)File:CrAreneScope4.png Pinacol coupling and the corresponding diamine couplingDavies, S. G.; Donohoe, T. J.; Williams, J. M. J. Pure Appl. Chem. 1992, 64, 379.
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Marshall, J. A.; Gung, W. Y. Tetrahedron Lett. 1989, 30, 7349. (7)File:StanAddScope1.png The use of chiral electrophiles is common and can provide "double diastereoselection" if the stannane is also chiral.Marshall, J. A.; Yashunsky, D. V. J. Org. Chem.
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Also, phenolic hydroxy groups in nitrogen- containing aromatics can be converted into the chlorine compounds.W. Kijrungphaiboon et al., Cl3CCN/PPh3 and CBr4/PPh3: two efficient reagent systems for the preparation of N-heteroaromatic halides, Tetrahedron Lett., 53, 674–677 (2006).
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A few exceptions exist, however.Posner, H.; Whitten, E.; Sterling, J.; Brunelle, J. Tetrahedron Lett., 1974, 2591. In the example below, conducting the reaction in THF led to transfer of the vinyl moiety, while other solvents promoted methyl transfer. (6)File:VDScope3.
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In 1900 Betti hypothesized that 2-naphthol would be a good carbon nucleophile to the imine produced from the reaction of benzaldehyde and aniline. This led to the Betti reaction.Cardellicchio, C.; Capozzi, M.A.M.; Naso, F. Tetrahedron: Asymmetry. 2010, 21, 507-517.
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Tetrahedron Letters, volume 45, issue 34, pp. 6381–6383. or glycerol. Rosaria Ciriminna and Mario Pagliaro (2003), One-Pot Homogeneous and Heterogeneous Oxidation of Glycerol to Ketomalonic Acid Mediated by TEMPO. Advanced Synthesis & Catalysis, volume 345, issue 3, Pages 383–388.
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Tetrahedron Letters 1997; 38(35):6111–6114. doi:10.1016/S0040-4039(97)01385-3Katoh T, et al. Synthetic studies on Sch 202596, an antagonist of the galanin receptor subtype GalR1: an efficient synthesis of (±)-geodin, the spirocoumaranone part of Sch 202596.
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The structure of zearalenone. Tetrahedron Lett. 27:3109-3114 Almost simultaneously, a second group isolated, crystallized, and studied the metabolic properties of the same compound and named it F-2.Christensen, C. M., G. H. Nelson, and C. J. Mirocha. 1965.
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Phytochemistry, volume 10, pages 1401-1402. It can be synthesized in various ways.Hisashi Nishikori, Katsuji Ito, and Tsutomu Katsuki (1998) A short-step synthesis of trans-whisky lactone by an asymmetric Michael reaction. Tetrahedron: Asymmetry, volume 9, pages 1165–1170.
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Keggin structure of phosphotungstate anion = Cage of the Keggin structure + Central tetrahedron The structure has full tetrahedral symmetry and is composed of one heteroatom surrounded by four oxygen atoms to form a tetrahedron. The heteroatom is located centrally and caged by 12 octahedral MO6-units linked to one another by the neighboring oxygen atoms. There are a total of 24 bridging oxygen atoms that link the 12 addenda atoms. The metal centres in the 12 octahedra are arranged on a sphere almost equidistant from each other, in four M3O13 units, giving the complete structure an overall tetrahedral symmetry.
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All silicates have a basic structural unit that is a tetrahedron with an oxygen ion O at each apex, and a silicon ion Si in the middle, forming (SiO4)4−. In tectosilicates (framework silicates) each oxygen ion is shared between two tetrahedra, linking all the tetrahedra together to form a framework. Since each O is shared between two tetrahedra only half of it "belongs" to the Si ion in either tetrahedron, and if no other components are present then the formula is SiO2, as in quartz. Aluminium ions Al, can substitute for some of the silicon ions, forming (AlO4)5− tetrahedra.
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Because the ideal regular tetrahedron, cube, octahedron, and dodecahedron all have dihedral angles that are integer fractions of 2\pi, they can all tile hyperbolic space, forming a regular honeycomb. In this they differ from the Euclidean regular solids, among which only the cube can tile space. The ideal tetrahedron, cube, octahedron, and dodecahedron form respectively the order-6 tetrahedral honeycomb, order-6 cubic honeycomb, order-4 octahedral honeycomb, and order-6 dodecahedral honeycomb; here the order refers to the number of cells meeting at each edge. However, the ideal icosahedron does not tile space in the same way.
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The difference between lone pairs and bonding pairs may also be used to rationalize deviations from idealized geometries. For example, the H2O molecule has four electron pairs in its valence shell: two lone pairs and two bond pairs. The four electron pairs are spread so as to point roughly towards the apices of a tetrahedron. However, the bond angle between the two O–H bonds is only 104.5°, rather than the 109.5° of a regular tetrahedron, because the two lone pairs (whose density or probability envelopes lie closer to the oxygen nucleus) exert a greater mutual repulsion than the two bond pairs.
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He guided around 85 students in their doctoral research which included such notable chemists as B. D. Tilak, B. S. Joshi, Nitya Anand and A. V. Rama Rao. His contributions are reported in the development of National Chemical Laboratory into one of World's leading research centre in dyestuff chemistry. He sat in the editorial boards of many journals which included Tetrahedron, Tetrahedron Letters and Indian Journal of Chemistry. Besides The Chemistry of Synthetic Dyes, he also edited another 612-page book, The Analytical Chemistry of Synthetic Dyes and these nine books remain reference texts in the discipline.
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What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The four relations given by this sine law further reduce the number of degrees of freedom, from 8 down to not 4 but 5, since the fourth constraint is not independent of the first three.
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The (red) side edges of tetragonal disphenoid represent a regular zig-zag skew quadrilateral A non-planar quadrilateral is called a skew quadrilateral. Formulas to compute its dihedral angles from the edge lengths and the angle between two adjacent edges were derived for work on the properties of molecules such as cyclobutane that contain a "puckered" ring of four atoms. Historically the term gauche quadrilateral was also used to mean a skew quadrilateral. A skew quadrilateral together with its diagonals form a (possibly non-regular) tetrahedron, and conversely every skew quadrilateral comes from a tetrahedron where a pair of opposite edges is removed.
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For a tetrahedron in a two-in, two-out state, the magnetization field is divergent-free; there is as much "magnetization intensity" entering a tetrahedron as there is leaving (see figure 3). In such a divergent-free situation, there exists no source or sink for the field. According to Gauss' theorem (also known as Ostrogradsky's theorem), a nonzero divergence of a field is caused, and can be characterized, by a real number called "charge". In the context of spin ice, such charges characterizing the violation of the two-in, two-out magnetic moment orientation rule are the aforementioned monopoles.
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In higher dimensions, Coxeter defined a quasiregular polytope or honeycomb to have regular facets and quasiregular vertex figures. It follows that all vertex figures are congruent and that there are two kinds of facets, which alternate. In Euclidean 4-space, the regular 16-cell can also be seen as quasiregular as an alternated tesseract, h{4,3,3}, Coxeter diagrams: = , composed of alternating tetrahedron and tetrahedron cells. Its vertex figure is the quasiregular tetratetrahedron (an octahedron with tetrahedral symmetry), . The only quasiregular honeycomb in Euclidean 3-space is the alternated cubic honeycomb, h{4,3,4}, Coxeter diagrams: = , composed of alternating tetrahedral and octahedral cells.
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Visualization diamond cubic Diamond cubic has a variety of visualizations. Instead of empty cells, every cell should be filled, with alternating inner tetrahedrons. For each tetrahedron inscribed in a cube, using the vertices of the cube and edges that cross the faces of the cube, the tetrahedron will occupy 4 points; the other 4 points form the corners of an inverted tetrahedron; the cubic cells are tiled such that the position of the cell (x+y+z+...) is odd, use one, else use the inverted; otherwise near cells would use a different diagonal to compute the intersection. Illustration of inverted inner Diamond Crystal Lattice cells Calculation of color based on a spacial texture system can be done using the current fragment position to select from a repeating texture based on the pairs of Texel_(graphics) coordinates (x,y), (y,z) and (x,z) and scaling those values by the absolute value of each respective component of the normal z, x, and y respectively.
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Danh, N. C.; Arnaud, C.; Huet, J. Bull. Soc. Chim. Fr. 1974, 1071. On the other hand, use of the bulky reagent LTBA leads to a high yield of the conjugate addition product (Eq. ()).Durand, J.; Anh, N. T.; Huet, J. Tetrahedron Lett.
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In solid [NH4][HF2], each ammonium cation is surrounded by four fluoride centers in a tetrahedron, with hydrogen- fluorine hydrogen bonds present between the hydrogen atoms of the ammonium ion and the fluorine atoms. Solutions contain tetrahedral [NH4]+ cations and linear [HF2]− anions.
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Imines may be reduced enantioselectively using stoichiometric amounts of chiral metal hydrides.Wallbaum, S.; Martens, J. Tetrahedron: Asymmetry 1992, 3, 1475. Such methods have the advantage that they are easy to implement. Reduction with hydrosilanes is a second alternative to transition-metal catalyzed hydrogenation.
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Stereoselective fluorinations may be either diastereoselective or enantioselective. Diastereoselective methods have focused on the use of chiral auxiliaries on the nucleophilic substrate. For fluorinations of carbonyl compounds, chiral oxazolidinones have been used with success.Davis, F. A.; Kasu, P. V. N. Tetrahedron Lett.
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The Whiting group, however, noticed that Brown's method was not applicable to more sterically hindered boronic esters (no reaction).Stewart, Sarah K., and Andrew Whiting. "Stereoselective synthesis of vinyl iodides from vinylboronate pinacol esters using ICI." Tetrahedron letters 36.22 (1995): 3929-3932.
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Even more challenging is the ability to take a racemic mixture and have only one chiral product left after a reaction. One method that has become an exceedingly useful tool is dynamic kinetic resolution (DKR).Pellissier, H. Tetrahedron 2008, 64, 1563–1601.
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Insertion into carbon–carbon or carbon–hydrogen bonds is possible in substrates lacking a double bond.Nakata, T.; Tahara, A. Tetrahedron Lett. 1976, 1515. The intramolecular version of this reaction forms fused carbocycles, although yields of reactions mediated by copper are typically moderate.
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Osladine is a high-intensity sweetener isolated from the rhizome of Polypodium vulgare. J Jizba, L Dolejs, V Herout & F Sorm, « The structure of osladin — The sweet principle of the rhizomes of Polypodium vulgare L. », dans Tetrahedron Lett., vol. 18, 1971, p.
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B. B. Liau, B. C. Milgram and M. D. Shair, J. Am. Chem. Soc., 2012, 134, 16765-16772Tatsuta, K.; Fukuda, T.; Ishimori, T.; Yachi, R.; Yoshida, S.; Hashimoto, H.; Hosokawa, S. Tetrahedron Lett., 2012, 53, 422-425Tatsuta, K.; Hosokawa, S. Chem. Rec.
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This article describes the importance of the Author/Speaker, Audience, Context, and Purpose to one another in an attempt to convince or persuade said audience, using the tools of the rhetorical triangle and tetrahedron to help simplify the abstract idea of rhetorical stance.
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It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope.
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CiteBank:47270 The Reeve tetrahedron shows that there is no analogue of Pick's theorem in three dimensions that expresses the volume of a polytope by counting its interior and boundary points. However, there is a generalization in higher dimensions via Ehrhart polynomials.
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Corey, E. J.; Cho, H.; Rücker C.; Hua, D. H. "Studies with trialkylsilyltriflates: new syntheses and applications." Tetrahedron Lett. 1981, 22, 3455–3458. Primary alcohols can be protected in less than one hour while some hindered alcohols may require days of reaction time.
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Lipton's main pillar brands are Lipton Yellow Label and Lipton Iced Tea. Other product lines include the Lipton pyramid (tetrahedron) range in Europe and North America, and Lipton Milk Tea in East Asia. In 2008, the brand launched Lipton Linea in Western Europe.
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Sarah Elizabeth Reisman is a Professor of Chemistry at the California Institute of Technology and winner of an Arthur C. Cope Scholar Award and Tetrahedron Young Investigator Award for Organic Synthesis. Reisman's research focuses on the total synthesis of complex natural products.
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If the area of the base is T_0 and the areas of the three other (right-angled) faces are T_1, T_2 and T_3, then :T_0^2=T_1^2+T_2^2+T_3^2. This is a generalization of the Pythagorean theorem to a tetrahedron.
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Tetrahedron, volume 58, issue 16, pp. 3287–3296 It has been used in chemical research, e.g. in the synthesis of mirror-image DNA. Hidehito Urata, Emiko Ogura, Keiko Shinohara, Yoshiaki Ueda and Masao Akagi (1992), Synthesis and properties of mirror-image DNA.
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Iwadare S, Shizuri Y, Yamada K, Hirata Y. Synthesis of trichotomine, a blue pigment obtained from Clerodendron trichotomum Thunb. Tetrahedron. 1978; 34(10):1457-1459. It is cultivated for its fragrant flowers, autumn colour, and ornamental berries. It is hardy but requires a sheltered position.
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HU-210 is a synthetic cannabinoid that was first synthesized in 1988 from (1R,5S)-myrtenolMechoulam, R., Lander, N., Breuer, A., Zahalka, J. Synthesis of the Individual, Pharmacologically Distinct, Enantiomers of a Tetrahydrocannabinol Derivative. Tetrahedron: Asymmetry. 1990. Vol 1, No 5. pp 315-318.
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Harmata, M.; Elahmad, S.; Barnes, C. L. Tetrahedron Lett. 1995, 36, 1397. (1)File:43Gen.png Recent developments have focused on expanding the scope of enantioselective (4+3) cycloadditions and the range of conditions available for generating the key oxyallyl cation (propenylium-2-olate) intermediate.
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Amurensin E is an oligostilbene found in Vitis amurensis.Four Novel Oligostilbenes from the Roots of Vitis amurensis. Kai-Sheng Huang, Mao Lin, Lin-Ning Yu and Man Kong, Tetrahedron, 3 March 2000, Volume 56, Issue 10, Pages 1321–1329, It is a pentamer of resveratrol.
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Vorbrüggen, H.; Bennua, B. Tetrahedron Lett. 1978, 1339. The combination of trifluoroacetic acid (TFA), trimethylsilyl chloride (TMSCl), and hexamethyldisilazide (HMDS) generates trimethylsilyl trifluoroacetate in situ, which accomplishes both the silylation of the heterocycle and its subsequent coupling with the sugar.Vorbrüggen, H.; Bennua, B. Chem. Ber.
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Diastereoselective conjugate addition reactions of chiral organocuprates provide β-functionalized ketones in high yield and diastereoselectivity. A disadvantage of these reactions is the requirement of a full equivalent of enantiopure starting material.Malmberg, H.; Nilsson, M.; Ullenius, C. Tetrahedron Lett. 1982, 23, 3823. (3)File:CopperStereo1.
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For example, in 3 dimensions, 4 of the 5 Platonic solids have central symmetry (cube/octahedron, dodecahedron/icosahedron), while the tetrahedron does not – however, the stellated octahedron has central symmetry, though the resulting symmetry group is the same as that of the cube/octahedron.
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A tetrahedroid In algebraic geometry, a tetrahedroid (or tétraédroïde) is a special kind of Kummer surface studied by , with the property that the intersections with the faces of a fixed tetrahedron are given by two conics intersecting in four nodes. Tetrahedroids generalize Fresnel's wave surface.
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53–54 As in the two-dimensional case, the centroid of the tetrahedron is the center of mass. However contrary to the two-dimensional case the centroid divides the medians not in a 2:1 ratio but in a 3:1 ratio (Commandino's theorem).
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It can be realized geometrically from a tetrahedron (the faces of which correspond to the four large faces in the drawing, three of which are between pairs of fragments and the fourth of which forms the exterior) by multiply truncating three of its vertices.
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2,3-dicarbonylated pyrroles can be synthesized by a version of the Hantzsch Pyrrole Synthesis.Moss, T.A.; Nowak, T. Tetrahedron Lett. 2012, 53, 3056-3060. These pyrroles are particularly useful for total synthesis because the carbonyl groups can be converted into a variety of other functional groups.
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The Austrian artist Martina Schettina created a tetrahedron using fluorescent lamps. It was shown at the light art biennale Austria 2010.Lightart-Biennale Austria 2010 It is used as album artwork, surrounded by black flames on The End of All Things to Come by Mudvayne.
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Wu et al. reported a C–X cross coupling using CuX2 (X= Br, Cl) and a silver catalyst to obtain aryl halides. Luo Y, Pan XL, Wu J. Silver-catalyzed decarboxylative halogenation of carboxylic acids. Tetrahedron Lett, 2010, 51: 6646–48 DOI: 10.1016/j.tetlet.2010.10.
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In targeting 10-epi-kedarcidin chromophore, Myers et al. sought to install the epoxide functionality syn to the adjacent C10 hydroxyl group. This was accomplished by vanadium-catalyzed epoxidation directed by the C10 hydroxyl group.Rossiter, B. E.; Verhoeven, T. R.; Sharpless, K. B. Tetrahedron Lett.
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In mineralogy, coordination polyhedra are usually considered in terms of oxygen, due its abundance in the crust. The base unit of silicate minerals is the silica tetrahedron – one Si4+ surrounded by four O2−. An alternate way of describing the coordination of the silicate is by a number: in the case of the silica tetrahedron, the silicon is said to have a coordination number of 4. Various cations have a specific range of possible coordination numbers; for silicon, it is almost always 4, except for very high-pressure minerals where the compound is compressed such that silicon is in six-fold (octahedral) coordination with oxygen.
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However, Empedocles of Acragas , is best known for having selected all elements as his archai and by the time of Plato , the four Empedoclian elements were well established. In the Timaeus, Plato's major cosmological dialogue, the Platonic solid he associated with fire was the tetrahedron which is formed from four triangles and contains the least volume with the greatest surface area. This also makes fire the element with the smallest number of sides, and Plato regarded it as appropriate for the heat of fire, which he felt is sharp and stabbing, (like one of the points of a tetrahedron).Plato, Timaeus, chap. 22–23; Gregory Vlastos, Plato’s Universe, pp. 66–82.
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If O_n is the nth octahedral number and T_n is the nth tetrahedral number then :O_n+4T_{n-1}=T_{2n-1}. This represents the geometric fact that gluing a tetrahedron onto each of four non-adjacent faces of an octahedron produces a tetrahedron of twice the size. Another relation between octahedral numbers and tetrahedral numbers is also possible, based on the fact that an octahedron may be divided into four tetrahedra each having two adjacent original faces (or alternatively, based on the fact that each square pyramidal number is the sum of two tetrahedral numbers): :O_n = T_n + 2T_{n-1} + T_{n-2}.
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The formation of a benzyne intermediate was determined by trapping studies, using benzene or anthracene to trap the benzyne as a Diels–Alder adduct. Ueda and co-workers further elaborated on this method in subsequent reports, trapping the benzyne using a variety of nucleophiles (oxygen, nitrogen, and sulfur-based), as well as synthesizing larger, fused-ring aromatic systems.Ueda, I.; Sakurai, Y.; Kawano, T.; Wada, Y.; Futai, M. Tetrahedron Lett., 1999, 40, 319 Miyawaki, K.; Kawano, T.; Ueda, I. Tetrahedron Lett., 2000, 41, 1447 K. Miyawaki, F. Ueno, I. Ueda, Heterocycles, 2001, 54, 887 Torikai, K.; Otsuka, Y.; Nishimura, M.; Sumida, M.; Kawai, T.; Sekiguchi, K.; Ueda, I. Bioorg. Med. Chem.
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If we consider a tetrahedron with a spin on each vertex pointing along the easy axis (that is, directly towards or away from the centre of the tetrahedron), then it is possible to arrange the four spins so that there is no net spin (Figure 3). This is exactly equivalent to having an antiferromagnetic interaction between each pair of spins, so in this case there is no geometrical frustration. With these axes, geometric frustration arises if there is a ferromagnetic interaction between neighbours, where energy is minimized by parallel spins. The best possible arrangement is shown in Figure 4, with two spins pointing towards the centre and two pointing away.
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In the presence of Lewis acids such as titanium tetrachloride, (E)-allylsilanes are highly selective for the syn diastereomer. (Z)-allylsilanes are much less selective (~60:40), but also favor the syn isomer.Green, J. R.; Majewski, M.; Alo, B. I.; Snieckus, V. Tetrahedron Lett. 1986, 27, 535.
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A significant disadvantage of kinetic resolution is a maximum yield of hydrolyzed product of 50%. However, if rapid racemization is occurring alongside hydrolysis (an example of dynamic kinetic resolution), a maximum yield of 100% is possible.Allen, J.; Williams, J. Tetrahedron Lett. 1996, 37, 1859. (8)File:EsteraseScope6.
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Nuclear Magnetic Resonance Configurational Proofs. Journal of Organic Chemistry, volume 28, issue 4, pages 894-900. E.O. von Lippmann (1901), Chemische Berichte, volume 34, pages 1159-1162. J.D. Ramanathan, J.S. Craigie, J. McLachlan, D.G. Smith, and A.G. McInnes (1966): Tetrahedron Letters, volume 1966, pages 1527-1531.
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Tetraiodonickelate is a complex ion of nickel with four iodide atoms [NiI4]2− arranged in a tetrahedron. [NiI4]2− is red in solution. This colour is due to absorption around 530 nm and below 450 nm. Maximum light transmission is around 620 nm, which is red.
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Equivalently, a graph is line perfect if and only if each of its biconnected components is either bipartite or of the form K4 (the tetrahedron) or K1,1,n (a book of one or more triangles all sharing a common edge). Every line perfect graph is itself perfect.
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"Closure of the oxide bridge in morphine biosynthesis." Tetrahedron letters 35.23 (1994): 3897-3900. This second step does not require an enzyme. Thebaine can then be converted to morphine through two slightly different biosynthetic routes, one of which makes use of the fourth enzyme codeinone reductase.
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Tetrahedral faceted spheres are targeted by beginning with simple spheres. In coordination with the faces of a tetrahedron, the sphere is sliced at four equal facets. Monte Carlo simulations were performed to determine different forms of α, the faceting amount. The particular faceting amount determines the lattice that assembles.
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National Toxicology Program"Summary of Data for Chemical Selection" The compound is prepared by acylation of 2-bromopyridine via the Grignard reagent.Trécourt, F.; Breton, G.; Bonnet, V.; Mongin, F.; Marsais, F.; Quéguiner, G., "New Syntheses of Substituted Pyridines via Bromine–Magnesium Exchange", Tetrahedron 2000, volume 56, pp. 1349-1360. .
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These two honeycombs, and three others using the ideal cuboctahedron, triangular prism, and truncated tetrahedron, arise in the study of the Bianchi groups, and come from cusped manifolds formed as quotients of hyperbolic space by subgroups of Bianchi groups. The same manifolds can also be interpreted as link complements.
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The most dominant primary minerals in soil are the silicate minerals. Silicate minerals consist of more than 90% of the minerals in the Earth's crust. There are six silica mineral groups, based on bonding arrangement, and silica tetrahedron. The silica groups include: nesosilicates, sorosilicates, cyclosilicates, inosilicates, phyllosilicates, and tectosilicates.
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Radical cyclizations are often reductive, which can cause undesired side reactions to occur if sensitive substrates are employed. The IMHR, on the other hand, can be run under reductive conditions if desired.Burns, B.; Grigg, R.; Ratananukul, P.; Sridharan, V.; Stevenson, P.; Worakun, T. Tetrahedron Lett. 1988, 29, 4329.
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Transmission electron microscopy image of a 5-fold twinned icosahedral Au nanoparticle. An icosahedral twin is a nanostructure appearing for atomic clusters. The these clusters are twenty-faced, made of ten interlinked dual- tetrahedron (bowtie) crystals, typically joined along triangular (e.g. cubic-(111)) faces having three-fold symmetry.
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The scheme below gives an example of two different products formed from the same nucleophile under α-regioselectivity (α) and γ-regioselectivity (γ). This regioselectivity does not appear to depend on conjugation or the degree of substitution. Isaac, M.B; Chan, T.H. Tetrahedron Lett. 1995, 36, 8957–8960. File:Example_of_ima_showing_alpha_vs_gamma.
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Gilman designed a tetrahedral truss different from Buckminster Fuller's famous 1953 truss in 1981. Gilman attributed the strength of his tetrahedral truss to "the inherent rigidity of the skeletal triangle"John J. Gilman, United States Patents, 1981, Google Patents, Description, p.1 and "the rigidity of the basic tetrahedron".
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Geometrically forms the edge set of a triangle, a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph as its skeleton. Every neighborly polytope in four or more dimensions also has a complete skeleton. through are all planar graphs.
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'Maha' signifies the Mahogany tree and 'ganitham' Mathematics. Concepts like the Golden ratio and Fibonacci series are used in the design. More than thousand mathematical entities and geometrical shapes are engraved in the sculpture. The five Platonic bodies - tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron - are present in it.
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3-simplex (3D) The K4 complete graph is often drawn as a square with all 6 possible edges connected, hence appearing as a square with both diagonals drawn. This graph also represents an orthographic projection of the 4 vertices and 6 edges of the regular 3-simplex (tetrahedron).
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2013, 15, 2179-2181. It can be changed from non-chelating to chelating diphosphine by tuning the length of the linking arm.Zong, J., J. T. Mague, and R. A. Pascal, Jr., "Encapsulation of non-hydrogen atoms by in, in-bis (triarylelement)-containing cyclophanes." Tetrahedron, 2017, 73, 455-460.
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Hidenori Watanabe et al., "The first synthesis of kaitocephalin based on the structure revision", Tetrahedron Letters 43 (2002): 861-864 In addition, three structure-activity relationship (SAR) studies of kaitocephalin have been performed.Yoko Yasuno et al., "Structure-activity relationship study at C9 position of kaitocephalin", Bioorg. Med. Chem. Lett.
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Tetrahedron, volume 51, issue 22, pages 6145-6166. They are both white, odorless, water-soluble solids.B.J. Brotherton "Boron: Inorganic Chemistry" in Encyclopedia of Inorganic Chemistry (1994) Ed. R. Bruce King, John Wiley & Sons This salt is widely used in laundry detergents, as one of the peroxide-based bleaches.
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He also found new fundamental reaction of organosilyl radicals. Especially he found the Hosomi reaction (the Hosomi-Sakurai) reaction using allylsilanes as one of the famous name reaction. This reaction is carbon-carbon formation using allylsilanes with carbon electrophiles promoted by Lewis acid (A.Hosomi, and H. Sakurai, Tetrahedron Lett.
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Tetrahedral packaging Aristotle claimed that tetrahedra could fill space completely. In 2006, Conway and Torquato showed that a packing fraction about 72% can be obtained by constructing a non- Bravais lattice packing of tetrahedra (with multiple particles with generally different orientations per repeating unit), and thus they showed that the best tetrahedron packing cannot be a lattice packing (with one particle per repeating unit such that each particle has a common orientation). These packing constructions almost doubled the optimal Bravais-lattice-packing fraction 36.73% obtained by Hoylman. In 2007 and 2010, Chaikin and coworkers experimentally showed that tetrahedron-like dice can randomly pack in a finite container up to a packing fraction between 75% and 76%.
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By Steinitz's theorem, the Goldner–Harary graph is a polyhedral graph: it is planar and 3-connected, so there exists a convex polyhedron having the Goldner–Harary graph as its skeleton. Geometrically, a polyhedron representing the Goldner–Harary graph may be formed by gluing a tetrahedron onto each face of a triangular dipyramid, similarly to the way a triakis octahedron is formed by gluing a tetrahedron onto each face of an octahedron. That is, it is the Kleetope of the triangular dipyramid.. Same page, 2nd ed., Graduate Texts in Mathematics 221, Springer-Verlag, 2003, .. The dual graph of the Goldner–Harary graph is represented geometrically by the truncation of the triangular prism.
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In the protonation of an enolate ion, the kinetic product is the enol and the thermodynamic product is a ketone or aldehyde. Carbonyl compounds and their enols interchange rapidly by proton transfers catalyzed by acids or bases, even in trace amounts, in this case mediated by the enolate or the proton source. In the deprotonation of an unsymmetrical ketone, the kinetic product is the enolate resulting from removal of the most accessible α-H while the thermodynamic product has the more highly substituted enolate moiety.Thermodynamic Product vs Kinetic ProductJean d'Angelo, Tetrahedron report number 25 : Ketone enolates: regiospecific preparation and synthetic uses, Tetrahedron, Volume 32, Issue 24, 1976, Pages 2979-2990, , The Chemistry of Carbanions.
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Parallel to these edge-linked octahedral chains are vertex-linked mixed chains of alternating octahedra and tetrahedra. The tetrahedra have a phosphorus ion P in the middle, and oxygen ions O at each of the four vertices, and the octahedra have an aluminium ion Al in the middle surrounded by six oxygen ions O, as in the octahedral chains. At each linked vertex one O is shared between a tetrahedron and an octahedron, and each tetrahedron and octahedron must have two linked vertices to form the mixed chain. Each octahedral chain is flanked by two mixed chains, one on either side, linked through the vertices of the chains, making an infinite triple chain.
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Trirectangular bipyramid with edges (240, 117, 44, 125, 244, 267, 44, 117, 240) The area of the base (a,b,c) is always (Gua) an irrational number. Thus a trirectangular tetrahedron with integer edges is never a perfect body. The trirectangular bipyramid (6 faces, 9 edges, 5 vertices) built from these trirectangular tetrahedrons and the related left-handed ones connected on their bases have rational edges, faces and volume, but the inner space-diagonal between the two trirectangular vertices is still irrational. The later one is the double of the altitude of the trirectangular tetrahedron and a rational part of the (proved)Walter Wyss, "No Perfect Cuboid", irrational space-diagonal of the related Euler-brick (bc, ca, ab).
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He also started researching Renaissance mathematics at this time. He also rekindled interest in a mistake that Aristotle made about tiling the universe with just the tetrahedron. It was first challenged in 1435. In 1926 Struik was offered positions both at the Moscow State University and the Massachusetts Institute of Technology.
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The two players must avoid to color a monochromatic tetrahedron (a three-dimensional pyramid with four triangular faces). The Ramsey number R(3,3,3)=17 implies that any three- coloring of the complete graph on 17 vertices must contain a monochromatic triangle. A corresponding Ramsey game uses pencils of three colors.
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Tetrahedron Lett. 33, 5791^5794 the cyclopentane derivatives was designed with a guanidino group replacing C4-hydroxyl position of DANA in the active site, similar to Zanamivir. Babu et al. found that the addition of n-butyl side chain makes the compound fit better to the hydrophobic region of the enzyme.
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Johnson solid J₇. In geometry, the elongated triangular pyramid is one of the Johnson solids (J7). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically (but not geometrically) self- dual.
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For IrB3−, vibrational progression from the stretching and breathing vibrations of IrB3 could be assigned in the overlaid spectra of both isomers present in the cluster beam. By comparison to computations, the minimum energy structure of IrB3 could then be formulated as a tetrahedron with an intact, aromatic B3+ moiety.
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A cuboctahedron can be obtained by taking an equatorial cross section of a four-dimensional 24-cell or 16-cell. A hexagon can be obtained by taking an equatorial cross section of a cuboctahedron. The cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron.
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A cube divided into six tetrahedra, with one tetrahedron shaded Marching tetrahedra is an algorithm in the field of computer graphics to render implicit surfaces. It clarifies a minor ambiguity problem of the marching cubes algorithm with some cube configurations. It was originally introduced in 1991.Akio Doi, Akio Koide.
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The reaction is usually carried out under inert atmosphere in anhydrous conditions, and the most common solvent used is dichloromethane. However, the enantioselectivity of asymmetric cyclopropanations may depend profoundly on the solvent.Doyle, M. P.; Zhou, Q.-L.; Charnsangavej, C.; Longoria, M. A.; McKervey, M. A.; Garcia, C. F. Tetrahedron Lett.
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The Book of Sydney Suburbs, Compiled by Frances Pollon, Angus & Robertson Publishers, 1990, Published in Australia , page 22 A number of tetrahedron tank traps can be found off Pittwater Road on the water side in Bayview. These tank traps are a little-known remnant of WWII defence along Sydney's Northern Beaches.
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To insist on their origin - and thereby differentiate them from the hydroxylated pulvinones found in Suillus grevillei - Seto and coworkers named these compounds Aspulvinones.N. Ojima, I. Takahashi, K. Ogura, S. Seto, Tetrahedron Lett. 1976, 17, 1013-1014.I. Takahashi, N. Ojima, K. Ogura, S. Seto, Biochemistry 1978, 17, 2696-2702.
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For versions designed for EXPRO-02 board, extra raster scaling and transformation effects are used throughout the game. In addition, player's cursor can move closer to the blocks to the point of touching them. Tetrahedron boss has longer lightning range. The Japanese version based on EXPRO-02 is published by Taito.
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It has 25 faces. It can be realized geometrically from a tetrahedron (the faces of which correspond to the four large nine-sided faces in the drawing, three of which are between pairs of fragments and the fourth of which forms the exterior) by multiply truncating three of its vertices.
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Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner.
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Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner.
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Tetrahedron Letters is a weekly international journal for rapid publication of full original research papers in the field of organic chemistry. According to the Journal Citation Reports, the journal has a 2014 impact factor of 2.379 and it is ranked 22nd out of 57 journals in the "Organic Chemistry" category.
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1966, 5239;H. Nozaki, H. Takaya, S. Moriuti, R. Noyori, Tetrahedron 1968, 24, 3655. ;1970: John Pople develops the Gaussian program greatly easing computational chemistry calculations.W. J. Hehre, W. A. Lathan, R. Ditchfield, M. D. Newton, and J. A. Pople, Gaussian 70 (Quantum Chemistry Program Exchange, Program No. 237, 1970).
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Melbourne-based Indigenous artist Reko Rennie was awarded an 200,000 commission to create 'Murri Totems', an installation located at the entrance of the LIMS building. The artwork consists of four vertical structures incorporating the five platonic forms – icosahedron, octahedron, star tetrahedron, hexahedron and dodecahedron – and painted with Rennie’s traditional pattern.
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First row (square): 00 10 01 11 Second row : 000 100 010 001 triad (triangle) 110 101 011 triad (triangle) 111 Third row 0000 1000 0100 0010 0001 tetrad (tetrahedron or 3-simplex) 1100 1010 1001 0110 0101 0011 hexany (octahedron) 1110 1101 1011 0111 tetrad 1111 The octahedron there is the edge dual of the tetrahedron, or rectified tetrahedron Fourth row 00000 10000 01000 00100 00010 00001 pentad (4-simplex or pentachoron – four- dimensional tetrahedron) 11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 2)5 dekany (10 vertices, rectified 4-simplex) 00111 01011 01101 01110 10011 10101 10110 11001 11010 11100 3)5 dekany (10 vertices) 01111 10111 11011 11101 11110 pentad 11111 The rectified 4-simplex which is the mathematical name for the geometrical shape of the dekany is also known as the dispentachoron Fifth row 000000 100000 010000 001000 000100 000010 000001 hexad (5-simplex or hexateron – five-dimensional tetrahedron) 110000 101000 100100 100010 100001 011000 010100 010010 010001 001100 001010 001001 000110 000101 000011 2)6 pentadekany (15 vertices, rectified 5-simplex) 111000 110100 110010 110001 101100 101010 101001 100110 100101 100011 011100 011010 011001 010110 010101 010011 001110 001101 001011 000111 eikosany (20 vertices birectified 5-simplex) 001111 010111 011011 011101 011110 100111 101011 101101 101110 110011 110101 110110 111001 111010 111100 4)6 pentadekany (15 vertices) 011111 101111 110111 111011 111101 111110 hexad 111111 The dekany is the edge dual of the 4-simplex. Similarly, the geometrical figure for the pentadekany is the edge dual of the 5-simplex. A dekany cam be made by joining together the midpoints of the edges of the 4-simplex, and similarly for the pentadekany and the 5-simplex. Similarly the dekany vertices when scaled by 1/2 move to the midpoints of the 4-simplex edges, and the pentadekany vertices move to the midpoints of the 5-simplex edges, and so on in all higher dimensions.
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His researches have been published as chapters to books authored by others as well as several peer- reviewed scientific articles; ResearchGate, an online repository of science articles, has listed 134 of them. He is the vice-chair of the National Organic Symposium Trust (NOST) and serves as an editor of Tetrahedron journal.
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However, another highly symmetric class of polyhedra, the Catalan solids, do not all have ideal forms. The Catalan solids are the dual polyhedra to the Archimedean solids, and have symmetries taking any face to any other face. Catalan solids that cannot be ideal include the rhombic dodecahedron and the triakis tetrahedron.; see .
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Cimino, G.; De Giulio, A.; De Rosa, S.; Di Marzo, V. Tetrahedron Lett. 1989,30 , 3563. The skeleton of this species is formed by a carbonated chain of 46 to 47 Carbon atoms.Férnandez-Trillo, 2004, PhD Dissertation, Universidad de Santiago de Compostela, Síntesis de Poliacetilenos de Origen Marino:Síntesis de Callyberinas A-C y (−)-Siphonodiol.
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Regular and semi-regular polytopes III, p. 315-316 They have been named with a demi- prefix to each hypercube name: demicube, demitesseract, etc. The demicube is identical to the regular tetrahedron, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered semiregular for having only regular facets.
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A clathrin cage with a single triskelion highlighted in blue. CryoEM map EMD_5119 was rendered in UCSF Chimera and one clathrin triskelion was highlighted. Each cage has 12 pentagons. Mini-coat (left) has 4 hexagons and tetrahedral symmetry as in a truncated triakis tetrahedron. Hexagonal barrel (middle) has 8 hexagons and D6 symmetry.
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The effect of anchoring substitution on the rate of an intramolecular diels-alder reaction with furan-diene Serge Cauwberghs and Pierre J. De Clercq B. Tinant and J. P. Declercq Tetrahedron Letters Volume 29, Issue 20 , 1988, Pages 2493-2496 tert-Butyl effect The tert-butyl effect is an example of steric hindrance.
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The bell tower was built after the war in the memory of Fr. Wachter. It consists of a concrete structure in the form of a tripod, topped with a tetrahedron-shaped roof. At the peak of the roof is a cross. The three bells of the tower must be rung by hand.
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Miyaura, M.; Sasaki, N.; Itoh, M.; Suzuki, A. Tetrahedron Lett. 1977, 173. Migration takes place with retention of configuration at the migrating carbonZweifel, G. in Aspects of Mechanism and Organometallic Chemistry, J. H. Bewster, Ed., Plenum, 1978, p. 229. and inversion of configuration at the migration terminus (provided it is sp3 hybridized).
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In Euclidean space, the hypervolume of an -facet of an -simplex is less than or equal to the sum of the hypervolumes of the other facets. In particular, the area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three sides.
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In the dialogue Timaeus Plato associated each of the four classical elements (earth, air, water, and fire) with a regular solid (cube, octahedron, icosahedron, and tetrahedron respectively) due to their shape, the so-called Platonic solids. The fifth regular solid, the dodecahedron, was supposed to be the element which made up the heavens.
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Lewis's early career was dedicated to magnetic properties of metal complexes. He achieved significant acclaim for contributions to metal carbonyl clusters. Together with his longtime collaborator Brian F. G. Johnson, his research group discovered many structurally unusual compounds. Illustrative of their achievements is their synthesis of super tetrahedron [Os10C(CO)24]2−.
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Electronographic investigations of HfCl4 in gas phase showed that the Hf-Cl internuclear distance is 2.33 Å and the Cl…Cl internuclear distance is 3.80 Å. The ratio of intenuclear distances r(Me-Cl)/r(Cl…Cl) is 1.630 and this value agrees well with the value for the regular tetrahedron model (1.633).
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248px Putting any of the four vertices in the role of O yields four such identities, but at most three of them are independent; if the "clockwise" sides of three of the four identities are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity. Three angles are the angles of some triangle if and only if their sum is 180° (π radians). What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°.
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Upon visiting the Tetra Pak factory in Lund in the 1950s, Danish Nobel Prize winner and Physics professor Niels Bohr called the tetrahedron "a perfect practical application of a mathematical problem", and the invention lay the foundation of the Tetra Pak success saga.Andersson, Peter and Larsson, Tommy, Tetra. Historien om dynastin Rausing, Stockholm: Norstedts 1998 (), p. 23 The Royal Swedish Academy of Engineering Sciences has called the Tetra Pak packaging system one of Sweden's most successful inventions of all times. The development of the triangular tetrahedron carton package, the rectangular Tetra Brik, was represented at the 2011 exhibition Hidden Heroes – The Genius of Everyday Things at the London Science Museum/Vitra Design Museum, celebrating "the miniature marvels we couldn’t live without".
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It is one of three semiregular 4-polytope made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a tetroctahedric for being made of tetrahedron and octahedron cells.Gosset, 1900 E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC5.
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Unsaturated carbonyl compounds may be reduced either to saturated or unsaturated alcohols by alkoxyaluminium hydride reagents. Addition of an unsaturated aldehyde to a solution of Red-Al afforded the saturated alcohol; inverse addition yielded the unsaturated alcohol product.Bazant, V.; Capka, M.; Cerny, M.; Chvalovský, V.; Kochloefl, K.; Kraus, M.; Málek, J. Tetrahedron Lett., 1968, 3303.
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The Li-arene reacted with water or deuterium to produce the dehalogenated product.Ramón, D.; Yus, M. Masked lithium bishomoenolates: Useful intermediates in organic synthesis, J. Org. Chem. 1991, 56, 3825-3831.Guijarro, A.; Ramón, D.; Yus, M. Naphthalene- catalysed lithiation of functionalized chloroarenes: regioselective preparation and reactivity of functionalized lithioarenes, Tetrahedron, 1993, 49, 469-482.
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The cubic distance-regular graphs have been completely classified. The 13 distinct cubic distance-regular graphs are K4 (or tetrahedron), K3,3, the Petersen graph, the cube, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the dodecahedron, the Desargues graph, Tutte 12-cage, the Biggs–Smith graph, and the Foster graph.
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Marcos, I. S.; Cubillo, M. A.; Moro, R. F.; Diez, D.; Basabe, P.; Sanz, F.; Urones, J. G. Tetrahedron Lett. 2003, 44, 8831-8835. The addition of the isopropyl group in the chemical synthesis was achieved with complete stereoselectivity. Acetylation to yield 15 required high temperatures due to the steric hindrance of the isopropyl group.
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Nitrile anions can also be involved in Michael-type additions to activated double bonds and vinylation reactions with a limited number of polarized, unhindered acetylene derivatives.Makosza, M. Tetrahedron Lett., 1966, 5489. (2)448px Arylation of nitrile anions is also possible and can take place through different mechanisms depending on the substrates and reaction conditions.
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The figure below is an example of an energy diagram for a compound with an (R) and (S) isomer.Hanefeld, U.; Veum, L. Tetrahedron: Asymmetry, 2004, 15, 3707–3709. Energy diagram of an (R) and (S) isomer. With addition of a catalyst, one transition state (TS) is lower and thus becomes the kinetically favored pathway.
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Gummadiol is a lignan hemiacetal.The structure of gummadiol - A lignan hemi- acetal. A.S.R. Anjaneyulu, A.Madhusudhana Rao, V.Kameswara Rao, L.Ramachandra Row, Andrew Pelter and Robert S. Ward, Tetrahedron Letters, 1975, Volume 16, Issues 22–23, Pages 1803–1806, It can be isolated from the heartwood of Gmelina arborea.Novel hydroxy lignans from the heartwood of gmelina arborea.
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2003, 43, 5845-5847. Other effort to make this a more selective reactions includes the use of oxidation reduction condition for the reaction.Otaka, A.; Watanabe, H.; Yukimasa, A.; Oishi, S.; Tamamura, H.; Fuji, N. New access to α- substituted (Z)-fluoroalkene dipeptide isosteres utilizing organocopper reagents under redoctive-oxidative alkylation (R-OA) conditions. Tetrahedron Lett.
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Triterpenoids of the Gramineae. Phytochemistry 9, 2137-2148.), which led many to view isoarborinol as a biomarker for higher plants. In the 1990s, a series of papersHauke, V., Graff, R., Wehrung, P., Trendel, J. M., Albrecht, P. (1992). Novel triterpene-derived hydrocarbons of arborane/fernane series in sediments. Part I. Tetrahedron 48, 3915-3924.
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Two more types of tetrahedron generalize the disphenoid and have similar names. The digonal disphenoid has faces with two different shapes, both isosceles triangles, with two faces of each shape. The phyllic disphenoid similarly has faces with two shapes of scalene triangles. Disphenoids can also be seen as digonal antiprisms or as alternated quadrilateral prisms.
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Cyphostemmin A is an oligostilbene found in Cyphostemma crotalarioides (Vitaceae).Cyphostemmins A-B, two new antifungal oligostilbenes from Cyphostemma crotalarioides (Vitaceae). Paul-Henri Ducrot, Albert Kollmann, Adil E. Bala, Amel Majira, Lucien Kerhoas, Robert Delorme and Jacques Einhorn, Tetrahedron Letters, Volume 39, Issue 52, 24 December 1998, Pages 9655–9658, It is a resveratrol dimer.
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Cyphostemmin B is an oligostilbene found in Cyphostemma crotalarioides (Vitaceae).Cyphostemmins A-B, two new antifungal oligostilbenes from Cyphostemma crotalarioides (Vitaceae). Paul-Henri Ducrot, Albert Kollmann, Adil E. Bala, Amel Majira, Lucien Kerhoas, Robert Delorme and Jacques Einhorn, Tetrahedron Letters, Volume 39, Issue 52, 24 December 1998, Pages 9655–9658, It is a resveratrol dimer.
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Maricite is an ionic double metal phosphate, with a space filling capacity of about 70% (Le Page, and Donnay, 1977). The structure of maricite contains a sodium cation enclosed by ten oxygen anions within 10 Å, in an irregular coordination. There is (2+2+2) type distorted tetrahedron around the iron (Bridson, et al., 1997).
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A direct computation shows that ∂2 = 0. In geometric terms, this says that the boundary of anything has no boundary. Equivalently, the abelian groups :(C_k, \partial_k) form a chain complex. Another equivalent statement is that Bk is contained in Zk. As an example, consider a tetrahedron with vertices oriented as w,x,y,z.
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The tetrahedron and the Császár polyhedron have no diagonals at all: every pair of vertices in these polyhedra forms an edge. It remains an open question whether there are any other polyhedra (with manifold boundary) without diagonals , although there exist non-manifold surfaces with no diagonals and any number of vertices greater than five .
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104 O and F. Fluorine and oxygen share the ability to often bring out the highest oxidation states among the elements. P and S (Se). Phosphorus reacts with sulfur and selenium (and oxygen) to form a large number of compounds. These compounds are characterized by structural analogies derived from the white phosphorus P4 tetrahedron.
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Potassium pyrosulfate contains the pyrosulfate anion which has a dichromate like structure. The geometry can be visualized as a tetrahedron with two corners sharing the SO4 anion's configuration and a centrally bridged oxygen atom. A semi-structural formula for the pyrosulfate anion is O3SOSO32−. The oxidation state of sulfur in this compound is +6.
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Instead, NMR spectroscopy was performed on derivatives of kaitocephalin. Other methods used to determine its absolute configuration included Mosher's method and NOESY.Hiroyuki Kobayashi et al., "Absolute configuration of a novel glutamate receptor antagonist kaitocephalin", Tetrahedron Letters 42 (2001): 4021-4023 Only small amounts of kaitocephalin are produced naturally, making it an attractive target for synthesis.
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Yukio Suzuki, Wakako Mori, Hiroyuki Ishizone, Koichi Naito, and Toshio Honda (1992) Concise Enantiospecific Syntheses of (+)-Eldanolide and (−)-cis-Whisky Lactone. Tetrahedron letters, volume 33, pages 4931-4932 The (3S,4R) isomer is extracted by the alcoholic beverage from some precursor substances in the oak wood.Masuda and Nishimura (1971), Branched nonalactones from some Quercus species.
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Since then, the use of cascade reactions has proliferated in the area of total synthesis. Similarly, the development of cascade-driven organic methodology has also grown tremendously. This increased interest in cascade sequences is reflected by the numerous relevant review articles published in the past couple of decades.Pellissier, H. Tetrahedron 2006, 62, 2143–2173.
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His second marriage was to Theodora Elizabeth De Kiewiet, PhD in South Africa. They had three children: Michael, Edith, and Richard. In 1957 Stephen and his family moved to Oxford. Stephen and Dora and together edited the organic chemistry journal Tetrahedron from 1957 up to his death in 1965, after which Dora continued on as editor.
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Typical orthosilicates tend to form blocky equant crystals, and are fairly hard., p. 573 Several rock-forming minerals are part of this subclass, such as the aluminosilicates, the olivine group, and the garnet group. The aluminosilicates –bkyanite, andalusite, and sillimanite, all Al2SiO5 – are structurally composed of one [SiO4]4− tetrahedron, and one Al3+ in octahedral coordination.
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It is usually found nestled in its food sponge Anthoarcuata graciae. This sponge is currently classified as a member of the genus Antho, family Clathriidae, but may actually belong to the family Hymedesmiidae as other Aldisa species specialise on sponges from this family. Ayer, S.W.; Andersen, R.J. Steroidal antifeedants from the dorid nudibranch Aldisa sanguinea cooperi. Tetrahedron Lett.
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Takeshi Akasaka, Koichiro Fukuoka, and Wataru Ando (1991): "Formation of 1,2-dioxolane in the singlet oxygenation of cyclopropane". Tetrahedron Letters, volume 32, issue 52, pages 7695-7698. He-Hua Liu, Hong- Xia Jin, and Yi-Kang Wu (2004): "Synthesis and cleavage studies of a 1,2-dioxolane-type peroxide". Chinese Journal of Chemistry, volume 22, issue 9, pages 1029-1033.
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Vanadium catalyzed epoxidations have been shown to be very sensitive to the steric bulk of the vinyl group. 500px Homoallylic alcohols are effective directing groups for epoxidations in both cyclic and acyclic systems for substrates which show hydrogen bonding. However these reactions tend to have lower levels of selectivity.Johnson, M. R.; Kishi, Y. Tetrahedron Lett., 1979, 4347-4350.
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Crich, D.; Natarajan, S.; Crich, J.Z. Tetrahedron 1997, 53, 7139–7158.() 650px The step shown above had a 70% yield (91% when the byproduct was converted to the Meyer-Schuster product in another step). The authors used the Meyer–Schuster rearrangement because they wanted to convert a hindered ketone to an alkene without destroying the rest of their molecule.
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Epoxidation of allyl- and vinylsilanes can be easily accomplished with peracids. Silyl epoxides can be converted into ketones, aldehydes, or alkenes after selective epoxide opening and elimination. When allylsilanes are combined with peracids, the intermediate epoxides are usually converted to allylic alcohols before isolation.Fleming, I.; Au-Yeung, B.-W. Tetrahedron 1981, 37, Supplement No. 1, 13. (12)File:SiAddScope6.
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This includes the computed surface normals and other graphics attributes at the intersection points. Each tetrahedron has sixteen possible configurations, falling into three classes: no intersection, intersection in one triangle and intersection in two (adjacent) triangles. It is straightforward to enumerate all sixteen configurations and map them to vertex index lists defining the appropriate triangle strips.
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5-hydroxy-4-pyrones can also be used to synthesize carbonyl ylides by an intramolecular hydrogen transfer.Garst, M. E.; McBride, B. J.; Douglass III, J. G. Intramolecular cycloadditions with 2-(ω-alkenyl)-5-hydroxy-4-pyrones Tetrahedron Lett. 1983, 24, 1675. After hydrogen transfer, the carbonyl ylide can then react with dipolarophiles to form oxygen-containing rings.
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Trichotomine is a bright blue pigment found in the berries of the plant Clerodendrum trichotomum, which is native to China and Japan. It has a novel chromophore structure which differs from previously studied plant pigments.Iwadare S, Shizuri Y, Yamada K, Hirata Y. Synthesis of trichotomine, a blue pigment obtained from clerodendron trichotomum thunb. Tetrahedron. 1978; 34(10):1457-1459.
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The order-6 cubic honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. It has a related alternation honeycomb, represented by ↔ . This alternated form has hexagonal tiling and tetrahedron cells. There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including the order-6 cubic honeycomb itself.
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Vertex figure: tetrahedron Net In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a C5, pentachoron,N.W. Johnson: Geometries and Transformations, (2018) Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249 pentatope, pentahedroid,Matila Ghyka, The geometry of Art and Life (1977), p.
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The sugar derivatives used for SHJ reactions should be purified, dried, and powdered before use. Intramolecular Friedel-Crafts reaction of the aromatic ring of a benzoate ester at the 2-position of 1-acetoxy ribose in the presence of a Lewis acid has been observed, and represents a potential side reaction.Martin, O. R. Tetrahedron Lett. 1985, 26, 2055.
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Her parents found out that mathematical conceptions, like cone and tetrahedron, were familiar to her before she reached the age of 6. She excelled at the primary school, moved up a class after only 3 months. After completing the Lyceum in HasseltDaubechies herself, quoted in Flanders Today #15, October 2014, available here. Retrieved 19 August 2017.
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Probert, J. M.; Rennex, D.; Bradley, M. (1996) Lanthionines for Solid Phase Synthesis. Tetrahedron Letters, 37, 1101-1104. The sulfur extrusion method is, however, the only pathway for lanthionine that has been employed in the total synthesis of a lantibiotic. Biosynthesis of the lanthionine bridge in peptidic natural products can be accomplished by through a number of different pathways.
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1987, 28, 2685. (9)File:DVCcarbylide.png Divinyl aziridines undergo a similar suite of reactions providing azepines or vinyl pyrrolines depending on the relative configuration of the aziridine starting material.Manisse, N.; Chuche, J. Tetrahedron, 1977, 33, 2399. Divinyl thiiranes can provide thiepines or dihydrothiophenes, although these reactions are slower than those of the corresponding nitrogen- and oxygen-containing compounds.
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In an interesting application of the transmetalation method, the chiral allylic stannane 1 adds to acrolein to yield the 1,5-syn diastereomer as a single stereoisomer. A subsequent sigmatropic rearrangement increased the distance between the stereocenters even further. This step was carried out en route to (±)-patulolide C.Dorling, E.K.; Thomas, E.J. Tetrahedron Lett. 1999, 40, 471.
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Polyhedra may be classified and are often named according to the number of faces. The naming system is based on Classical Greek, for example tetrahedron (a polyhedron with four faces), pentahedron (five faces), hexahedron (six faces), triacontahedron (30 faces), and so on. For a complete list of the Greek numeral prefixes see , in the column for Greek cardinal numbers.
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A proposed mechanism for chorismate mutase. Clark, T., Stewart, J.D. and Ganem, B. Transition-state analogue inhibitors of chlorismate mutase. Tetrahedron 46 (1990) 731–748. © IUBMB 2001 Chorismate mutase is an intramolecular transferase and it catalyzes the conversion of chorismate to prephenate, used as a precursor for L-tyrosine and L-phenylalanine in some plants and bacteria.
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Because of this, inhalation of beryllium metal dust by people leads to the development of the fatal condition of berylliosis. Be(OH)2 dissolves in strongly alkaline solutions. In basic beryllium acetate the central oxygen atom is surrounded by a tetrahedron of beryllium atoms. Beryllium difluoride, unlike the other alkaline earth difluorides, is very soluble in water,.
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A uniform antiprism has, apart from the base faces, 2n equilateral triangles as faces. Uniform antiprisms form an infinite class of vertex- transitive polyhedra, as do uniform prisms. For we have the regular tetrahedron as a digonal antiprism (degenerate antiprism), and for the regular octahedron as a triangular antiprism (non-degenerate antiprism). Dual polyhedra of antiprisms are trapezohedra.
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In geometry, a compound of four tetrahedra can be constructed by four tetrahedra in a number of different symmetry positions. One compound can be constructed rotating a tetrahedron by 45 degree turns along an axis of the middle of an edge. It has dihedral symmetry, D4 It has the same vertex arrangement as the convex octagonal prism.
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Basic three-dimensional cell shapes The basic 3-dimensional element are the tetrahedron, quadrilateral pyramid, triangular prism, and hexahedron. They all have triangular and quadrilateral faces. Extruded 2-dimensional models may be represented entirely by the prisms and hexahedra as extruded triangles and quadrilaterals. In general, quadrilateral faces in 3-dimensions may not be perfectly planar.
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S. Yao, M. Johannsen, R.G. Hazell, K.A. Jorgensen, J. Org. Chem., 63, 118-121. Dihydroactinidiolide is a pheromone for a variety of insects;Pherobase listing for dihydroactinidiolide for example, it is one of the three components of the pheromone for queen recognition of the workers of the red fire ant.Rocca, J.R. Tumlinson, J.H., Glancey, B.M., Lofgren, C.S., Tetrahedron Lett.
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Organic trans-hyponitrites R1−O−N=N−O−R2 can be obtained by reacting trans silver(I) hyponitrite with various alkyl halides. For example, reaction with t-butyl chloride yields trans di-tert-butyl hyponitrite.H. Kiefer and T. G. Traylor (1966), Tetrahedron Lett., page 6163.R. L. Huang, T. W. Lee, and S. H. Ong (1969), J. Chem. Soc.
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So there are 16 edges, 32 triangles, 24 octahedra, and 8 24-cells meeting at every vertex. The edge figure is a tetrahedron, so there are 4 triangles, 6 octahedra, and 4 24-cells surrounding every edge. Finally, the face figure is a triangle, so there are 3 octahedra and 3 24-cells meeting at every face.
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DNA tetrahedron described in Goodman, 2005. Models of this type are useful for ensuring that tertiary structure constraints do not cause excessive strain to the molecule. Geometrical models of nucleic acids are used to predict tertiary structure. This is important because designed nucleic acid complexes usually contain multiple junction points, which introduces geometric constraints to the system.
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S. Takahashi, T. Shibano, and N. Hagihara: The dimerization of butadiene by palladium complex catalysts. In: Tetrahedron Letters 8.26 (1967): 2451-2453. Products of the Telomerisation of 1,3-Butadiene The formation of several isomers are possible. In addition to 1,3-butadiene also substituted dienes such as isoprene or cyclic dienes such as cyclopentadiene can be used.
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The third spin cannot simultaneously minimize its interactions with both of the other two. Since this effect occurs for each spin, the ground state is sixfold degenerate. Only the two states where all spins are up or down have more energy. Similarly in three dimensions, four spins arranged in a tetrahedron (Figure 2) may experience geometric frustration.
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Tetrahedron Lett. 2003, 44, 3569-3572. Structure 5 was desymmetrized through an enzymatic transesterification process, using an immobilized lipase PS 30 enzyme to give structure 6, which was formed with a 99% enantiomeric excess.Mehta, G.; Islam, K. Enantioselective Total Synthesis of (−)-epoxyquinols A and B. Novel, convenient access to chiral epoxyquinone building blocks through enzymatic desymmetrization. Tetrahedron Lett. 2004, 45, 3611-3615. :File:IntegrasoneSynthesis1.gif The stereochemistry of the hydroxyl group at carbon 6 in the final integrasone molecule (3) was determined by reduction, which was both regio and stereo selective due to the directing effects of the primary hydroxyl group (carbon 8) and the epoxide ring (carbons 4 and 5). The hydroxy group on carbon 8 is then selectively protected with as the triethylsilyl (TES) ether to give structure 8.
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The symmetry group of a right n-sided antiprism with regular base and isosceles side faces is Dnd of order 4n, except in the case of a tetrahedron, which has the larger symmetry group Td of order 24, which has three versions of D2d as subgroups, and the octahedron, which has the larger symmetry group Oh of order 48, which has four versions of D3d as subgroups. The symmetry group contains inversion if and only if n is odd. The rotation group is Dn of order 2n, except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has three versions of D2 as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D3 as subgroups.
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In 2013, Nature Chemical Biology published an article about Jerala's achievement that paves a path to designing and producing completely new protein shapes using reprogrammed bacteria by synthesizing protein that folds itself into a tetrahedron — a pyramid with a triangular base measuring just 5 nanometres along each edge - which can be used as container for delivering drugs on the nanoscale. Genetically modified Escherichia coli bacteria were drafted in to synthesize the protein.Protein gets in on DNA's origami act: Engineered bacteria make self-assembling tetrahedra, Nature, 28 April 2013 Dek Woolfson, a biochemist from Centre for Nanoscience and Quantum Information, UK, described this kind of engineering with the following words: Jerala's team is trying to double the size of the coiled coils in the tetrahedron, and thinking about making other shapes, such as prisms and bipyramids.
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A number of designs were used, including a tetrahedron which could be close-packed without waste space, and could not be knocked over accidentally (slogan: "No more crying over spilt milk"). However, the industry eventually settled on a design similar to that used in the United States.Milk and Juice Cartons Fact Sheet, Waste Wise WA, zerowastewa.com.au. Retrieved June 21, 2009.
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The Dürer graph is a well-covered graph, meaning that all of its maximal independent sets have the same number of vertices, four. It is one of four well-covered cubic polyhedral graphs and one of seven well-covered 3-connected cubic graphs. The only other three well-covered simple convex polyhedra are the tetrahedron, triangular prism, and pentagonal prism.; .
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It was introduced by . The three tetrahedra of a desmic system are contained in a pencil of quartic surfaces. The name "desmic" comes from the Greek word δεσμός, meaning band or chain, referring to the pencil of quartics. Every line that passes through two vertices of two tetrahedra in the system also passes through a vertex of the third tetrahedron.
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The chlorate groups take the shape of a distorted tetrahedron. At , the chlorine-oxygen distances in each chlorate ion are 1.498, 1.488 and 1.468 Å, with the longest being the oxygen next to copper. The ∠O-Cu-O (angle subtended at copper by oxygen atoms) is 105.2°, 108.3°, and 106.8°. At lower temperatures (), the water molecules and copper-chlorate distance shrink.
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Since many substrates are not stable to sodium hydride, several procedures have been developed using milder bases. Masamune and Roush have developed mild conditions using lithium chloride and DBU.Blanchette, M. A.; Choy, W.; Davis, J. T.; Essenfeld, A. P.; Masamune, S.; Roush, W. R.; Sakai, T. Tetrahedron Letters 1984, 25, 2183–2186. Rathke extended this to lithium or magnesium halides with triethylamine.
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In some cases, the protoplast skips the thread phase and remains isodiametric. The nuclei then migrate to four points of a tetrahedron and the protoplast cleaves into a tetrad of lobes. Soon after, the tetrad undergoes mitosis and splits again into an octette of haploid cells. Each cell releases a swarmcell with either one long flagellum, 2 unequal flagella, or 2 long flagella.
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A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar.
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201 and 228–239. The equilateral triangle is the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes) (the heptagonal triangle being the only obtuse one).Leon Bankoff and Jack Garfunkel, "The heptagonal triangle", Mathematics Magazine 46 (1), January 1973, 7–19. A regular tetrahedron is made of four equilateral triangles.
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According to a model advanced by Felkin,Chérest, M.; Felkin, H.; Prudent, N. Tetrahedron Lett., 1968, 2199. diastereoselectivity is controlled by the relative energy of the three transition states I, II, and III. Transition state I is favored in the absence of polar groups on the α carbon, and stereoselectivity increases as the size of the achiral ketone substituent (R) increases.
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Anhydrous BaI2 can be prepared by treating Ba metal with 1,2-diiodoethane in ether.Duval, E.; Zoltobroda, G.; Langlois, Y.; A new preparation of BaI2: application to (Z)-enol ether synthesis. Tetrahedron Letters, 2000, 41, 337-339 BaI2 reacts with alkyl potassium compounds to form organobarium compounds.Walter, M. D.; Wolmershauser, G.; Sitzmann, H.; Calcium, Strontium, Barium, and Ytterbium Complexes with Cyclooctatetraenyl or Cyclononatetraenyl Ligands.
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The molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. Cartoon representation of the protein Calmodulin in its two conformation: "closed" on the left (derived from PDB id: 1CFD) and "open" on the right (derived from PDB id: 3CLN). The open conformation is represented bound with 4 calcium ions (orange spheres).
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In five and higher dimensions, there are 3 regular polytopes, the hypercube, simplex and cross-polytope. They are generalisations of the three-dimensional cube, tetrahedron and octahedron, respectively. There are no regular star polytopes in these dimensions. Most uniform higher-dimensional polytopes are obtained by modifying the regular polytopes, or by taking the Cartesian product of polytopes of lower dimensions.
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E. Gil-Av, B. Feibush, R. Charles-Sigler: Separation of Enantiomers by Gas Liquid Chromatography With an Optically Active Stationary Phase, Tetrahedron Lett. 7 (1966) 1009–1015. Many racemic compounds, amenable for enantioselective interaction via hydrogen bonding with the CSP, could be analytically enantioseparated by GC.E. Gil-Av: Present Status of Enantiomeric Analysis by Gas Chromatography, J. Mol. Evol. 6 (1975) 131–144.
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The two most common problems in this class of reactions are low ortho/para selectivities and dearomatization (the latter is a particularly significant problem for phenols).Zupan, M.; Iskra, J.; Stavber, S. Bull. Chem. Soc. Jpn. 1995, 68, 1655. :File:EFScope5.png Enol ethers and glycals are nucleophilic enough to be fluorinated by Selectfluor.Albert, M.; Dax, K.; Ortner, J. Tetrahedron 1998, 54, 4839.
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Keck, G. E., Geraci, L. S., Tetrahedron Lett. 1993, 34, 7827-7828. Keck's early success with stereoselectivity and the simplicity of the catalyst preparation led to many improvements in reaction design, including development of other structural analogs of BINOL, use of stoichiometric additives to enhance the reaction rate, and broadening the scope of the reaction to include substituted stannane nucleophiles.
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In addition to toys such as the Hoberman sphere, Hoberman created the BrainTwist, a hard plastic tetrahedron that folds, stellates, and becomes self-dual while having a component that rotates similarly to a Rubik's Cube. Likewise, Chuck Hoberman's Pocket Flight Ring is a folding, throwable toy resembling a chakram. Hoberman has also created the Expandagon Construction System, a kind of construction toy.
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The compound can be synthesised from veratrole alcohol upon acid catalysis. :Cyclotriveratrylene synthesis Operating via the same intermediates, an alternative synthesis involves the acid-catalyzed reaction of veratrole and formaldehyde. The method is similar to the reactions that give phenol formaldehyde resins except that the degree of polymerization is moderated by the methoxy substituents.Collet, A., "Cyclotriveratrylenes and cryptophanes", Tetrahedron 1987, vol.
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The phosphidosilicates or phosphosilicides are inorganic compounds containing silicon bonded to phosphorus and one or more other kinds of elements. In the phosphosilicates each silicon atom is surrounded by four phosphorus atoms in a tetrahedron. The triphosphosilicates have a SiP3 unit, that can be a planar triangle like carbonate CO3. The phosphorus atoms can be shared to form different patterns e.g.
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Another interpretation can represent this solid as a hexahedron, by considering pairs of trapezoids as a folded regular hexagon. It will have 6 faces (4 triangles, and 2 hexagons), 12 edges, and 8 vertices. It could also be seen as a folded tetrahedron also seeing pairs of end triangles as a folded rhombus. It would have 8 vertices, 10 edges, and 4 faces.
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In 2009 the zebrafish screening operations was acquired by Evotec for £0.5 Million. In 1996, he became Professor of Chemistry and in 2006, Waynflete Professor of Chemistry. Davies is founder and editor-in-chief for Tetrahedron: Asymmetry. Davies along with Malcolm Green and Michael Mingos have compiled a set of rules that summarize where nucleophilic additions will occur on pi ligands.
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Ruitenberg, K.; Vermeer, P. Tetrahedron Lett. 1984, 25, 3019. These compounds react similarly to allylstannanes to afford homopropargyl alcohols, and any of the three reaction modes described above can be used with this class of reagents as well. (9)File:StanAddScope3.png Imines are less reactive than the corresponding aldehydes, but palladium catalysis can be used to facilitate addition into imines.
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The ketimine is then subjected to the Bucherer-Bergs reagents, resulting in a 5,5'- disubstituted hydantoin 3. This reaction has three points of chemical diversity as the structure of the aziridine starting compound, the organometallic reagent, and the electrophile can all be varied to synthesize a different hydantoin.Montagne, C.; Shiers, J. J.; Shipman, M. Tetrahedron Lett. 2006, 47, 9207-9209.
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In graph theory this operation creates a medial graph. The rectification of any regular self-dual polyhedron or tiling will result in another regular polyhedron or tiling with a tiling order of 4, for example the tetrahedron {3,3} becoming an octahedron {3,4}. As a special case, a square tiling {4,4} will turn into another square tiling {4,4} under a rectification operation.
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These alkynes react in the annulation reaction to form resorcinol monosilyl ethers which can be de-protected under mild reaction conditions. Base-promoted dehydrohalogenation of (Z)-2-halovinyl ethers to form alkoxyacetylenes is one of the most well established routes of alkyne synthesis (Scheme 10).Danheiser, R.L., Nishida, A., Savariar, S., Trova, M. P. “Trialkylsiloxyalkynes: Synthesis and Aromatic Annulation Reactions. Tetrahedron Lett.
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Also, while not very useful for display on a physical computer screen, by adding extra terms, a Bézier tetrahedron or Bézier polytope results. Due to the nature of the equation, the entire triangle will be contained within the volume surrounded by the control points, and affine transformations of the control points will correctly transform the whole triangle in the same way.
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The mechanism proceeds in two stages: β-nucleophilic addition to the unsaturated carbonyl compound, followed by electrophilic substitution at the α-carbon of the resulting enolate. When the nucleophile is an organometallic reagent, the mechanisms of the first step can vary. Whether reactions take place by ionic or radical mechanisms is unclear in some cases.Corey, J.; Boaz, W. Tetrahedron Lett.
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Using a more complicated construction, proved that every tetrahedron can be dissected into at most 12 orthoschemes. proved that this is optimal: there exist tetrahedra that cannot be dissected into fewer than 12 orthoschemes. In the same paper, Böhm also generalized Lenhard's result to three-dimensional spherical geometry and three-dimensional hyperbolic geometry. In four dimensions, at most 500 orthoschemes are needed.
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The Barnette–Bosák–Lederberg graph has a similar construction to the Tutte graph but is composed of two Tutte fragments, connected through a pentagonal prism, instead of three connected through a tetrahedron. Without the constraint of having exactly three edges at every vertex, much smaller non-Hamiltonian polyhedral graphs are possible, including the Goldner–Harary graph and the Herschel graph.
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In 1959, the food industry saw the advent of the use of paper-foil-plastic laminated containers called tetrahedron. In 1962, a Swedish company, Tetra Pak, introduced this container to the United States market. They sold pasteurized milk and beverages in the containers. Roy Graves' company started sterilizing this container with chlorine and were able to aseptically fill and hermetically seal the container.
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748 Simeïsa (prov. designation: or ) is a very large Hilda asteroid from the outermost regions of the asteroid belt, approximately in diameter. It was discovered on 14 March 1913, by Russian astronomer Grigory Neujmin at the Simeiz Observatory on the Crimean peninsula. The dark P-type asteroid has a rotation period of 11.9 hours and a shape that is reminiscent of a tetrahedron.
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A company history with a difference, Lund: Tetra Pak International 1996 (), p. 27f In March 1944 he filed for a patent, and in 1951 Tetra Pak was created as a subsidiary to Åkerlund & Rausing. The tetrahedron package made Tetra Pak one of the world's most successful companies. The package is still sold today under the name of Tetra Classic Aseptic.
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Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all concurrent at a point called the centroid of the tetrahedron.Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong University Press, 1994, pp. 53–54 In addition the four medians are divided in a 3:1 ratio by the centroid (see Commandino's theorem).
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At some airfields, a large frame in the shape of a tetrahedron with two sides covered with a thin material is mounted on a rotating pivot and always points into the wind. It is built big enough to be seen from the air and is sometimes illuminated. Its purpose is to serve as a reference to pilots indicating wind direction..
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For example, the electron configuration of Fe is determined by starting at H, which is 1s1, and counting in atomic number order. This gives a configuration of 1s2 2s2 2p6 3s2 3p6 4s2 3d6 or, in short form, [Ar] 4s2 3d6. The four blocks of the Adomah table can be rearranged such that they fit, equidistantly spaced, inside a regular tetrahedron.
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The Soundfield microphone is an audio microphone composed of four closely spaced subcardioid or cardioid (unidirectional) microphone capsules arranged in a tetrahedron. It was invented by Michael Gerzon and Peter Craven, and is a part of, but not exclusive to, Ambisonics, a surround sound technology. It can function as a mono, stereo or surround sound microphone, optionally including height information.
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Use of fluorophores appended to macrocycles provides advantages over other techniques for selective and sensitive metal ion detection. Izatt demonstrated that certain 8-¬hydroxyquinoline derivatives attached to diazamacrocycles elicit a strong fluorescent response when complexed to selected closed-shell metal ions.Prodi, L. et al Characterization of 5-Chloro-8-Methoxyquinoline appended diaza 18-Crown-6 as a chemosensor for cadmium. Tetrahedron Lett.
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240px Conformational analysis of medium rings begins with examination of cyclooctane. Spectroscopic methods have determined that cyclooctane possesses three main conformations: chair-boat, chair-chair, and boat-boat. Cyclooctane prefers to reside in a chair-boat conformation, minimizing the number of eclipsing ethane interactions (shown in blue), as well as torsional strain.Petasis, N. A.; Patane, M.A. Tetrahedron 1992, 48, 5757-5821.
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Low- temperature Ba2TiO4 \- TiO4 tetrahedra shaded blue; Ba atoms green; O atoms red The solid has two known phases: a low-temperature (β) phase with P21/n symmetry and a high-temperature (α′) phase with P21nb symmetry. The structure of Ba2TiO4 is unusual among the titanates because its titanium atoms sit in a four-oxygen tetrahedron rather than a six-oxygen octahedron.
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It can be obtained by successive condensations of 2,5-bis(hydroxymethyl)pyrrole and pyrrole, with a tripyrrole intermediate.Shozo Taniguchi, Hikaru Hasegawa, Shoko Yanagiya, Yusuke Tabeta, Yoshiharu Nakano, and Masahiko Takahashi (2001): "The first isolation of unsubstituted porphyrinogen and unsubstituted 21-oxaporphyrinogen by the ‘3+1’ approach from 2,5-bis(hydroxymethyl)pyrrole and tripyrrane derivatives". Tetrahedron, volume 57, issue 11, pages 2103-2108.
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Rausing studied chemistry at the University of Lund and began his career as the head of the research laboratory at Åkerlund & Rausing, his father's company, where he was in charge of the team developing suitable materials for the newly invented tetrahedron package."Gad Rausing", The Economist, 3 February 2000 The tetrahedron subsequently became the central product of Tetra Pak, which was founded in 1951 as a subsidiary to Åkerlund & Rausing."Our History", Tetrapak.com, retrieved 8 November 2011 Rausing joined Tetra Pak as deputy managing director in 1954. Over the years the company evolved from a small family business with six full-time employees, in 1954, into a multi-national corporation with over 20,000 employees (2011), a development much of which has been credited to the leadership of Rausing and his brother throughout the 1960s and 1970s.
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Johnson, A. W., Smith, R. M., Guthrie, R. D."Vancosamine: the Structure and Configuration of a Novel Amino-sugar from Vancomycin." J. C. S. Perkin I, 1972, 2153-2159. Epivancosamine was subsequently isolated in 1988 by Hunt et al at Eli LillyHunt, A. H., Molloy, R. M., Debono, M., Occolowitz, J. L. "Isolation and Characterization of 4-epi- vancosamine." Tetrahedron Lett, 1988, 29, 1223-1226.
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Although again not derived from acetylenediol, a variety of structurally related compounds are known. Examples include the diethers diisopropoxyethyne ((CH3)2CH)-O-C≡C-O-(CH(CH3)2) and di-tert-butoxyethyne ((CH3)3C)-O-C≡C-O-(C(CH3)3).Anna Bou, Miquel A. Pericàs and Félix Serratosa (1981), Diisopropoxy- and di-tert-butoxyethyne : Stable acetylene diethers. Tetrahedron, Volume 37, Issue 7, Pages 1441-1449.
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Aegirine, an iron- sodium clinopyroxene, is part of the inosilicate subclass. The base unit of a silicate mineral is the [SiO4]4− tetrahedron. In the vast majority of cases, silicon is in four-fold or tetrahedral coordination with oxygen. In very high- pressure situations, silicon will be in six-fold or octahedral coordination, such as in the perovskite structure or the quartz polymorph stishovite (SiO2).
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The tetrahedral hypothesis is an obsolete scientific theory attempting to explain the arrangement of the Earth's continents and oceans by referring to the geometry of a tetrahedron. Although it was a historically interesting theory in the late 19th and early 20th century, it was superseded by the concepts of continental drift and modern plate tectonics. The theory was first proposed by William Lowthian Green in 1875.
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One can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the 16-cell honeycomb and has Schläfli symbol {3,3,4,3}. Hence, the 16-cell has a dihedral angle of 120°. Each 16-cell has 16 neighbors with which it shares a tetrahedron, 24 neighbors with which it shares only an edge, and 72 neighbors with which it shares only a single point.
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Low E, Nair S, Shiau T, Belisle B, Debabov D, Celeri C, Zuck M, Najafi R, Georgopapadakou N, Jain R. "N,N-Dichloroaminosulfonic acids as novel topical antimicrobial agents." Bioorg Med Chem Lett. 2009;19: 196–198. doi: 10.1016/j.bmcl.2008.10.114. Wang L, Khosrovi B, Najafi R. “N-Chloro-2,2-dimethyltaurine: a new class of remarkably stable N-chlorotaurines.” Tetrahedron Lett. 2008;49:2193–2195. doi:10.1016/j.tetlet.2008.02.038.
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Tetrahedron 2009, 65, 8185 300px MBH reaction can involve proline derivative as a cocatalyst. It was proposed that imidazole nucleophilic catalyst and proline effect the reaction via iminium intermediate.Chem. Eur, J. 2009, 15, 1734 With (S)-proline and DABCO, α-amido sulfones and α,β-unsaturated aldehydes undergo a highly enantioselective aza-MBH reaction (46-87% yield, E/Z 10:1-19:1, 82-99% ee).
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Because only reactive electrophiles undergo reduction, selectivity is possible in reactions of substrates with multiple reducible functional groups. Chiral Lewis acids and metal complexes may be used for the enantioselective reduction of ketones with hydrosilanes.Hayashi, T.; Hayashi, C.; Uozumi, Y. Tetrahedron: Asymmetry 1995, 6, 2503. (1)File:HSGen.png Upon the generation of a carbocation, rate-determining hydride transfer from the organosilane occurs to yield a reduced product.
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This equation is satisfied for the tetrahedron with h = 0 and v = 4, and for the Császár polyhedron with h = 1 and v = 7. The next possible solution, h = 6 and v = 12, would correspond to a polyhedron with 44 faces and 66 edges, but it is not realizable as a polyhedron. It is not known whether such a polyhedron exists with a higher genus .
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A necessary condition for two lines to intersect is that they are in the same plane--that is, are not skew lines. Satisfaction of this condition is equivalent to the tetrahedron with vertices at two of the points on one line and two of the points on the other line being degenerate in the sense of having zero volume. For the algebraic form of this condition, see .
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The Pyraminx Duo in the middle of a twist, showing how the puzzle can be scrambled. The Pyraminx Duo is a puzzle in the shape of a tetrahedron, divided into 4 corner pieces and 4 face centre pieces. Each corner piece has three colours, while the centre pieces each have a single colour. Each face of the puzzle contains one face centre piece and three corner pieces.
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Cyclic molecules often exist in much more rigid conformations than their linear counterparts. Even very large macrocycles like erythromycin exist in defined geometries despite having many degrees of freedom. Because of these properties, it is often easier to achieve asymmetric induction with macrocyclic substrates rather than linear ones. Early experiments performed by W. Clark StillStill, W. C.; Galynker, I. Tetrahedron 1981, 37, 3981-3996.
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The Rainy River originates in a small unnamed lake just north of Tetrahedron Peak. The river, shortly after exiting the lake, likely drops over a fairly tall waterfall as it drops down a headwall. The river flows southeast for about 3.6 km before turning south and flowing that way for about 9.8 km to its mouth at Port Mellon. The river does not receive any major tributaries.
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1985, 50, 3615. The combination of an allylsilane with an α,β-unsaturated ketone in the presence of a Lewis acid is known as the Sakurai reaction. The enolate intermediate that results from addition can be functionalized with a separate electrophile or simply protonated. Intramolecular Sakurai reactions provide ene-6-onesMajetich, G.; Hull, K.; Defauw, J.; Shawe, T. Tetrahedron Lett., 26, 2755 (1985). (9)File:SiAddScope3.
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Gigantine is an tetrahydroisoquinoline alkaloid found in Carnegiea gigantea and other related cacti. Gigantine was first discovered along with macromerine in 1967."Two New Alkaloids in Cacti" [macromerine and gigantine] J E Hodgkins, Tetrahedron Lett., 14, 1321 (1967) It is found in significant quantities in many mescaline-containing cactus species, but it is unclear whether gigantine contributes in any way to their psychoactive effects.
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A study showed that this is because alkyl shifts on cyclohexadienes proceed through a different mechanism. First the ring opens, followed by a [1,7] shift, and then the ring reforms electrocyclically:Schiess, P.; Dinkel, R. Tetrahedron Letters, 1975, 16, 29, 2503. :alkyl shift on cyclohexadiene This same mechanistic process is seen below, without the final electrocyclic ring-closing reaction, in the interconversion of lumisterol to vitamin D2.
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Stamos, D. P.; Taylor, A. G.;Kishi, Y; Tetrahedron Lett. 1996, 37 (48), 8647-8650 They observed retention of olefin geometry in some vinyl silyl substrates while inversion in others. They reasoned that the R group's size had an effect on the geometry of the olefin. If the R group is small, the solvent acetonitrile can participate in the reaction leading to inversion of the olefin's geometry.
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Waterhouseite has a unique asymmetrical structure consisting of a dense, complex framework of Mn(O, OH)6 octahedra and PO4 tetrahedra which are linked by both edges and corners. It is highly unusual for the PO4 tetrahedron to share two of its edges with the Mn(O, OH)6 octahedral. There are only two other known arsenates that have the same edge sharing as waterhouseite.
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Light Art Biennale in Linz, Manfred Kielnhofer: Guardians of Time (Art) Eric Michel: Seven Keys Martina Schettina: Tetrahedron Manabu Shimada: Paper floaty cube, 2010 Heidulf Gerngross: Archiquants lamp Artwork by Mounty R. P. Zentara The Light Art Biennale Austria 2010 (German: Biennale für Lichtkunst Austria 2010) is the first biennale for light art in Austria. About 60 artists from 21 countries and 4 continents participate.
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Technical report of the Technological University of Nagaoka 25: 105–111. a dimeric naphthoquinone derivative, whose structure was determined by spectroscopic means in the 1960s R. L. Edwards and N. Kale, Tetrahedron 1965, 21, 2095–2107 and later confirmed by X-ray crystallography.Saikawa Y, Watanabe T, Hashimoto K, Nakata M. (2000). Absolute configuration and tautomeric structure of xylindein, a blue–green pigment of Chlorociboria species.
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A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles. In graph theory, a line perfect graph is a graph whose line graph is a perfect graph. Equivalently, these are the graphs in which every odd-length simple cycle is a triangle.
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P2O5 crystallizes in at least four forms. The most familiar polymorph (see figure) comprises molecules of P4O10. The other polymorphs are polymeric, but in each case the phosphorus atoms are bound by a tetrahedron of oxygen atoms, one of which forms a terminal P=O bond. The O-form adopts a layered structure consisting of interconnected P6O6 rings, not unlike the structure adopted by certain polysilicates.
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1999, 96, 79.Mitcheltree, M. J.; Konst, Z. A.; Herzon, S. B. Tetrahedron 2013, 69, 5634. These reactions employ electron-deficient sulfonyl azides, which undergo chemoselective, uncatalyzed [3+2] cycloaddition to the silyl enol ether, followed by loss of dinitrogen, and alkyl migration to give ring- contracted products in good yield. These reactions may be directed by substrate stereochemistry, giving rise to stereoselective ring-contracted product formation.
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In 1960 Lawrence drew up a new tubular steel chassis based on a tetrahedron design that he had used for an earlier project. The front suspension was from a VW beetle, as was the transaxle. The engine was a 1000 cc 105E Ford Kent engine. Lawrence took two DS101s to the first race at Goodwood in 1960 but found that his new cars were already outmatched.
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Bangladeshi samosas, snack food Both flat-shaped (triangular) and full-shaped (tetrahedron/triangular pyramid) samosas are popular snacks in Bangladesh. A Bengali version of the full-shaped samosa is called a সিঙাড়া (shingara) and is normally smaller than the standard variety. The shingara is usually filled with pieced potatoes, vegetables, nuts etc. However, shingaras filled with beef liver are very popular in some parts of the country.
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For example, the tetrahedron is an alternated cube, h{4,3}. Diminishment is a more general term used in reference to Johnson solids for the removal of one or more vertices, edges, or faces of a polytope, without disturbing the other vertices. For example, the tridiminished icosahedron starts with a regular icosahedron with 3 vertices removed. Other partial truncations are symmetry-based; for example, the tetrahedrally diminished dodecahedron.
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In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non- compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by . The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3.
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In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by . The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3.
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From sulfonic acids, the corresponding sulfochlorides are formed analogously.O. Chantarasriwong et al., A practical and efficient method for the preparation of sulfonamides utilizing Cl3CCN/PPh3, Tetrahedron Lett., 47, (42), 7489–7492 (2006). In an analogous manner, the activation of diphenylphosphoric acid with Cl3CCN/PPh3 and reaction with alcohols or amines proceeds to the corresponding phosphoric acid esters or amides in a gentle and efficient one-pot reaction.
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For Euclidean 3-space, there are 3 simple and related Goursat tetrahedra, represented by [4,3,4], [4,31,1], and [3[4 . They can be seen inside as points on and within a cube, {4,3}. In geometry, a Goursat tetrahedron is a tetrahedral fundamental domain of a Wythoff construction. Each tetrahedral face represents a reflection hyperplane on 3-dimensional surfaces: the 3-sphere, Euclidean 3-space, and hyperbolic 3-space.
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The following sections show all of the whole number Goursat tetrahedral solutions on the 3-sphere, Euclidean 3-space, and Hyperbolic 3-space. The extended symmetry of each tetrahedron is also given. The colored tetrahedal diagrams below are vertex figures for omnitruncated polytopes and honeycombs from each symmetry family. The edge labels represent polygonal face orders, which is double the Coxeter graph branch order.
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Addition of tetrabutylammonium bromide as a phase transfer catalyst dramatically increases the rate of reaction. The co-catalyst is presumed to increase the concentration of the peroxide oxidant in the organic phase enabling more efficient access to the reactive ternary complex.Geller, T.; Gerlach, A.; Kruger, C. M.; Militzer, H. C., "Novel conditions for the Julia–Colonna epoxidation reaction providing efficient access to chiral, nonracemic epoxides". Tetrahedron Lett.
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Four points in any dimension have only finitely many central configurations. The number of configurations in this case is at least 32 and at most 8472, depending on the masses of the points. The only convex central configuration of four equal masses is a square. The only central configuration of four masses that spans three dimensions is the configuration formed by the vertices of a regular tetrahedron.
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The central angle between any two vertices of a perfect tetrahedron is arccos(−), or approximately 109.47°. Water, , also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel more than the single O–H bonds. Quaternary phase diagrams in chemistry are represented graphically as tetrahedra.
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The A-Format is the signals from four microphones on the faces of a tetrahedron. The first set, the A-Format, is produced by the Soundfield microphone itself and consists of the four signals from the microphone capsules. These four signals are not intended to be used without further processing. The A-Format is normally transformed into the second set of audio signals, the B-Format.
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Solvation numbers of 3 - 6 have been found for lithium aqua ions. Solvation numbers less than 4 may be the result of the formation of contact ion pairs. The higher solvation numbers may be interpreted in terms of water molecules that approach [Li(H2O)4]+ through a face of the tetrahedron, though molecular dynamic simulations may indicate the existence of an octahedral aqua ion.Richens, p. 126.
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The student body is small and combines with nearby towns to make up teams for some sports. Although tiny, somewhat rustic, and usually quiet, Lyle has attracted some attention because it lies very near several popular windsurfing beaches, including Doug's Beach. The area is also home to six boutique wineries: Cor Cellars, Domaine Pouillon, Jacob Williams Winery, Klickitat Canyon Winery, Tetrahedron Wines and Syncline.
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Marine nematocides: tetrahydrofurans from a southern Australian brown alga, Notheia anomala. Tetrahedron, 54(10), 2227-2242. highlighted for the first time that tetrahydrofurans from Notheia act as potent and selective inhibitors of the larval development of parasitic nematodes, which may be a positive factor that Hormosira receives from this symbiotic relationship. A small, branching Notheia epiphyte living on its host, Hormosira banksii, in a Kaikoura tide pool.
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Up to some extent amorphous metals and quasicrystals can also be modeled by close packing of spheres. The local atomic order is well modeled by a close packing of tetrahedra, leading to an imperfect icosahedral order. A regular tetrahedron is the densest configuration for the packing of four equal spheres. The dense random packing of hard spheres problem can thus be mapped on the tetrahedral packing problem.
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Other elements also substitute in the structure, most notably iron and zinc, along with less common silver, mercury and lead. Bismuth also substitutes for the antimony site and bismuthian tetrahedrite or annivite is a recognized variety. The related, silver dominant, mineral species freibergite, although rare, is notable in that it can contain up to 18% silver. Tetrahedrite gets its name from the distinctive tetrahedron shaped cubic crystals.
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The configuration is named after August Ferdinand Möbius, who in 1828 proved that, if two tetrahedra have the property that seven of their vertices lie on corresponding face planes of the other tetrahedron, then the eighth vertex also lies on the plane of its corresponding face, forming a configuration of this type. This incidence theorem is true more generally in a three-dimensional projective space if and only if Pappus's theorem holds for that space (Reidemeister, Schönhardt), and it is true for a three-dimensional space modeled on a division ring if and only if the ring satisfies the commutative law and is therefore a field (Al- Dhahir). By projective duality, Möbius' result is equivalent to the statement that, if seven of the eight face planes of two tetrahedra contain the corresponding vertices of the other tetrahedron, then the eighth face plane also contains the same vertex.
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The largest strictly-convex deltahedron is the regular icosahedron This is a truncated tetrahedron with hexagons subdivided into triangles. This figure is not a strictly-convex deltahedron since coplanar faces are not allowed within the definition. In geometry, a deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle.
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In 1979, Jeremy's brother Peter, met Bil Bilson and Richard Burgman in Sydney and formed a band, Shy Imposters, which broke up in early 1980. Jeremy joined and they formed a new band, Sunnyboys. The band's name came from a Sunnyboy, an orange-flavoured water ice in a tetrahedron shaped 'tetra-pack'. According to Richard Burgman, the band chose the name because it represented ‘bright, happy, young, fun’.
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With this construction it is given the Wythoff symbol: . 50px A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, and six rectangles. While its edges are unequal, this solid remains vertex-uniform: the solid has the full tetrahedral symmetry group and its vertices are equivalent under that group. The edges of a cuboctahedron form four regular hexagons.
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Examples of digital dark field bowtie/butterfly detection. Dark field analysis of dual-tetrahedron crystal pairs. Icosahedral twinning has been seen in face-centered-cubic metal nanoparticles that have nucleated: (i) by evaporation onto surfaces, (ii) out of solution, and (iii) by reduction in a polymer matrix. Quasicrystals are un-twinned structures with long range rotational but not translational periodicity, that some initially tried to explain away as icosahedral twinning.
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More significantly, the value of the coefficients of each term can be computed directly from the exponents. The formula is: , where x, y, z are the exponents of A, B, C, respectively, and "!" means factorial (e.g.: ). The exponent formulas for the 4th layer are: The exponents of each expansion term can be clearly seen and these formulae simplify to the expansion coefficients and the tetrahedron coefficients of layer 4.
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In most cases, an excess of the diene is employed to prevent isomerization of the oxyallyl cation intermediate. Increasing the covalent character of the metal-oxygen bond (by, for instance, employing iron carbonyl reducing agents instead of sodium) also stabilizes the oxyallyl cation, leading to cleaner reactions. Strongly electrophilic allyl cations tend to give products of electrophilic substitution rather than cycloaddition.Henning, R.; Hoffmann, H. M. R. Tetrahedron Lett.
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Oh is only insignificantly higher, indicating that the minimum on the energy surface is very shallow. 129Xe and 19F NMR spectroscopy indicates that in solution the compound assumes a tetrameric structure: four equivalent xenon atoms are arranged in a tetrahedron surrounded by a fluctuating array of 24 fluorine atoms that interchange positions in a "cogwheel mechanism". Six polymorphs of are known. including one that contains XeF ions with bridging F ions.
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The bacterium Pseudomonas diminuta will hydrolyze this compound but not its stereoisomer trans-2,3-Butylene carbonate, yielding cis-2,3-butanediol. This has been proposed as an efficient route to produce the latter from a racemic mixture of 2,3-butylene carbonates. Kazutsugu Matsumoto, Youichi Sato, Megumi Shimojo and Minoru Hatanaka (2000), Highly enantioselective preparation of C2-symmetrical diols: microbial hydrolysis of cyclic carbonates. Tetrahedron: Asymmetry, volume 11, issue 9, pages 1965-1973.
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Because the stereoselectivity of carbocupration is extremely high, the reaction has been applied to the synthesis of pheromones in which the geometric purity of double bonds is critical. One example is the insect pheromone of Cossus cossus, which is synthesized by syn-selective carbocupration of acetylene and alkylation of the resulting organocuprate in the presence of added phosphite.Cahiez, G.; Alexakis, A.; Normant, J. F. Tetrahedron Lett. 1978, 2027.
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Copper(II) acetate is not necessary in this case because of the high resonance stabilization of the intermediate diphenylmethyl radical.Yang, F. Z.; Trost, M. K.; Fristad, W. E. Tetrahedron Lett. 1987, 28, 1493. (7)File:MnCoupleScope4.png When alkenes or carbonyl compounds containing pendant unsaturated moiety are treated with manganese(III) acetate, tandem intramolecular cyclization reactions may occur. Generally, exo cyclization of terminal double bonds is favored, as shown in equation (10).
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If the faces of a disphenoid are equilateral triangles, it is a regular tetrahedron with Td tetrahedral symmetry, although this is not normally called a disphenoid. When the faces of a disphenoid are isosceles triangles, it is called a tetragonal disphenoid. In this case it has D2d dihedral symmetry. A sphenoid with scalene triangles as its faces is called a rhombic disphenoid and it has D2 dihedral symmetry.
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August Wilhelm von Hofmann is credited with the first physical molecular model around 1860 (Fig. 1). Note how the size of the carbon appears smaller than the hydrogen. The importance of stereochemistry was not then recognised and the model is essentially topological (it should be a 3-dimensional tetrahedron). Jacobus Henricus van 't Hoff and Joseph Le Bel introduced the concept of chemistry in space—stereochemistry in three dimensions.
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Each of the Platonic solids occurs naturally in one form or another. The tetrahedron, cube, and octahedron all occur as crystals. These by no means exhaust the numbers of possible forms of crystals (Smith, 1982, p212), of which there are 48. Neither the regular icosahedron nor the regular dodecahedron are amongst them, but crystals can have the shape of a pyritohedron, which is visually almost indistinguishable from a regular dodecahedron.
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Adult E. soleae lays tetrahedral eggs that have a tanned protein shell. A detachable operculum constitutes one of the tetrahedron corners of the egg. The cap-like operculum is bonded to the rest of the egg-shell by a thin cement layer. The opercular bond is strong, and operculum detachment only occurs through the actions of oncomiracidium and the secretion of hatching fluid by glands from the head region.
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Enantioselective benzylic functionalization methods use the complexed chromium tri(carbonyl) moiety essentially as a chiral auxiliary.Uemura, M.; Hayashi, Y.; Hayashi, Y. Tetrahedron: Asymmetry, 1993, 4, 2291. Approach of the functionalizing reagent anti to the ----chromium tri(carbonyl) fragment leads to a single diastereomer of the product complex. After removal of the chromium group with lightMishchenko, O. G.; Klementeva, S. V.; Maslennikov, S. V.; Artemov, A. N.; Spirina, I. V. Rus.
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However, low concentrations of a radical scavenger establish thermodynamic control and provide access to 6-endo products—not via 6-endo cyclization, but by 5-exo cyclization followed by 3-exo closure and subsequent fragmentation (Dowd-Beckwith rearrangement). Whereas at high concentrations of the exo product is rapidly trapped preventing subsequent rearrangement to the endo productBeckwith, A.; O'Shea, D. Tetrahedron Lett. 1986, 27, 4525. Aryl radicals exhibit similar reactivity.
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This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead).
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The four phosphorus atoms are at the corners of a tetrahedron surrounding the palladium(0) center. This structure is typical for four-coordinate 18 e− complexes. The corresponding complexes Ni(PPh3)4 and Pt(PPh3)4 are also well known. Such complexes reversibly dissociate PPh3 ligands in solution, so reactions attributed to Pd(PPh3)4 often in fact arise from Pd(PPh3)3 or even Pd(PPh3)2.
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A 2-dimensional pyramid is a triangle, formed by a base edge connected to a noncolinear point called an apex. A 4-dimensional pyramid is called a polyhedral pyramid, constructed by a polyhedron in a 3-space hyperplane of 4-space with another point off that hyperplane. Higher-dimensional pyramids are constructed similarly. The family of simplices represent pyramids in any dimension, increasing from triangle, tetrahedron, 5-cell, 5-simplex, etc.
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Dissection of a square and equilateral triangle into each other. No such dissection exists for the cube and regular tetrahedron. In two dimensions, the Wallace–Bolyai–Gerwien theorem states that any two polygons of equal area can be cut up into polygonal pieces and reassembled into each other. David Hilbert became interested in this result as a way to axiomatize area, in connection with Hilbert's axioms for Euclidean geometry.
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A recent example of the Fürst-Plattner rule can be seen from Chrisman et al. where limonene is epoxidized to give a 1:1 mixture of diastereomers. Exposure to a nitrogen nucleophile in water at reflux provides only one ring opened product in 75-85% ee.Chrisman, W.; Camara, J. N; Marcellini, K.; Singaram, B.; Goralski, C. T.; Hasha, D. L.; Rudolf, P. R.; Nicholson, L.W.; Borodychuck, K. K. Tetrahedron Lett.
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The initial surface area of the (iteration-0) tetrahedron of side-length L is L2. The next iteration consists of four copies with side length , so the total area is 4()2 = 4L2· = L2 again. Meanwhile the volume of the construction is halved at every step and therefore approaches zero. The limit of this process has neither volume nor surface but, like the Sierpinski gasket, is an intricately connected curve.
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Silica-grafted polyleucine has been shown to effectively catalyze epoxidation of α,β-unsaturated aromatic ketones. The silica graft allows for the catalyst to be easily recovered with only mild loss of activity and is particularly useful for scale-up reactions.Yi, H.; Zou, G.; Li, Q.; Chen, Q.; Tang, J.; He, M. Y., "Asymmetric epoxidation of alpha,beta-unsaturated ketones catalyzed by silica-grafted poly-(L)-leucine catalysts". Tetrahedron Lett.
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Charge equilibria and pK values of 5-carboxypyranomalvidin-3-glucoside (vitisin A) by electrophoresis and absorption spectroscopy. Robert E. Asenstorfer and Graham P. Jones, Tetrahedron, Volume 63, Issue 22, 28 May 2007, Pages 4788-4792, Effect of acetaldehyde and several acids on the formation of vitisin A in model wine anthocyanin and colour evolution. Romero C. and Bakker J., International journal of food science & technology, 2000, vol. 35, no.
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"Advancing Parallel Solution Phase Library Synthesis through Efficient Purification, Quantitation, and Characterization Techniques" Journal of Laboratory Automation, 48 (1992) 3789. In order to handle the vast number of structural possibilities, researchers often create a 'virtual library', a computational enumeration of all possible structures of a given pharmacophore with all available reactants.E. V.Gordeeva et al. "COMPASS program - an original semi-empirical approach to computer-assisted synthesis" Tetrahedron, 48 (1992) 3789.
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A system of synergetics coordinates uses only one type of simplex (triangle, tetrahedron, pentachoron, ..., n-simplex) as space units, and in fact uses a regular simplex, rather like Cartesian coordinates use hypercubes (square, cube, tesseract, ..., n-cube.) Synergetics coordinates in two dimensions The n Synergetics coordinates axes are perpendicular to the n defining geometric objects that define a regular simplex; 2 end points for line segments, 3 lines for triangles, 4 planes for tetrahedrons etc.. The angles between the directions of the coordinate axes are Arc Cosine (-1/(n-1)). The coordinates can be positive or negative or zero and so can their sum. The sum of the n coordinates is the edge length of the regular simplex defined by moving the n geometric objects in increments of the height of the n-1 dimensional regular simplex that has an edge length of one. If the sum of the n coordinates is negative the triangle (n = 3) or tetrahedron (n = 4) is upside down and inside out.
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In a tetrahedral molecular geometry, a central atom is located at the center with four substituents that are located at the corners of a tetrahedron. The bond angles are cos−1(−) = 109.4712206...° ≈ 109.5° when all four substituents are the same, as in methane () as well as its heavier analogues. Methane and other perfectly symmetrical tetrahedral molecules belong to point group Td, but most tetrahedral molecules have lower symmetry. Tetrahedral molecules can be chiral.
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Asbestiform tremolite, part of the amphibole group in the inosilicate subclass Inosilicates consist of tetrahedra repeatedly bonded in chains. These chains can be single, where a tetrahedron is bound to two others to form a continuous chain; alternatively, two chains can be merged to create double-chain silicates. Single-chain silicates have a silicon:oxygen ratio of 1:3 (e.g. [Si2O6]4−), whereas the double-chain variety has a ratio of 4:11, e.g. [Si8O22]12−.
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For example, methyllithium, ethyllithium and tert-butyllithium all exist in the tetramer [RLi]4. Methyllithium exists as tetramers in a cubane- type cluster in the solid state, with four lithium centers forming a tetrahedron. Each methanide in the tetramer in methyllithium can have agostic interaction with lithium cations in adjacent tetramers. Ethyllithium and tert- butyllithium, on the other hand, do not exhibit this interaction, and are thus soluble in non-polar hydrocarbon solvents.
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The high stability of Heck reaction catalysts permits the synthesis of highly strained compounds at elevated temperatures. In the example below, the arene and alkene must both be in energetically unfavorable axial positions in order to react.Wünsch, B.; Diekmann, H.; Höfner, G. Tetrahedron: Asymmetry 1995, 6, 1527. (7)File:HeckScope2.png Endo cyclization is observed most often when small or large rings are involved. For instance, 5-endo cyclization is generally preferred over 4-exo cyclization.
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MER lander petals opening (Courtesy NASA/JPL-Caltech) The spacecraft lander is a protective shell that houses the rover, and together with the airbags, protects it from the forces of impact. The lander is a tetrahedron shape, whose sides open like petals. It is strong and light, and made of beams and sheets. The beams consist of layers of graphite fiber woven into a fabric that is lighter than aluminium and more rigid than steel.
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After the lander stopped bouncing and rolling on the ground, it came to rest on the base of the tetrahedron or one of its sides. The sides then opened to make the base horizontal and the rover upright. The sides are connected to the base by hinges, each of which has a motor strong enough to lift the lander. The rover plus lander has a mass of about 533 kilograms (1,175 pounds).
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1985, 107, 1448-1449. doi 10.1021/ja00291a077Curran, D. P.; Rakiewicz, D. M., “Radical-initiated polyolefinic cyclizations in linear triquinane synthesis. Model studies and total synthesis of hirsutene” Tetrahedron 1985, 41, 3943-3958. doi 10.1016/S0040-4020(01)97175-3 the first of many such syntheses with cascade radical reactions, is today regarded as a classic.Nicolaou, K. C.; Sorenson, E. J., Classics in Total Synthesis, VCH: Weinheim, 1995; Ch 23, pp 381-420, “Hirstutene and Capnellene”.
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The molecular structure of ZSM-5 zeolite, showing well defined pores and channels in the zeolite. Yellow balls represent Si and red balls represent O. The structure of ZSM-5 zeolite by showing the coordination tetrahedron. Electron micrograph of MFI zeolite with the structural model overlaid in the right-bottom corner. ZSM-5, Zeolite Socony Mobil–5 (framework type MFI from ZSM-5 (five)), is an aluminosilicate zeolite belonging to the pentasil family of zeolites.
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The primary radars of RAF Fylingdales are active electronically scanned array (AESA) phased array radars, mounted on each face of a truncated tetrahedron, typically referred to as the "pyramid" or the SSPAR (Solid State Phased Array). This makes Fylingdales unique amongst its peers in that it covers a full 360 degrees. Each of the three arrays is across and contains around 2560 transmit/receive modules; mean power output is about , with a tracking range of .
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They are generally more methods in making β-vinyl iodides versus α-vinyl iodides using hydrometalation (with aluminum with DIBAL-H (hydroalumination), with boron (hydroboration), with HZrCp2Cl (hydrozirconation)).Chong, J.; Darwish, Alla. Tetrahedron, Volume 68, Issue 2, 14 Jan 2012, pages 654-658 However, hydrometalation with alkyne with various functional groups often react poorly with side products. The Chong groups have demonstrated using hydrostannation, using Bu3SnH with palladium catalyst with high E stereoselectivity.
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The key step in one synthesis of barbaralone is the selective intramolecular cyclopropanation of a cycloheptatriene.Doering, W.; Ferrier, B.; Fossel, E.; Hartenstein, J.; Jones, Jr., M.; Klumpp, G.; Rubin, R.; Saunders, M. Tetrahedron 1967, 23, 3943. (3)File:MCPScope1.png α,β-Cyclopropyl ketones may act as masked α,β-unsaturated ketones. In one example, intramolecular participation of an aryl group leads to the formation of a polycyclic ring system with complete diastereoselectivity.
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The previous theorem has further interesting consequences other than the aforementioned generalization of Commandino's theorem. It can be used to prove the following theorem about the centroid of a tetrahedron, first described in the Mathematische Unterhaltungen by the German physicist Friedrich Eduard Reusch:Friedrich Joseph Pythagoras Riecke (Hrsg.): Mathematische Unterhaltungen. Zweites Heft. 1973, S. 100, 128In den Mathematische Unterhaltungen (Zweites Heft, S. 128) wird auf die S. 36 von Reuschs Abhandlung Der Spitzbogen verwiesen.
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A specific case of Reusch's theorem where all four vertices of a tetrahedron are coplanar and lie on a single plane, thereby degenerating into a quadrilateral, Varignon's theorem, named after Pierre Varignon, states the following:Coxeter, op. cit., S. 242DUDEN: Rechnen und Mathematik. 1985, S. 652 :Let a quadrilateral in \R^2 be given. Then the two midlines connecting opposite edge midpoints intersect in the centroid of the quadrilateral and are divided in half by it.
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The structure of the fluorophosphate anion consists of phosphorus at the center of a tetrahedron defined by three oxygen atoms and one fluorine. Formal representations depict a double bond between one oxygen atom and phosphorus, with single bonds for the other two oxygen atoms and the fluorine. In this very formal depiction, negative charge is localized on the O atoms of the single P-O bonds. MFP is similar to and isoelectronic with Na2SO4.
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Propargyl methanesulfinates are useful substrates for the synthesis of allenes from stoichiometric organocopper complexes. In this case, the complexes were generated in situ through the combination of a Grignard reagent, copper(I) bromide, and lithium bromide. Organocopper complexes very often need Lewis acid activation in order to react efficiently; magnesium bromide generated in situ serves as an activating Lewis acid in this case.Kleijn, H.; Elsevier, C. J.; Westmijze, H.; Meijer, J.; Vermeer, P. Tetrahedron Lett.
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OSCAR IV ( OSCAR 4) is the fourth amateur radio satellite launched by Project OSCAR and the first targeted for Geostationary orbit on 12 December 1965. The satellite was launched piggyback with three United States Air Force satellites on a Titan IIIC launch vehicle. The satellite was regular tetrahedron, 48 cm on each side. It had four independent monopole antennas and like OSCAR 3 contained a tracking beacon transmitter and a communications repeater.
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Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls.
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It is impossible to partition the Schönhardt polyhedron into tetrahedra whose vertices are vertices of the polyhedron. More strongly, there is no tetrahedron that lies entirely inside the Schönhardt polyhedron and has vertices of the polyhedron as its four vertices. For, among any four vertices of the Schönhardt polyhedron, at least one pair of vertices from these four vertices must be a diagonal of the polyhedron, which lies entirely outside the polyhedron.
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Ruppert's algorithm can be naturally extended to three dimensions, however its output guarantees are somewhat weaker due to the sliver type tetrahedron. An extension of Ruppert's algorithm in two dimensions is implemented in the freely available Triangle package. Two variants of Ruppert's algorithm in this package are guaranteed to terminate for a poor- quality threshold of about 26.5 degrees. In practice these algorithms are successful for poor-quality thresholds over 30 degrees.
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Parts IV and V contain two short manuals of Pāśaka kevalī, or cubomancy, i.e., the art of foretelling a person's future by means of the cast of dice, a ritualistic practice found in Tibetan manuscripts. Part IV is almost complete, while the manual constituting Part V is markedly more fragmentary and defective. The dice is stated to be a group of three die, each with four faces (tetrahedron) numbered 1, 2, 3 and 4.
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He was the recipient of the Arthur C. Cope Award in 2003, and he was awarded the Tetrahedron Prize for Creativity in Organic Chemistry for 2008. Overman's research is focused on the development of new chemical reactions, particularly transition metal catalyzed reactions, and the application of those reactions toward the synthesis of natural products. Overman is most known for the Overman rearrangement, a Claisen rearrangement of allylic alcohols to give allylic trichloroacetamides.
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The 3-D structures of a number of rubredoxins have been solved. The fold belongs to the α+β class, with 2 α-helices and 2-3 β-strands. Rubredoxin active site contains an iron ion which is coordinated by the sulfurs of four conserved cysteine residues forming an almost regular tetrahedron. This is sometimes denoted as a [1Fe-0S] or an Fe1S0 system, in analogy to the nomenclature for iron-sulfur proteins.
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Enantioselectivity is maximized by poly-amino acid sequences containing the greatest α-helical content; these include poly-leucine and poly-alanine. Both poly-L- and poly-D-amino acids are available and cause the opposite stereoinduction.Bentley, P. A.; Cappi, M. W.; Flood, R. W.; Roberts, S. M.; Smith, J. A., Towards a mechanistic insight into the Julia-Colonna asymmetric epoxidation of α,β-unsaturated ketones using discrete lengths of poly-leucine. Tetrahedron Lett.
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The coordinates of all the atoms. The distribution of silicon and aluminium along the tetrahedral sites mixed with the nature of the barium atom makes an impact on the surrounding silicate framework (Newham and Megaw, 1960). The Si-Al bonds are partially ordered, and in some cases the aluminium substitute's silicon. The order in celsian is very simple, each aluminium tetrahedron is surrounded by four silicon tetrahedra, and vice versa (Newham and Megaw, 1960).
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The Logos Foundation is a professional artistic organisation founded in 1968. It focuses on the promotion of new musics and audio related arts by means of new music production, concerts, performances, composition, technological research projects and other contemporary music related activities. The Logos Foundation (and its concert hall, the Logos Tetrahedron) is based in Ghent, Belgium (Flanders region). Godfried-Willem Raes is the founder and major driving force behind the Logos Foundation.
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List of Fellows of the American Mathematical Society, retrieved 2013-07-07. Rudin is best known in topology for her constructions of counterexamples to well-known conjectures. In 1958, she found an unshellable triangulation of the tetrahedron. Most famously, Rudin was the first to construct a Dowker space, which she did in 1971, thus disproving a conjecture of Clifford Hugh Dowker that had stood, and helped drive topological research, for more than twenty years.
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As the Stetter reaction is an example of umpolung chemistry, the aldehyde is converted from an electrophile to a nucleophile under the reaction conditions.Albright, J. D. Tetrahedron 1983, 39, 3207. This is accomplished by activation from some catalyst - either cyanide (CN−) or thiazolium salt. For the use of either catalyst, the mechanism is very similar; the only difference is that with thiazolium salts, the catalyst must be deprotonated first to form the active catalytic species.
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The numbers of the tetrahedron are derived from trinomial expansion. The nth layer is the detached coefficient matrix (no variables or exponents) of a trinomial expression (e.g.: A + B + C) raised to the nth power. The nth power of the trinomial is expanded by repeatedly multiplying the trinomial by itself: Each term in the first expression is multiplied by each term in the second expression; and then the coefficients of like terms (same variables and exponents) are added together.
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Gaseous clean agents are installed in places where water or dry chemical extinguishing agents will do more harm than good. They are mostly installed in server or computer rooms where water or fine dust particles will cause vast amounts of damage. The system works by flooding an area with an inert or a mix of inert gasses to break the fire tetrahedron. Due to the inert gases displacing oxygen, there is a suffocation risk when it is activated.
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The Wichita VOR is a few miles northwest of the airport, and a non-directional beacon (NDB) a few miles southwest of the field. The field has no instrument landing system. A wind sock is in the center of the field, west of the runways, and a wind tetrahedron has been seen near the north end of the runway, though its status is unknown. Prevaililng winds are from the south, and Runway 35 is the most commonly used approach.
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Hypothetical zeolites are combinatorial models of potential structures of the minerals known as zeolites. They are four-regular periodic graphs, with vertices representing the tetrahedral atom (usually Si or Al) and the edges representing the bridging oxygen atoms. Alternatively, by ignoring the tetrahedral atoms, zeolites can be represented by a six-valent periodic graph of oxygen atoms, with each O-O edge defining an edge of a tetrahedron. At present there are millions of hypothetical zeolite frameworks known.
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One of the more classic applications of DKR is Noyori's asymmetric hydrogenation.Noyori, R.; Tokunaga, M.; Kitamura, M.; Ohkuma, T. Tetrahedron: Asymmetry, 1990, 1, 1–4. The presence of an acidic center between two carbonyl groups allows for easy epimerization at the chiral center under basic conditions. To select for one of the four possible stereoisomers, a BINAP-Ru catalyst is used to control the outcome of the reaction through the steric bulk of the phosphorus ligand.
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Atomic structure of coesite Coesite is a tectosilicate with each silicon atom surrounded by four oxygen atoms in a tetrahedron. Each oxygen atom is then bonded to two Si atoms to form a framework. There are two crystallographically distinct Si atoms and five different oxygen positions in the unit cell. Although the unit cell is close to being hexagonal in shape ("a" and "c" are nearly equal and β nearly 120°), it is inherently monoclinic and cannot be hexagonal.
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Raymond and coworkers developed a supramolecular host M4L6 (4 gallium ions and 6 ligands for each complex) that self-assembles via metal-ligand interaction in aqueous solution. This container molecule is polyanionic and thus its tetrahedron- shaped cavity is capable of encapsulating and stabilizing a cationic molecule. Consequently, encapsulated molecule can be easily protonated as a resulting carbocation from protonation is stabilized by the surrounding anions. Raymond utilized this property to perform acid-catalyzed Nazarov cyclization.
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Saraceno has developed a Solar Bell flying sculpture made of lightweight and sustainable materials. Its design is based on the modular tetrahedron, or four-sided pyramid, invented by Alexander Graham Bell during his early investigations into manned flight. Bell made important discoveries in the field of aviation and frame construction, and happened upon the strongest geometrical structure known in the cosmos, the octet truss. This was the same spaceframe that Buckminster Fuller later followed for his Geodesic dome.
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Thus, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of an isotoxal polyhedron (having equivalent edges) is also isotoxal. Duality is closely related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
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Gund, T. M.; Osawa, E.; Van Zandt, W. Jr.; Schleyer, P. v. R.: Diamantane. 1. Preparation of Diamantane. Physical and Spectral Properties. In: Journal of Organic Chemistry, Band 39, Nr. 20, 1974, S. 2979–2987, doi:10.1021/jo00934a009.Gund, T. M.; Osawa, E.; Van Zandt, W. Jr.; Schleyer, P. v. R.: A Convenient, High-Yield Preparation of Diamantane (Congressane). In: Tetrahedron Letters, 1970, Band 11 , Nr. 44, S. 3877–3880, doi:/10.1016/S0040-4039(01)98613-7.
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The red represents partially negatively charged regions. Ammonia, NH3, is a molecule whose three N−H bonds have only a slight polarity (toward the more electronegative nitrogen atom). The molecule has two lone electrons in an orbital that points towards the fourth apex of an approximately regular tetrahedron, as predicted by the (VSEPR theory). This orbital is not participating in covalent bonding; it is electron-rich, which results in a powerful dipole across the whole ammonia molecule.
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In his Princeton lecture notes, Thurston noted a method for describing the geometric shape of each hyperbolic tetrahedron by a complex number and a set of nonlinear equations of complex variables whose solution would give a complete hyperbolic metric on the 3-manifold. These equations consist of edge equations and cusp (completeness) equations. SnapPea uses an iterative method utilizing Newton's method to search for solutions. If no solution exists, then this is reported to the user.
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Identification of a new lepidopteran sex pheromone in picogram quantities using an antennal biodetector: (8E,10Z)-tetradeca-8,10-dienal from Cameraria ohridella. Tetrahedron Letters, 40: 7011-7014 In any case, infestation levels could diminish over time as Cameraria ohridella starts to recruit generalist members of the local parasitoid wasp community.Girardoz, S., Kenis M., & Quicke D. L. J. 2006. Recruitment of native parasitoids by an exotic leaf miner, Cameraria ohridella: host - parasitoid synchronization and influence of the environment.
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This would greatly increase the sensitivity of the technique, and open the door for magnetic resonance imaging or MRI. ;1965:Robert Burns Woodward and Roald Hoffmann propose the Woodward–Hoffmann rules, which use the symmetry of molecular orbitals to explain the stereochemistry of chemical reactions. ;1966:Hitoshi Nozaki and Ryōji Noyori discovered the first example of asymmetric catalysis (hydrogenation) using a structurally well-defined chiral transition metal complex.H. Nozaki, S. Moriuti, H. Takaya, R. Noyori, Tetrahedron Lett.
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It follows from Euler's formula that any simplicial 2-sphere with n vertices has 3n − 6 edges and 2n − 4 faces. The case of n = 4 is realized by the tetrahedron. By repeatedly performing the barycentric subdivision, it is easy to construct a simplicial sphere for any n ≥ 4. Moreover, Ernst Steinitz gave a characterization of 1-skeleta (or edge graphs) of convex polytopes in R3 implying that any simplicial 2-sphere is a boundary of a convex polytope.
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In the total synthesis of spiroketal ionophore antibiotic routiennocin 1 (Fig. 1), the central spiroketal skeleton was constructed by a multistep tandem reaction (Fig. 2).Diez-Martin, D. Kotecha, N. R.; Ley, S. V.; Mantegani, S.; Menendez, J. C.; Organ, H. M.; White, A. D., Tetrahedron, 1992, 48, 1899-7938. Fragment A and fragment B were coupled in a single step to form the key intermediate G that could be further elaborated to afford the final product routiennocin. Fig.
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Further work extended the scope of this reaction to include aliphatic aldehydes as well.Moore, J. L.; Kerr, M. S.; Rovis, T. Tetrahedron 2006, 62, 11477. Subsequently, it was shown that the olefin geometry of the Michael acceptor dictates diastereoselectivity in these reactions, whereby the catalyst dictates the enantioselectivity of the initial carbon bond formation and allylic strain minimization dictates the diastereoselective intramolecular protonation.de Alaniz, J. R.; Rovis, T. J. Am. Chem. Soc. 2005, 127, 6284. Scheme 10.
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The gameplay involves elements of both luck and strategy. Movements are determined by rolling a set of four-sided, tetrahedron-shaped dice. Two of the four corners of each die are marked and the other two are not, giving each die an equal chance of landing with a marked or unmarked corner facing up. The number of marked ends facing upwards after a roll of the dice indicates how many spaces a player may move during that turn.
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New synthetic methods were developed, involving the reaction of ambident nucleophilic reagents (prepared by metalation of esters, nitriles and amides of alkanoic and arylacetic acids) with N-acylated aldimines and ketimines. The resulting new derivatives of 3-acylaminopropanoic acids were transformed into various heterocyclic compounds: beta-lactams, 1,3-oxazines, 2-imidazolidinones, etc. Bojilova, Anka; Nikolova, R.; Ivanov, Christo; Rodios, Nestor A.; Terzis, A.; Raptopoulou, C. P.; A Comparative Study of the Interaction of Salicylaldehydes with Phosphonoacetates under Knoevenagel Reaction Conditions. Synthesis of 1,2-Benzoxaphosphorines and their Dimers, 1996, Tetrahedron, 52 (38), pp. 12597–12612. Ivanov and his lab studied in detail the reaction of 2-benzylbenzazoles with aromatic aldehydes under phase-transfer catalysis or in aprotic solvents in the presence of a water solution of sodium hydroxide leading to 2-styrylbenzazols or alcohols having a benzazole ring.Bojilova, Anka; Trendafilova, Antoaneta; Ivanov, Christo; Rodios, Nestor A.; Cyclopropanation Reaction of 3-Acyl-2H-1-Benzopyran-2-ones with Phenacylbromide in Phase Transfer Systems, 1993, Tetrahedron, 49 (11), pp. 2275–2286.
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The materials paradigm represented in the form of a tetrahedron A material is defined as a substance (most often a solid, but other condensed phases can be included) that is intended to be used for certain applications."For Authors: Nature Materials" There are a myriad of materials around us—they can be found in anything from buildings to spacecraft. Materials can generally be further divided into two classes: crystalline and non-crystalline. The traditional examples of materials are metals, semiconductors, ceramics and polymers.
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Two desmic tetrahedra. The third tetrahedron of this system is not shown, but has one vertex at the center and the other three on the plane at infinity. The Reye configuration with the same 12 vertices as a desmic system In projective geometry, a desmic system is a set of three tetrahedra in 3-dimensional projective space, such that any two are desmic, (i.e. related such that each edge of one cuts a pair of opposite edges of the other).
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The Tetra Brik package was introduced in 1963, after a long process of development and built on the previous Tetra Classic tetrahedron package that laid the foundation for Tetra Pak. Despite the revolutionary character of the new retail system, by the end of the 1950s Ruben Rausing and the Tetra Pak management team realised that the Tetra Pak package portfolio needed to be supplemented by an additional rectangular model to continue to be competitive.Leander, Lars, Tetra Pak. A Vision Becomes Reality.
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A few molecules have a tetrahedral geometry with no central atom. An inorganic example is tetraphosphorus () which has four phosphorus atoms at the vertices of a tetrahedron and each bonded to the other three. An organic example is tetrahedrane () with four carbon atoms each bonded to one hydrogen and the other three carbons. In this case the theoretical C−C−C bond angle is just 60° (in practice the angle will be larger due to bent bonds), representing a large degree of strain.
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The shortest distance between each lead ion is 4.48 Å. The octahedron shares two of its opposite faces with that of neighbouring vanadinite units, forming a continuous chain of octahedrons. Each vanadium atom is surrounded by four oxygen atoms at the corners of an irregular tetrahedron. The distance between each oxygen and vanadium atom is either 1.72 or 1.76 Å. Three oxygen tetrahedrons adjoin each of the lead octahedrons along the chain. Crystals of vanadinite conform to a hexagonal system of symmetry.
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Tetrahedron Letters, volume 30, issue 50, pages 6985-6988. A. J. Bloodworth and Neil A. Tallant (1992): "1,2,4-Trioxane versus 1,2-dioxolane formation in the mercury(II) acetate- mediated cyclisation of hemiperoxyacetals derived from allylic hydroperoxides". Journal of the Chemical Society, Chemical Communications,volume 1992, issue 5, Tsan H. Lay and Joseph W. Bozzelli (1997): "Enthalpies of formation of cyclic alkyl peroxides: dioxirane, 1,2-dioxetane, 1,2-dioxolane, and 1,2-dioxane". Chemical Physics Letters, volume 268, issues 1–2, pages 175-179.
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In geometry, tetrahedron packing is the problem of arranging identical regular tetrahedra throughout three-dimensional space so as to fill the maximum possible fraction of space. The currently densest known packing structure for regular tetrahedra is a double lattice of triangular bipyramids and fills 85.63% of space Currently, the best lower bound achieved on the optimal packing fraction of regular tetrahedra is 85.63%. Tetrahedra do not tile space, and an upper bound below 100% (namely, 1 − (2.6...)·10−25) has been reported.
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The two-dimensional surface of a sphere S^2 has an open cover by two contractible sets, open neighborhoods of opposite hemispheres. However these two sets have an intersection that forms a non-contractible equatorial band. To form a good cover for this surface, one needs at least four open sets. A good cover can be formed by projecting the faces of a tetrahedron onto a sphere in which it is inscribed, and taking an open neighborhood of each face.
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It is a lithium ion conductor. Na2B6O9F2 contains a unit with two boroxine rings linked with a shared oxygen. Each ring has an extra fluorine, and each connects to two other units through shared oxygen. Formula weight 292.84; monoclinic P21/c; a=8.196 b=13.001 c=7.896 β=90.750° V =841.3, Z=4. K3B6O9F3(KBF), two dimensional sheets composed of units that have a pair of boroxine rings fused at one boron, one links to a tetrahedron BF2O2 bridge unit.
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Research into alternate cathode material for lithium batteries often examines the complex variations in stoichiometry and tetrahedron layered structure of copper sulfides. Advantages include limited toxicity and low costs. The high electrical conductivity of covellite (10−3 S cm−1) and a high theoretical capacity (560 mAh g−1) with flat discharge curves when cycled versus Li+/Li has been determined to play critical roles for capacity. The variety of methods of formations is also a factor of the low costs.
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Construction from the vertices of a truncated octahedron, showing internal rectangles. The coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted. This construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector (ϕ, 1, 0), where ϕ is the golden ratio.
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James Byron Friauf (1896 1972) was an American electrical engineer who first determined the crystal structure of MgZn2 in 1927, while he was a professor of physics at the Carnegie Institute of Technology, now Carnegie Mellon University. The phase consists of intra-penetrating icosahedra, which coordinate the Zn atoms, and 16-vertex polyhedra that coordinate the Mg atoms. The latter type of polyhedron is called a Friauf polyhedron and is, actually, an inter-penetrating tetrahedron and a 12-vertex truncated polyhedron.
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An early idea of abstract polyhedra was developed in Branko Grünbaum's study of "hollow-faced polyhedra." Grünbaum defined faces to be cyclically ordered sets of vertices, and allowed them to be skew as well as planar.. The graph perspective allows one to apply graph terminology and properties to polyhedra. For example, the tetrahedron and Császár polyhedron are the only known polyhedra whose skeletons are complete graphs (K4), and various symmetry restrictions on polyhedra give rise to skeletons that are symmetric graphs.
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This rearrangement reaction is named after its discoverer, Lars Skattebøl, Professor emeritus at the University of Oslo. It proceeds through a carbene reaction intermediate: Skattebøl rearrangement When the cyclopropane ring is fitted with a 2-vinyl group, a cyclopentadiene is formed through a so-called foiled carbene intermediate.Chemistry of gem- dihalocyclopropanes—VI A novel synthesis of cyclopentadienes and fulvenes L. Skattebøl Tetrahedron Volume 23, Issue 3 , 1967, Pages 1107-1117 .Leo A. Paquette and Mark L. McLaughlin Organic Syntheses, CV 8, 223 Link.
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The four basis vectors stem from the center of a regular tetrahedron and go to its four corners. Their coordinate addresses are (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) respectively. These may be scaled and linearly combined to span conventional XYZ space, with at least one of the four coordinates unneeded (set to zero) in any given quadrant. The normalization scheme is somewhat unusual in keeping all coordinates non- negative.
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Graph theory has close links to group theory. This truncated tetrahedron graph is related to the alternating group A4. Graph theory, the study of graphs and networks, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right.Graphs on Surfaces, Bojan Mohar and Carsten Thomassen, Johns Hopkins University press, 2001 Graphs are one of the prime objects of study in discrete mathematics.
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Structure of [14.22]adz Structure of hexamethylenetetramine Adamanzanes (abbreviated Adz) are compounds containing four nitrogen atoms linked by carbons (analogous to adamantane with nitrogen at the branched position). Often coordinated to a central ligand, the nitrogens occupy the vertices of a tetrahedron, with potentially four faces and six edges, with the carbon chains running approximately along the edges. They can have a "bowl" or "cage" structure, with varying lengths or omission of the carbon chains. In the nomenclature of Springborg et al.
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A: Chemical, 2003, 191, 1-8 2-thiophenylacetophenone (36) was reacted with iodine gas and pyridine in quantitative yield to generate acylmethylpyridinium iodide 37. Reaction with a chiral cyclic α, β-unsaturated ketone derived from 2-(+)-carene yielded the desired N, S-type ligand 38. center Novel, chiral P, N-ligands have been prepared using the Kröhnke method.Andrei V. Malkov, Marco Bella, Irena G. Stara, P. Kocovsky "Modular pyridine-type P,N-ligands derived from monoterpenes: application in asymmetric Heck addition" Tetrahedron Lett.
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A single 30-tetrahedron ring Boerdijk–Coxeter helix within the 600-cell, seen in stereographic projection Regular tetrahedra can be stacked face-to-face in a chiral aperiodic chain called the Boerdijk–Coxeter helix. In four dimensions, all the convex regular 4-polytopes with tetrahedral cells (the 5-cell, 16-cell and 600-cell) can be constructed as tilings of the 3-sphere by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface.
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Pyraminx in the middle of a twist The Pyraminx was first conceived by Mèffert in 1970. He did nothing with his design until 1981 when he first brought it to Hong Kong for production. Uwe is fond of saying had it not been for Erno Rubik's invention of the cube, his Pyraminx would have never been produced. The Pyraminx is a puzzle in the shape of a regular tetrahedron, divided into 4 axial pieces, 6 edge pieces, and 4 trivial tips.
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Although not strictly speaking switchable the compound is able to take up cations after a photochemical trigger and exposure to acetonitrile gives back the open form. :Anthracene Crown Desvergne 1978 In 1980 Yamashita et al.Synthetic macrocyclic ligands. II. Synthesis of a photochromic crown ether Tetrahedron Letters, Volume 21, Issue 6, 1980, Pages 541-544 Isamu Yamashita, Mieko Fujii, Takahiro Kaneda, Soichi Misumi and Tetsuo Otsubo construct a crown ether already incorporating the anthracene units (an anthracenophane) and also study ion uptake vs photochemistry.
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Thus, any cascade reaction is also a one-pot procedure, while the reverse does not hold true. Although often composed solely of intramolecular transformations, cascade reactions can also occur intermolecularly, in which case they also fall under the category of multicomponent reactions.Pellissier, H. Tetrahedron 2006, 62, 1619–1665. The main benefits of cascade sequences include high atom economy and reduction of waste generated by the several chemical processes, as well as of the time and work required to carry them out.
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In iron tetracarbonyl hydride the Fe(CO)4 group has C2v molecular symmetry with a geometry intermediate between octahedral and tetrahedral. Viewed as an octahedral complex, the hydride ligands are cis. Viewed as a tetrahedral Fe(CO)4 complex, the hydrides occupy adjacent faces of the tetrahedron. Although the structure of tetracarbonyliron with the hydrogen atoms bound as a single H2 ligand has been proposed as an intermediate in some rearrangement reactions, the stable state for the compound has the two atoms as independent ligands.
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The spacecraft descent was assisted by rockets that brought the spacecraft to a dead stop above the Martian surface. Radar altimeter unit: A radar altimeter unit was used to determine the distance to the Martian surface. The radar's antenna is mounted at one of the lower corners of the lander tetrahedron. When the radar measurement showed the lander was the correct distance above the surface, the Zylon bridle was cut, releasing the lander from the parachute and backshell so that it was free and clear for landing.
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The Bohr model of the chemical bond took into account the Coulomb repulsion - the electrons in the ring are at the maximum distance from each other. Thus, according to this model, the methane molecule is a regular tetrahedron, in which center the carbon nucleus locates, and in the corners - the nucleus of hydrogen. Chemical bond between them form four two-electron rings, rotating around the lines connecting the center with the corners. The Bohr model of the chemical bond could not explain the properties of the molecules.
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Geek Pop '09 was launched on the 6 March 2009. Alongside the two virtual stages imagined in the podcast from the previous year, the reproductive stage and the Tesla Tent, the festival introduced a new main stage which will be called the Tetrahedron stage (in homage to Glastonbury's Pyramid stage). The major difference for the 2009 festival from the previous year was its relocation to a standalone website and the introduction of an interactive festival map that allowed listeners to choose songs to listen to.
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Polymer derived from BINOL Yoshida and coworkers developed a synthesis of dendritic binaphthols that serve as homogenous, easily recoverable catalyst systems, and showed that they were amenable to forming homoallylic alcohols using Keck's allylation conditions.Yamago, S., Furukawa, M., Azuma, A., Yoshida, J., Tetrahedron Lett., 1998, 39, 3783-3786. Maruoka and Kii developed a bidentate Ti(IV) binol ligand for the allylation of aldehydes with the aim of restricting M-O bond rotation between the lewis acid and the aldehyde in order to improve enantiomeric excesses.
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The bidentate ligand contains two titaniums, binols, and an aromatic diamine connecting moiety, gave enantiomeric excesses of up to 99%.Kii, S., Maruoka, K., Tetrahedron Lett., 2001, 42, 1935-1939. Improved stereoselectivity is proposed to come from double activation of the carbonyl from the titaniums, a hypothesis supported by C13 NMR and IR spectroscopy studies on 2,6-γ-pyrone substrate. The most convincing evidence that the M-O rotation is restricted comes from NOE NMR studies on trans-4-methoxy-3-buten-2-one.
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Derek H.R. Barton, Neerja Yadav Bhatnagar, Jean-Pierre Finet, William B. Motherwell "Pentavalent organobismuth reagents. Part vi. Comparative migratory aptitudes of aryl groups in the arylation of phenols and enols by pentavalent bismuth reagents" Tetrahedron 1986, Volume 42, Pages 3111-3122 Bismuth(III) iodide is a catalyst in the Mukaiyama aldol reaction. Bi(III) is also used in a Barbier type allylation of carbonyl compounds in combination with a reducing agent such as zinc or magnesium, possibly forming the active Bi(0) catalyst in situ.
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Infusions of the bulbs have been used for medicinal and hallucinogenic purposes. Brunsvigia species are rich in alkaloids, particularly brunsvigine or 5,11b-methanomorphanthridine, and may be extremely toxic.South African National Biodiversity Institute"Alkaloids of the amaryllidaceae: Brunsvigine" - Tetrahedron, Volume 31, Issue 16, Pages 1855-1859 - R.C. Clark, F.L. Warren, K.G.R. Pachler B. bosmaniae was first described in 1932 by the South African botanist Frances Margaret Leighton (1909–2006) in the journal South African Gardening, and the specific epithet honours Marie Bosman, who collected the plant for Leighton.
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50, 47–65.Totté, N., Charon, L., Rohmer, M., Compernolle, F., Baboeuf, I., Geuns, J.M.C., 2000. Biosynthesis of the diterpenoid steviol, an ent-kaurene derivative from Stevia rebaudiana Bertoni, via the methylerythritol phosphate pathway Tetrahedron Letters 41, 6407–6410 Two molecules derived from primary metabolism, Pyruvate and Glyceraldehyde 3-Phosphate, are the initial molecules for this pathway. Elongation to GGPP from IPP and DMAPP Upon forming IPP and DMAPP, the diterpene GGPP is formed by via head-to-tail addition by an Sn1 mechanism.
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Some representative examples of Crich’s β-mannosylation are shown in Scheme 3.Crich, D.; Sun, S. Tetrahedron 1998, 54, 8321. It is noteworthy that, with this method in hand, primary, secondary, and tertiary alcohols (9, 12, and 13) all serve as glycosyl acceptors effectively in terms of yields and selectivity. In a recent version, the β-mannosylation of thioglycoside 14 and its analogues were examined to prepare sterically hindered glycosides, in which PhSOTf (or other newly developed sulfur-type oxidantsCrich, D.; Smith, M. Org. Lett.
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Ammonia is moderately basic; a 1.0 M aqueous solution has a pH of 11.6, and if a strong acid is added to such a solution until the solution is neutral (pH = 7), 99.4% of the ammonia molecules are protonated. Temperature and salinity also affect the proportion of NH4+. The latter has the shape of a regular tetrahedron and is isoelectronic with methane. The ammonia molecule readily undergoes nitrogen inversion at room temperature; a useful analogy is an umbrella turning itself inside out in a strong wind.
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Pseudoproline dipeptides have proven particularly effective in the synthesis of intractable peptides, long peptides/small proteins, and cyclic peptides, enabling in many cases the production of peptides that otherwise could not be made. These dipeptides are extremely easy to use: simply substitute a serine or threonine residue together with the preceding amino acid residue in the peptide sequence with the appropriate pseudoproline dipeptide (see the figure on your right). The native sequence is regenerated on cleavage and deprotection.T. Haack & M. Mutter (1992) Tetrahedron Lett.
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Wall temperature readings also serve as a maintenance tool, alerting the system to any irregularities. It is crucial to monitor the position of the wear line in order to avoid hot metal or slag from breaking through the lining and cause damage to the operators, the furnace and nearby equipment. In addition, the gaseous product leaving a furnace mostly comprises out of phosphorus tetrahedron (P4) and carbon monoxide. It still needs to be purified of any dust and is typically sent to an electrostatic gas purification system.
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"The Six Platonic Solids", an image that humorously adds the Utah teapot to the five standard Platonic solids One famous ray-traced image, by James Arvo and David Kirk in 1987, shows six stone columns, five of which are surmounted by the Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron). The sixth column supports a teapot. The image is titled "The Six Platonic Solids", with Arvo and Kirk calling the teapot "the newly discovered Teapotahedron". This image appeared on the covers of several books and computer graphic journals.
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At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time. Using a structure called a simplex, it divides spacetime into tiny triangular sections. A simplex is the generalized form of a triangle, in various dimensions. A 3-simplex is usually called a tetrahedron, and the 4-simplex, which is the basic building block in this theory, is also known as the pentatope, or pentachoron.
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Every second aluminum tetrahedron is attached to a hydroxide ion and the structure is interrupted. The oxygen tetrahedra connect in ladder-like chains to form large 10-membered rings as well as 8-membered, 6-membered, and two types of 4-membered rings. The calcium atoms and water molecules cross through the structure parallel to the c axis and sit in large channels created by the linked aluminum and silicon tetrahedral ringed structure, the main property of zeolites that allow for their absorption and dehydration abilities.
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The prototypical examples of exceptional objects arise in the classification of regular polytopes: in two dimensions, there is a series of regular n-gons for n ≥ 3\. In every dimension above 2, one can find analogues of the cube, tetrahedron and octahedron. In three dimensions, one finds two more regular polyhedra — the dodecahedron (12-hedron) and the icosahedron (20-hedron) — making five Platonic solids. In four dimensions, a total of six regular polytopes exist, including the 120-cell, the 600-cell and the 24-cell.
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By the isoperimetric inequality, the curve of constant width in the Euclidean plane with the largest area is a circle. The perimeter of a curve of constant width w is \pi w, regardless of its shape; this is Barbier's theorem. It is unknown which surfaces of constant width in three- dimensional space have the minimum volume. Bonnesen and Fenchel conjectured in 1934 that the minimizers are the two Meissner bodies obtained by rounding some of the edges of a Reuleaux tetrahedron, but this remains unproven.
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There exist tetrahedra having integer-valued edge lengths, face areas and volume. These are called Heronian tetrahedra. One example has one edge of 896, the opposite edge of 990 and the other four edges of 1073; two faces are isosceles triangles with areas of and the other two are isosceles with areas of , while the volume is . A tetrahedron can have integer volume and consecutive integers as edges, an example being the one with edges 6, 7, 8, 9, 10, and 11 and volume 48.
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Early work on the biosynthesis of the moenomycins focused on the 25-carbon lipid tail derived from moenocinol; the tail was of particular interest given that it appears to break the isoprene rule at C8, containing a quaternary carbon. Feeding studies revealed the moenocinol lipid tail originates from a 15-carbon farnesyl precursor and a 10-carbon geranyl pyrophosphate.Arigoni, D., et al. Tetrahedron Letters, 2001, 42, 3835–37 More recently, the biosynthetic gene cluster for moenomycin A was first described in 2007 in Streptomyces ghanaensis.
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It belongs to the hexagonal crystal system, but the crystal class is unknown. As with all phyllosilicates, the basic structural element is a triple layer, called a t-o-t layer, where "t" stands for a tetrahedral sheet and "o" stands for an octahedral sheet. The tetrahedral sheets comprise (Si,Al)O4 tetrahedra, linked together in nearly hexagonal rings. Three of the oxygens in each tetrahedron form links to other tetrahedra in the sheet, and the fourth oxygen, the apical oxygen, points away from the sheet.
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Conduritol or 1,2,3,4-cyclohexenetetrol is any of the organic compounds with chemical formula C6H10O4, that can be seen as derivatives of cyclohexene with four hydroxyl groups (OH) replacing hydrogen atoms on the four carbon atoms not adjacent to the double bond. They are therefore cyclic polyols or cyclitols. (1990) Tetrahedron, Vol 46, No 11, 3715 - 3742. The compounds in this group exhibit cis–trans isomerism, with six isomers that differ by the relative positions of the hydroxyls compared to the mean plane of the ring.
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By ordering the solids selectively—octahedron, icosahedron, dodecahedron, tetrahedron, cube—Kepler found that the spheres could be placed at intervals corresponding to the relative sizes of each planet's path, assuming the planets circle the Sun. Kepler also found a formula relating the size of each planet's orb to the length of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius. However, Kepler later rejected this formula, because it was not precise enough.Caspar. Kepler, pp.
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Leander, Lars, Tetra Pak. A Vision Becomes Reality. A company history with a difference, Lund: Tetra Pak International 1996 (), p. 18 At the time, non-carbonated drinks such as milk and juices were sold in heavy and cumbersome glass bottles and Rausing was determined to find a way of implementing the new, modern packaging technology, spending a lot of resources on developing new concepts.Leander, Lars, Tetra Pak. A Vision Becomes Reality. A company history with a difference, Lund: Tetra Pak International 1996 (), p. 18 With the aim of producing a carton container for liquid foods, similar to the hygienic and practical wax-coated paper cartons Rausing had seen overseas, Åkerlund & Rausing created the plastic-coated carton tetrahedron, patented on 27 March 1944.Idem p. 27Sedig, Kjell, Swedish Innovations, Stockholm: The Swedish Institute 2002 (), p. 45 In 1951 there was still no viable packaging material for the new package, and efforts to develop packaging material increased.Leander, Lars, Tetra Pak. A Vision Becomes Reality. A company history with a difference, Lund: Tetra Pak International 1996 (), appendix p. vii Finally, in 1952, the first machine producing tetrahedron cream packages was sold to Lundabygdens Mejeri, a local dairy.
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Rebecca Kamen in front of her installation The Platonic Solids at the Chemical Heritage Foundation, 2011 The Platonic Solids was inspired by Plato's conception of the five classical elements: earth, air, fire, water, and ether. In Plato's work Timaeus (ca. 350 BCE), the five forms of matter are related to elemental solids and shapes (the cube, the octahedron, the tetrahedron, the icosahedron, and the dodecahedron). In Kamen's work these regular polyhedra, created from fiberglass rods and sheets of mylar, are held against the larger plane of the wall, demonstrating "tension and compression".
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From 1992 until 2000, Zometool produced many kits which included connector nodes and blue, yellow, and red struts. In 2000, Zometool introduced green lines created in France by the architect Fabien VienneFabien Vienne, which were designed to allow the user to build, among other things, models of the regular tetrahedron and octahedron. In 2003, Zometool changed the style of the struts slightly. The struts "with clicks" have a different surface texture and they also have longer nibs which allow for a more robust connection between connector node and strut.
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Jean-Paul Behr has spent most of his career at the University of Strasbourg. After a doctorate in physical organic chemistry under the supervision of Jean-Marie Lehn (1973) followed by a postdoctoral internship in England, he founded the Genetic Chemistry Laboratory at the Faculty of Pharmacy in Strasbourg. His research there focused mainly on the development of molecules capable of encapsulating DNA and transporting it inside living cells. Jean-Paul Behr developed the first effective lipid vectors,Behr, J.P., « DNA strongly binds to micelles and vesicles containing lipopolyamines or lipointercalants », Tetrahedron Lett, 1986.
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Firefighters had to focus their efforts on saving the adjacent church instead of this burning building, an abandoned convent in Massueville, Quebec, Canada Firefighter carrying out a ladder slide A fire burns due to the presence of three elements: fuel, oxygen and heat. This is often referred to as the fire triangle. Sometimes it is known as the fire tetrahedron if a fourth element is added: a chemical chain reaction which can help sustain certain types of fire. The aim of firefighting is to deprive the fire of at least one of those elements.
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Alforsite exhibits Lau symmetry 6/m. It is concluded that the space group is P63/m because every single apatite compound possess that space group. The parameters of the unit cell are a=10.25 Å and c=7.64 Å. One barium atom in the unit cell is enclosed by nine oxygen atoms with a mean distance of 2.824 Å. Another barium atom is surrounded by six oxygen atoms with a mean distance of 2.791 Å and 3.230 Å, which creates two crystallographically distinctive barium atoms. The phosphate group in alforsite forms a regular tetrahedron.
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The plot revolves around a fictitious Onion television news anchorman, Norm Archer (Len Cariou), who is forced to face the inevitability of a corporate takeover by the Onion's perennial fictitious multinational, Global Tetrahedron. Onion news is described as "fair and balanced" in the context of the film. The plot serves as a springboard for various comedy sketches featuring The Onion's satirical humor. Vignettes include parodies of music videos reminiscent of Britney Spears' work, and Steven Seagal appearing as a parody of the type of action hero he normally portrays.
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There also exist polyhedra with four edges per vertex that cannot be realized as ideal polyhedra. If a simplicial polyhedron (one with all faces triangles) has all vertex degrees between four and six (inclusive) then it has an ideal representation, but the triakis tetrahedron is simplicial and non-ideal, and the 4-regular non-ideal example above shows that for non-simplicial polyhedra, having all degrees in this range does not guarantee an ideal realization.; quote this result, but incorrectly omit the qualifier that it holds only for simplicial polyhedra.
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Most commonly, software is used to generate the starting position, as was used in the 2019 World Fischer Random Championship. Ingo Althofer If this is not available, there are several other procedures for generating random starting positions with equal probability. In 1998 Ingo Althöfer proposed in 1998 the "single die method", a method that requires only a single standard die. If a full set of polyhedral dice is available (a tetrahedron (d4), cube (d6), octahedron (d8), dodecahedron (d12), and a icosahedron (d20)), one never needs to reroll any dice.
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A silicate mineral is generally an ionic compound whose anions consist predominantly of silicon and oxygen atoms. In most minerals in the Earth's crust, each silicon atom is the center of an ideal tetrahedron, whose corners are four oxygen atoms covalently bound to it. Two adjacent tetrahedra may share a vertex, meaning that the oxygen atom is a bridge connecting the two silicon atoms. An unpaired vertex represents an ionized oxygen atom, covalently bound to a single silicon atom, that contributes one unit of negative charge to the anion.
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They were filled with a mixture of argon and carbon dioxide at . Eight chambers were arranged in a tray and connected in parallel; four trays were located in a tetrahedron around the experimental assembly, recording the gamma radiation around the sphere, sufficiently close to give a signal and sufficiently far away to not be destroyed by the blast before they could record the required information. The initiation of the explosives was initially performed by a multipoint Primacord system. The results were erratic, as the detonations weren't sufficiently synchronized.
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The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance. PDB id:11LFL) and "relaxed (R)" on the right corresponding to the oxy form (derived from PDB id:1LFT). As described in the historical section, the most famous example of cooperative binding is hemoglobin. Its quaternary structure, solved by Max Perutz using X-ray diffraction, exhibits a pseudo- symmetrical tetrahedron carrying four binding sites (hemes) for oxygen.
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GJK makes use of Johnson's distance subalgorithm, which computes in the general case the point of a tetrahedron closest to the origin, but is known to suffer from numerical robustness problems. In 2017 Montanari, Petrinic, and Barbieri proposed a new subalgorithm based on signed volumes which avoids the multiplication of potentially small quantities and achieved a speedup of 15% to 30%. GJK algorithms are often used incrementally in simulation systems and video games. In this mode, the final simplex from a previous solution is used as the initial guess in the next iteration, or "frame".
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The oligostilbenes amurensin A, B, C, D, E, F,Four Novel Oligostilbenes from the Roots of Vitis amurensis. Kai-Sheng Huang, Mao Lin, Lin-Ning Yu and Man Kong, Tetrahedron, 3 March 2000, Volume 56, Issue 10, Pages 1321–1329, G, H, I, J, K, L and MAnti-inflammatory tetramers of resveratrol from the roots of Vitis amurensis and the conformations of the seven-membered ring in some oligostilbenes. Kai-Sheng Huang, Mao Lin, and Gui- Fang Cheng, Phytochemistry, September 2001, Volume 58, Issue 2, Pages 357–362, can be found in V. amurensis.
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Dry chemical works by directly inhibiting the chemical chain reaction which forms one of the four sides of the fire tetrahedron (heat + oxygen + fuel + chemical chain reaction = fire). To a much smaller degree it also has a smothering effect by excluding oxygen from the fire. "Dry chemical" extinguishers, such as Purple-K, are different from "dry powder" extinguishers that are used to fight Class D flammable metal fires. Purple-K was developed by the United States Naval Research Laboratory in 1959, as an improvement over sodium bicarbonate for extinguishing oil and gasoline fires.
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Although the groups of E. Tagliavini and K. Mikami reported the catalysis of this reaction using a Ti(IV)–BINOL complex in the same year as the Keck group,Aoki, S., Mikami, K., Terada, M., Nakai, T., Tetrahedron 1993, 49, 1783-1792.Costa, A. L., Piazza, M. G., Tagliavini, E., Trombini, C., Umani-Ronchi, A., J. Am. Chem. Soc., 1993, 115, 7001-7002. Keck's publication reported higher enantio- and diastereoselectivity, and did not require the use of 4 Angstrom molecular sieves as in Mikami's procedure or an excess of BINOL as in Tagliavini's procedure.
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It became a local tourist attraction as a result. Coach tours to the nearby coastal town of Whitby drove past the site, at which point drivers would typically switch the radio on and allow passengers to listen to the interference caused by the radars. Between 1989 and 1992, Raytheon, the US defence contractor, completed a contract that saw the domes replaced by the current tetrahedron ('pyramid') structure, housing the AESA phased array radar. The site is above sea level and the structure is nine floors high rising from its ground level to high.
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The reaction conditions are harsh: medium molten potassium hydroxide at temperatures in the range of 250 to 300 °C. The reaction has been of some importance in structure elucidation of certain fatty acids, but has little practical synthetic value.Organic reactions in strong alkalis-I : Fission of ethylenic acids (the Varrentrapp reaction) Tetrahedron, Volume 8, Issues 3–4, 1960, Pages 221–238 R. G. Ackman, Patrick Linstead, B. J. Wakefield and B. C. L. Weedon The original 1840 Varrentrapp reaction concerned the conversion of oleate to palmitate and acetate.
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Marching tetrahedra computes up to nineteen edge intersections per cube, where marching cubes only requires twelve. Only one of these intersections cannot be shared with an adjacent cube (the one on the main diagonal), but sharing on all faces of the cube complicates the algorithm and increases memory requirements considerably. On the other hand, the additional intersections provide for a slightly better sampling resolution. The number of configurations, determining the size of the commonly used lookup tables, is much smaller, since only four rather than eight separate vertices are involved per tetrahedron.
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Some famous vinyl iodide synthesis methods involve conversion of aldehyde or ketone to vinyl iodide. Barton's hydrazone iodination method involves addition of hydrazines to aldehyde or ketone to form hydrazone. Then the hydrazone is converted to vinyl iodide by addition of iodide and DBU.Barton, D. H. R. , R. E. O'Brien and S. Sternhell Journal of the Chemical Society,1962, 470 - 476Barton, D. H. R.; Bashiardes, G.; Fourrey, J.-L. Tetrahedron 1988, 44, 147 This method has been used in natural product synthesis of Taxol by DanishefskyDanishefsky, Samuel J., et al.
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Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number () can be modelled with 35 billiard balls and the standard triangular billiards ball frame that holds 15 balls in place. Then 10 more balls are stacked on top of those, then another 6, then another three and one ball at the top completes the tetrahedron. When order- tetrahedra built from spheres are used as a unit, it can be shown that a space tiling with such units can achieve a densest sphere packing as long as .
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B-type procyanidin (catechin dimer) can be converted to A-type procyanidin by radical oxidation.Conversion of procyanidin B-type (catechin dimer) to A-type: evidence for abstraction of C-2 hydrogen in catechin during radical oxidation. Kazunari Kondo, Masaaki Kurihara, Kiyoshi Fukuhara, Takashi Tanaka, Takashi Suzuki, Naoki Miyata and Masatake Toyoda, Tetrahedron Letters, Volume 41, Issue 4, 22 January 2000, Pages 485-488, Dimeric proanthocyanidins can also be synthesized with procyanidin-rich grape seed extracts reacted with flavan-3-ols under acid catalysis.New Approach for the Synthesis and Isolation of Dimeric Procyanidins.
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Lam, L. K. P.; Hui, R. A. H. F.; Jones, J. B. J. Org. Chem. 1986, 51, 2047. This switch in enantioselectivity is accurately predicted by the active site model given above. (4)File:EsteraseScope2.png An opposite trend is observed in desymmetrizing hydrolyses of 2-methyl malonates, which afford the (S) enantiomer when the other substituent on C-2 is small, and the (R) enantiomer when the other C-2 substituent is large.Björkling, F.; Boutelje, J.; Gatenbeck, S.; Hult, K.; Norin, T.; Szmulik, P. Tetrahedron 1985, 41, 1347. (5)File:EsteraseScope3.
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Among the awards that Reetz has received are the Leibniz Prize (1989), the Nagoya Gold Medal Award of Organic Chemistry (2000), the RSC Centenary Prize (2002), the (2005), the (2006), the ACS Arthur C. Cope Award (2009), the Tetrahedron Prize (2011), the Otto Hahn Prize (2011), and the International Kyoto Conference on New Aspects of Organic Chemistry (IKCOC) Prize (2012). He was elected to the German Academy of Sciences Leopoldina in 1997. He was elected a foreign member of the Royal Netherlands Academy of Arts and Sciences in 2005.
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Each ball is drilled with as many holes as its conventional valence (C: 4; N: 3; O: 2; H: 1) directed towards the vertices of a tetrahedron. Single bonds are represented by (fairly) rigid grey rods. Double and triple bonds use two longer flexible bonds which restrict rotation and support conventional cis/trans stereochemistry. Fig. 4. Beevers ball and stick model of ruby (Cr- doped corundum) made with acrylic balls and stainless steel rods by Miramodus Ltd.. However, most molecules require holes at other angles and specialist companies manufacture kits and bespoke models.
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Maricite or marićite is a sodium iron phosphate mineral (NaFe2+PO4), that has two metal cations connected to a phosphate tetrahedron. It is structurally similar to the much more common mineral olivine. Maricite is brittle, usually colorless to gray, and has been found in nodules within shale beds often containing other minerals. Maricite is most commonly known to be found in the Big Fish River area of the Yukon Territory, Canada, but it has also been found in Eastern Germany, as well as inside of various meteorites around the world.
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As a secondary path, she researched algebraic differentials, studying transversals, including Liouville's and Reiss' theorems, as well as the invariant theory of Enrico Bompiani. She was also one of the first scholars in Italy to study integral geometry. She published works which summarized the results of Morgan Crofton, Henri Lebesgue, and Luis Santaló, presenting new formulas for determining ovals and ellipses. Also concerned with the history of mathematics, Biggiogero published studies on the geometry of the triangle and the tetrahedron, in conjunction with Virginio Retali for the Encyclopedia of Elementary Mathematics.
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The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of which was that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy. He also discovered the Kepler solids.
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A right pyramid with a regular base has isosceles triangle sides, with symmetry is Cnv or [1,n], with order 2n. It can be given an extended Schläfli symbol ( ) ∨ {n}, representing a point, ( ), joined (orthogonally offset) to a regular polygon, {n}. A join operation creates a new edge between all pairs of vertices of the two joined figures.N.W. Johnson: Geometries and Transformations, (2018) Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms The trigonal or triangular pyramid with all equilateral triangle faces becomes the regular tetrahedron, one of the Platonic solids.
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Ring contraction through anionic (A), cationic (B), and carbenoid (C) reactive intermediates. Ring contractions are useful for making smaller, more strained rings from larger rings. The impetus for making these rings comes from the difficulty associated with making a fully elaborated small ring when such a ring could more easily be made from an elaborated larger ring, from which an atom can be excised, or that the original larger scaffold is more accessible.Silva, L.F. Tetrahedron 2002, 9137–9161 Ring contractions are easily characterized simply by the reactive intermediate which performs the contraction.
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In chemistry, a trigonal pyramid is a molecular geometry with one atom at the apex and three atoms at the corners of a trigonal base, resembling a tetrahedron (not to be confused with the tetrahedral geometry). When all three atoms at the corners are identical, the molecule belongs to point group C3v. Some molecules and ions with trigonal pyramidal geometry are the pnictogen hydrides (XH3), xenon trioxide (XeO3), the chlorate ion, , and the sulfite ion, . In organic chemistry, molecules which have a trigonal pyramidal geometry are sometimes described as sp3 hybridized.
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By ordering the solids correctly—octahedron, icosahedron, dodecahedron, tetrahedron, and cube—Kepler found that the spheres correspond to the relative sizes of each planet's path around the Sun, generally varying from astronomical observations by less than 10%. Kepler also found a formula relating the size of each planet's orbit to the length of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius. However, Kepler later rejected this formula because it was not precise enough.Caspar. Kepler, pp.
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He was awarded the 1975 Ernest Guenther Award and the 1994 Arthur C. Cope Scholar Award by the American Chemical Society (ACS). He gave the RSC Centenary Lecture in 1994 and the Royal Society Bakerian Lecture in 1996. He took the Tetrahedron Prize (1995), the RSC Natural Products Award (1996), the Welch Award in Chemistry (2000), the Royal Society's Davy Medal (2001), the Queen's Royal Medal of the Royal Society of Edinburgh (2001), and the ACS Nakanishi Prize (2003). He was Texas Scientist of the Year in 2002.
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The Bamberger triazine synthesis in organic chemistry is a classic organic synthesis of a triazine first reported by Eugen Bamberger in 1892.Hassner, A., Stumer, C., Organic Synthesis Based on Name Reactions: 2nd. Ed. Tetrahedron Organic Chemistry Series, Volume 22 Pergamon, Oxford Bamberger triazine synthesis The reactants are an aryl diazonium salt obtained from reaction of the corresponding aniline with sodium nitrite and hydrochloric acid and the hydrazone of pyruvic acid. The azo intermediate converts to the benzotriazine in the third step with sulfuric acid in acetic acid.
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Fuller was most famous for his lattice shell structures – geodesic domes, which have been used as parts of military radar stations, civic buildings, environmental protest camps and exhibition attractions. An examination of the geodesic design by Walther Bauersfeld for the Zeiss-Planetarium, built some 28 years prior to Fuller's work, reveals that Fuller's Geodesic Dome patent (U.S. 2,682,235; awarded in 1954) is the same design as Bauersfeld's. Their construction is based on extending some basic principles to build simple "tensegrity" structures (tetrahedron, octahedron, and the closest packing of spheres), making them lightweight and stable.
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The Kröhnke method is featured in a solvent-free synthesis of triarylpyridines that proceeds via a homo-coupling of two diaryl substituted α, β-unsaturated carbonyl compounds.Adib, M.; Tahermansouri, H.; Koloogani, S. A.; Mohammadi, B.; Bijanzadej, H. R. Tetrahedron Lett.2006, 47, 5957-5960 This strategy offers a facile means for preparation of pyridnyl aryl systems that are important fragments of many useful drug scaffolds. center In 1992, Robinson and co- workers developed a similar pyridine synthesis using enamino nitriles as one of the three-carbon fragments in place of an α-pyridinium methyl ketone.
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In this context, a simplex in d-dimensional Euclidean space is the convex hull of d+1 points that do not all lie in a common hyperplane. For example, a 2-dimensional simplex is just a triangle (the convex hull of three points in the plane) and a 3-dimensional simplex is a tetrahedron (the convex of four points in three-dimensional space). The points that form the simplex in this way are called its vertices. An orthoscheme, also called a path simplex, is a special kind of simplex.
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Rhetorical Triangle Aristotle established the classic triad of ethos, pathos and logos (the Aristotelian triad of appeals) that serves as the foundation of the rhetorical triangle. The rhetorical triangle has evolved from its original, sophistic model into what rhetorician, Sharon Crowley, describes as the "postmodern" rhetorical triangle, the rhetorical tetrahedron. The expanded rhetorical triangle now emphasizes context by integrating situational elements. The original version includes only 3 points: the writer/speaker (ethos), the audience (pathos), and the message itself (logos), as shown in the bottom image to the right.
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Erskine was sympathetic to the American cause, but worried that might he lose his workers to the army. He organised them into a militia and was appointed as a militia captain in August 1775. Once the war broke out in earnest, the rebels were concerned that the British warships would go up the Hudson River to attack northern forts and separate New England from the rest of the colonies. Erskine, ever the engineer, designed a tetrahedron-shaped marine cheval-de-frise, a defensive barrier to prevent warships from sailing upriver.
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The complete row of Pascal's triangle for the hypercube in this construction runs 1 (single vertex), 4 (tetrahedron tetrad), 6 (hexany), 4 (another tetrad), 1. The idea generalises to other numbers of dimensions, for instance, the cross-sections of a five-dimensional cube give two versions of the dekany, a ten-note scale rich in tetrads, triads and dyads, which also contains many hexanies. In six dimensions the same construction gives the twenty-note eikosany, which is even richer in chords. It has pentads, tetrads, and triads as well as hexanies and dekanies.
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If the glass is extruded and cooled quickly at this temperature, it will be unable to form an ordered structure. In the polymer it forms SiO4 groups which are configured as a tetrahedron with the silicon atom at the center, and four oxygen atoms at the corners. These atoms then form a network bonded at the corners by sharing the oxygen atoms. The vitreous and crystalline states of silica (glass and quartz) have similar energy levels on a molecular basis, also implying that the glassy form is extremely stable.
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In the Bloch sphere representation of a qubit, the states of a SIC-POVM form a regular tetrahedron. Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions. A symmetric, informationally complete, positive operator-valued measure (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism.
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In the worst case, every pair of vertices is connected, giving a complete graph. To immerse the complete graph K_n with all the edges having unit length, we need the Euclidean space of dimension n-1. For example, it takes two dimensions to immerse K_3 (an equilateral triangle), and three to immerse K_4 (a regular tetrahedron) as shown to the right. :\dim K_n = n-1 In other words, the dimension of the complete graph is the same as that of the simplex having the same number of vertices.
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A polyhedron (a solid object in three-dimensional space, bounded by two- dimensional faces) may have two different types of diagonals: face diagonals on the various faces, connecting non-adjacent vertices on the same face; and space diagonals, entirely in the interior of the polyhedron (except for the endpoints on the vertices). Just as a triangle has no diagonals, so also a tetrahedron (with four triangular faces) has no face diagonals and no space diagonals. A cuboid has two diagonals on each of the six faces and four space diagonals.
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Phase I is noted as the superprotonic phase (strong conducting phase), which triggers an extreme growth in proton conductivity by four orders of magnitude, reaching 10 mS/cm. This makes the conductivity of CsHSO4 ten-fold stronger than the conductivity of a sodium chloride aqueous solution. In the superprotonic phase, the movement of an SO4 tetrahedron generates a disruption of the hydrogen bond network, which accelerates proton transfer. The tetragonal anions available in the structure are accountable for the arrangement of the hydrogen bonds with the moving protons.
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In cascade reactions, isolation of intermediates is not required, as each reaction composing the sequence occurs spontaneously. In the strictest definition of the term, the reaction conditions do not change among the consecutive steps of a cascade and no new reagents are added after the initial step.Padwa, A.; Bur, S. K. Tetrahedron 2007, 63, 5341–5378. By contrast, one-pot procedures similarly allow at least two reactions to be carried out consecutively without any isolation of intermediates, but do not preclude the addition of new reagents or the change of conditions after the first reaction.
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"Möbius Dick" is the fifteenth episode of the sixth season of the animated sitcom Futurama, and originally aired August 4, 2011 on Comedy Central. The episode was written by Dan Vebber and directed by Dwayne Carey-Hill. In the episode, the Planet Express crew pass through an area in space known as the Bermuda Tetrahedron, where many other ships passing through the area have mysteriously disappeared, including that of the first Planet Express crew. While exploring the area, a mysterious four-dimensional space whale devours the ship's engine, leaving them stranded in the area.
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The Tetrahedron Computer Methodology was a short lived journal that was published by Pergamon Press (now Elsevier) to experiment with electronic submission of articles in the ChemText format, and the sharing source code to enable reproducibility. It was the first chemical journal to be published electronically, with issues distributed in print and on floppy disks. It is likely it was also the first journal to accept submissions in a non-paper format (on floppy disks). The journal ceased publication owing to technical and non-technical reasons, and may have lacked sufficient institutional support.
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The formation of chiral quaternary carbon centers has been a synthetic challenge. Chemists have developed asymmetric Diels–Alder reactions, Nicolaou, K. C.; Vassilikogiannakis, G.; Mägerlein, W.; Kranich, R Angew. Chem. Int. Ed. Volume 2001, Issue 40 , Pages 2482–2486 Heck reaction, Enyne cyclization, cycloaddition reactions, Quasdorf, K.W.; Overman, L. E. Nature Volume 2014, Volume 516, Pages 181 C–H activation, Allylic substitution, Pauson–Khand reaction, Ishizaki, M.; Niimi, Y.; Hoshino, O.; Hara, H.; Takahashi, T. Tetrahedron Volume 2001, Issue 61, Pages 4053–4065 etc. to construct asymmetric quaternary carbons.
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Adding a seventh sphere gives a new cluster consisting in two "axial" balls touching each other and five others touching the latter two balls, the outer shape being an almost regular pentagonal bi- pyramid. However, we are facing now a real packing problem, analogous to the one encountered above with the pentagonal tiling in two dimensions. The dihedral angle of a tetrahedron is not commensurable with 2; consequently, a hole remains between two faces of neighboring tetrahedra. As a consequence, a perfect tiling of the Euclidean space R3 is impossible with regular tetrahedra.
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The materials paradigm represented as a tetrahedron Materials science is a relatively new, interdisciplinary field which deals with the study of matter and its properties; as well as the discovery and design of new materials. Originally developed through the field of metallurgy, the study of the properties of materials and solids has now expanded into all materials. The field covers the chemistry, physics and engineering applications of materials including metals, ceramics, artificial polymers, and many others. The core of the field deals with relating structure of material with it properties.
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Harn Museum architecture The original architecture for the Harn Museum was designed by Kha Le-Huu, a native of South Vietnam and a 1982 architecture alumni of the University of Florida. Le-Huu reportedly designed the building to include buddhist sensibilities in the entrance garden along with featuring his contemporary aesthetic for geometric elements including use of the tetrahedron. Kha Le-Huu & Partners of Orlando, Florida also designed the additional wing for Asian art, completed in 2011. The Harn Museum features several outdoor gardens including an Asian Water Garden and an Asian Rock Garden designed by Hoichi Kurisu of Kurisu International.
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Chapter 5 discusses normal surfaces, surfaces that intersect the tetrahedra of a triangulation of a manifold in a controlled way. By parameterizing these surfaces by how many pieces of each possible type they can have within each tetrahedron of a triangulation, one can reduce many questions about manifolds such as the recognition of trivial knots and trivial manifolds to questions in number theory, on the existence of solutions to certain Diophantine equations. The book uses this tool to prove the existence and uniqueness of prime decompositions of manifolds. Chapter 6 concerns Heegaard splittings, surfaces which split a given manifold into two handlebodies.
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The first multi-anvil press design was a tetrahedral press, using four anvils to converge upon a tetrahedron-shaped volume. The cubic press was created shortly thereafter to increase the volume to which pressure could be applied. A cubic press is typically smaller than a belt press and can more rapidly achieve the pressure and temperature necessary to create synthetic diamond. However, cubic presses cannot be easily scaled up to larger volumes: the pressurized volume can be increased by using larger anvils, but this also increases the amount of force needed on the anvils to achieve the same pressure.
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They also reported a glassy, disordered packing at densities exceeding 78%. For a periodic approximant to a quasicrystal with an 82-tetrahedron unit cell, they obtained a packing density as high as 85.03%. In late 2009, a new, much simpler family of packings with a packing fraction of 85.47% was discovered by Kallus, Elser, and Gravel. These packings were also the basis of a slightly improved packing obtained by Torquato and Jiao at the end of 2009 with a packing fraction of 85.55%, and by Chen, Engel, and Glotzer in early 2010 with a packing fraction of 85.63%.
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Aspidophytine is an indole alkaloid that has attracted a lot of attention from synthetic chemists."Enantioselective Total Synthesis of Aspidophytine" He, F.; Bo, Y.; Altom, J. D.; Corey, E. J. J. Am. Chem. Soc. 1999, 121(28), 6771–6772 ()"Stereocontrolled total synthesis of (−)-aspidophytine" Sumi, S.; Matsumoto, K.; Tokuyama, H.; Fukuyama, T. Tetrahedron 2003, 59, 8571–8587 ()"Application of the Rh(II) Cyclization/Cycloaddition Cascade for the Total Synthesis of (±)-Aspidophytine" Mejia-Oneto, J. M., Padwa, A. Org. Lett. 2006, 8, 3275–3278 ()"A Concise Asymmetric Total Synthesis of Aspidophytine" Nicolaou, K. C., Dalby, S. M., Majumder, U. J. Am. Chem. Soc.
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In the late 1970s Connelly and D. Sullivan formulated the bellows conjecture stating that the volume of a flexible polyhedron is invariant under flexing. This conjecture was proved for polyhedra homeomorphic to a sphere by using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by . The proof extends Piero della Francesca's formula for the volume of a tetrahedron to a formula for the volume of any polyhedron. The extended formula shows that the volume must be a root of a polynomial whose coefficients depend only on the lengths of the polyhedron's edges.
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Ancient philosophy posited a set of classical elements to explain observed patterns in nature. These elements originally referred to earth, water, air and fire rather than the chemical elements of modern science. The term 'elements' (stoicheia) was first used by the Greek philosopher Plato in about 360 BCE in his dialogue Timaeus, which includes a discussion of the composition of inorganic and organic bodies and is a speculative treatise on chemistry. Plato believed the elements introduced a century earlier by Empedocles were composed of small polyhedral forms: tetrahedron (fire), octahedron (air), icosahedron (water), and cube (earth).
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The structural zinc binding motif in alcohol dehydrogenase from an MD simulation Mammalian alcohol dehydrogenases also have a structural zinc site. This Zn ion plays a structural role and is crucial for protein stability. The structures of the catalytic and structural zinc sites in horse liver alcohol dehydrogenase (HLADH) as revealed in crystallographic structures, which has been studied computationally with quantum chemical as well as with classical molecular dynamics methods. The structural zinc site is composed of four closely spaced cysteine ligands (Cys97, Cys100, Cys103, and Cys111 in the amino acid sequence) positioned in an almost symmetric tetrahedron around the Zn ion.
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In the year 1999, the Machine Empire of Baranoia, led by Emperor Bacchushund, invades Earth with the intention of wiping out all human life and bringing about machine rule. Chief Counsellor Miura revives super energies that had been born of the lost civilization of Pangaea. Assembling pieces of a stone plate uncovered three years previously, he reveals the secrets of "Super Power". Enlisting an elite five-man team of the United Airforce's finest pilots, Miura builds a pyramid to generate Tetrahedron power in order to allow five UAOH officers to transform into the Ohrangers and stop Baranoia's invasion.
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The order-5 cubic honeycomb has a related alternated honeycomb, ↔ , with icosahedron and tetrahedron cells. The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space: There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the regular form: The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures. It is also in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.
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331 (JSTOR) The theorem is attributed to Commandino, who stated, in his work De Centro Gravitatis Solidorum (The Center of Gravity of Solids, 1565), that the four medians of the tetrahedron are concurrent. However, according to the 19th century scholar Guillaume Libri, Francesco Maurolico (1494–1575) claimed to have found the result earlier. Libri nevertheless thought that it had been known even earlier to Leonardo da Vinci, who seemed to have used it in his work. Julian Coolidge shared that assessment but pointed out that he couldn't find any explicit description or mathematical treatment of the theorem in da Vinci's works.
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Aluminium, like its congener boron, is less electronegative than carbon (Al, 1.61; C, 2.55); thus, aluminium-bound carbons in organoalanes possess partial negative charge and are nucleophilic as a result. Generally, however, organoalanes are not nucleophilic enough to transfer an organic group on their own (the exception being when carbonyl and enone acceptors are used, due to the high oxophilicity of aluminiumCollins, P. W.; Dajani, E. Z.; Bruhn, M. S.; Brown, C. H.; Palmer, J. H.; Pappo, R. Tetrahedron Lett. 1975, 4217.). In most cases, nucleophilic activation of organoalanes is necessary for group transfer to take place.
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The kissing number problem can be generalized to the problem of finding the maximum number of non-overlapping congruent copies of any convex body that touch a given copy of the body. There are different versions of the problem depending on whether the copies are only required to be congruent to the original body, translates of the original body, or translated by a lattice. For the regular tetrahedron, for example, it is known that both the lattice kissing number and the translative kissing number are equal to 18, whereas the congruent kissing number is at least 56.
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Ilan Marek is editorial board member in many different journals such as Chemical Communications, (RSC); Organic and Biomolecular Chemistry, (RSC); European Journal of Organic Chemistry, (WILEY-VCH); The Chemical Record, (Wiley-VCH); Advanced Synthesis and Catalysis, (Wiley-VCH); Synthesis, (Thieme ); Synlett, Thieme; Angewandte Chemie International Edition, (WILEY- VCH); Chemistry, A European Journal, (WILEY-VCH); Helvetica, (WILEY-VCH); Chemical Reviews (ACS); ACS Central Science (ACS). He is editor of numerous journals and revues : Comprehensive Organic Synthesis (Second Edition. Elsevier); Beilstein Journal of Organic Chemistry; Israel Journal of Chemistry, (Wiley-VCH); Tetrahedron (Elsevier); “The Chemistry of Functional groups” Patai series, (WILEY-Chichester).
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More than 1000 atoms are available in the unit cell, which is built up by large structure units such as two supertetrahedra T(1) and T(2) and one superoctahedron O(1). As shown in figure 24a, T(1) consists of 4 icosahedra I(1) which have no direct bonding but are bridged by four B and C20 atoms. These atoms also form tetrahedron centered by the Si2 sites. The supertetrahedron T(2) that consists of 4 icosahedra I(2) is the same as shown in figure 18b; its mixed-occupancy sites B and C6 directly bond with each other.
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The special perpendicular is employed to compute the volume of a pyramid (p 35), an equation on skew lines that reduces to zero when they are coplanar, a property of a spherical triangle, and the coincidence of the perpendiculars in a tetrahedron. Chapter four describes equations of geometric figures: line, plane, circle, sphere. The definition of a conic section is taken from Kelland and Tait: "the locus of a point which move so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line." Ellipse, hyperbola, and parabola are then illustrated.
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There are phenomena like supercooling, in which the water is cooled below its freezing point, but the water remains liquid, if there are too few defects to seed crystallization. One can therefore observe a delay until the water adjusts to the new, below-freezing temperature. Supercooled liquid water must become ice at minus 48 C (minus 55 F) not just because of the extreme cold, but because the molecular structure of water changes physically to form tetrahedron shapes, with each water molecule loosely bonded to four others. This suggests the structural change from liquid to "intermediate ice".
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Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist. The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue Timaeus 360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid.
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However, it differs from rock-salt structure in how the two lattices are positioned relative to one another. The zincblende structure has tetrahedral coordination: Each atom's nearest neighbors consist of four atoms of the opposite type, positioned like the four vertices of a regular tetrahedron. Altogether, the arrangement of atoms in zincblende structure is the same as diamond cubic structure, but with alternating types of atoms at the different lattice sites. Examples of compounds with this structure include zincblende itself, lead(II) nitrate, many compound semiconductors (such as gallium arsenide and cadmium telluride), and a wide array of other binary compounds.
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Koolhaas & Obrist (2011), p. 507 The former of these was composed of capsules plugged onto six point frames and was assembled in just six days; the latter was a space frame composed of tetrahedron modules, based upon his Helix City that could grow in 14 different directions and resemble organic growth.Koolhaas & Obrist (2011), p. 528-530 Expo '70 has been described has the apotheosis of the Metabolist movement. But even before Japan's period of rapid economic growth ended with the world energy crisis, critics were calling the Expo a dystopia that was removed from reality.Sasaki (1970), p.
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It is named after the English chemist and mineralogist William Hyde Wollaston (1766–1828). Some of the properties that make wollastonite so useful are its high brightness and white coloration, low moisture and oil absorption, and low volatile content. Wollastonite is used primarily in ceramics, friction products (brakes and clutches), metalmaking, paint filler, and plastics. Despite its chemical similarity to the compositional spectrum of the pyroxene group of minerals—where magnesium and iron substitution for calcium ends with diopside and hedenbergite respectively—it is structurally very different, with a third SiO4 tetrahedron in the linked chain (as opposed to two in the pyroxenes).
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In an equatorial attack, the nucleophile approaches by eclipsing a neighboring hydrogen atom and subsequently pushes the carbonyl substituents into eclipsing positions as it pyramidalizes the carbonyl carbon.Wigfield, D. C. Tetrahedron 1979, 35, 449 In an axial approach, the nucleophile approaches gauche to neighboring hydrogen atoms and so does not cause eclipsing interactions while pyramidalizing the carbonyl carbon. It is this torsional strain—the energy cost of rotating the bonds out of an eclipsed position—that favors axial over equatorial approach. In the case of substituted norbornones, stereoselectivity may be explained by electrostatic interactions between substituents and nucleophiles.
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For equal masses, one possible central configuration places the masses at the vertices of a regular polygon (forming a Klemperer rosette), a Platonic solid, or a regular polytope in higher dimensions. The centrality of the configuration follows from its symmetry. It is also possible to place an additional point, of arbitrary mass, at the center of mass of the system without changing its centrality. Placing three masses in an equilateral triangle, four at the vertices of a regular tetrahedron, or more generally masses at the vertices of a regular simplex produces a central configuration even when the masses are not equal.
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The equatorial OH groups [O(1)H] are coordinated by two Zn(1) cations and one Zn(2) cation, whereas the apical OH groups [O(2)H] are coordinated by three Zn(1) cations. As Zn(1) is six- coordinated and Zn(2) is four-coordinated, the local bond-valence requirements require the Zn(1)-O(1) bonds to be considerably longer than the Zn(1)-O(2) bonds. The [Zn(OH)3Cl] tetrahedron has three short Zn(2)-O(1) bonds (at 1.950 Å) and one long Zn(2)-Cl bond (2.312 Å) (Figure 1).
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Meffert also produces a similar puzzle called the Tetraminx, which is the same as the Pyraminx except that the trivial tips are removed, turning the puzzle into a truncated tetrahedron. Scrambled Pyraminx The purpose of the Pyraminx is to scramble the colors, and then restore them to their original configuration. The 4 trivial tips can be easily rotated to line up with the axial piece they are respectively attached to, and the axial pieces are also easily rotated so that their colors line up with each other. This leaves only the 6 edge pieces as a real challenge to the puzzle.
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In 1953, after rationing in the UK ended, Tetley launched the tea bag to the UK, and it was an immediate success. The "pyramid tea bag" (or sachet), introduced by Lipton and PG Tips/Scottish Blend in 1996, attempts to address one of the connoisseurs' arguments against paper tea bags by way of its three-dimensional tetrahedron shape, which allows more room for tea leaves to expand while steeping. However, some types of pyramid tea bags have been criticised as being environmentally unfriendly, since their synthetic material is not as biodegradable as loose tea leaves and paper tea bags.
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Many of methylene's electronic states lie relatively close to each other, giving rise to varying degrees of radical chemistry. The ground state is a triplet radical with two unpaired electrons (X̃3B1), Isaiah Shavitt (1985), Geometry and singlet-triplet energy gap in methylene: A critical review of experimental and theoretical determinations. Tetrahedron, volume 41, issue 8, page 1531 and the first excited state is a singlet non-radical (ã1A1). With the singlet non-radical only 38 kJ above the ground state, a sample of methylene exists as a mixture of electronic states even at room temperature, giving rise to complex reactions.
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It is a practical exercise to try to pack table tennis balls in order to form only tetrahedral configurations. One starts with four balls arranged as a perfect tetrahedron, and try to add new spheres, while forming new tetrahedra. The next solution, with five balls, is trivially two tetrahedra sharing a common face; note that already with this solution, the fcc structure, which contains individual tetrahedral holes, does not show such a configuration (the tetrahedra share edges, not faces). With six balls, three regular tetrahedra are built, and the cluster is incompatible with all compact crystalline structures (fcc and hcp).
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Euler's line (red) is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red). In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron.
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John L. Wood (born December 4, 1961) is a Robert A. Welch Distinguished Professor of Chemistry and Biochemistry; Cancer Prevention Research Institute of Texas Scholar; and Associate Editor for the Americas, Tetrahedron Letters at Baylor University. Wood received a B.A. degree from the University of Colorado in 1985 and a Ph.D. from the University of Pennsylvania in 1991. He then moved to Harvard University as an American Cancer Society postdoctoral fellow and continued studying natural products synthesis in the laboratories of Stuart Schreiber. He joined the faculty at Yale University in 1993 as an assistant professor and was promoted to full professor in 1998.
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The simplest affine plane contains only four points; it is called the affine plane of order 2. (The order of an affine plane is the number of points on any line, see below.) Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines. It corresponds to a tetrahedron where non- intersecting edges are considered "parallel", or a square where not only opposite sides, but also diagonals are considered "parallel". More generally, a finite affine plane of order n has n2 points and lines; each line contains n points, and each point is on lines.
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The crystal structure for chrisstanleyite has two different polyhedra structures that intersect and support each other, which is the same as jagueite. An AgSe4 (or CuSe4) tetrahedral creates a grooved (100) layer that are grouped in dimers of Ag2Se6, which share four vertices with adjacent dimers. Oriented alternatively above and below the layer are the two remaining vertices for each tetrahedron, resulting in the corrugation of the silver-based layer as well the sharing of Se atoms with Pd polyhedra (Topa et al. 2006). The second framework consists of single coordination squares of Pd1 and paired Pd2 polyhedra, which create a zig-zag composition.
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Manhattan Community Board 7 members blamed Trump for failing to build the proposed enhancement and monument at Freedom Place. However, the Riverside South Planning Corporation said that the Freedom Place plan was merely a concept for an arts program that was not included in the final project. A street called Freedom Place South, along the same axis as Freedom Place, runs southward from 64th Street to 59th Street, where the historic, full-block IRT Powerhouse and Riverside Center building are located. Adjacent to the Powerhouse, and visible from Freedom Place South, is a new tetrahedron-shaped building at 625 West 57th Street, also known as VIA 57 West.
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Douglas W. Phillipson and Kenneth L. Rinehart Jr. (1983): "Antifungal peroxide-containing acids from two Caribbean sponges". Journal of the American Chemical Association, volume 105, issue 26, pages 7735–7736. Peng Dai, Tony K. Trullinger, Xuejun Liu, and Patrick H. Dussault (2006): "Asymmetric synthesis of 1,2-dioxolane-3-acetic acids: Synthesis and configurational cssignment of plakinic acid A". Journal of Organic Chemistry, volume 71, issue 6, pages 2283–2292. A. J. Bloodworth, Brian D. Bothwell, Andrew N. Collins, and Nicola L. Maidwell (1996): "A short synthesis of naturally occurring and other analogues of plakinic acids that contain the 1,2-dioxolane group". Tetrahedron Letters, volume 37, issue 11, pages 1885-1888.
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Curran has held various positions in the American Chemical Society (ACS) Division of Organic Chemistry hierarchy, culminating as Chair of the Division in 2000. He was an Associate Editor of Organic Reactions from 1991–2001 and Editor of Tetrahedron Letters from 1995–2001. He is a member of Organic Syntheses, Inc and edited Volume 83 in 2006. He was a member of the Executive Committee of the symposium "Gomberg•2000, A Century of Radical Chemistry",Gomberg•2000, A Century of Radical Chemistry and he spearheaded the presentation on June 25, 2000 of an ACS National Chemical Landmark to the University of Michigan for Gomberg's 1900 discovery of organic free radicals.
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The Herschel graph is named after British astronomer Alexander Stewart Herschel, who wrote an early paper concerning William Rowan Hamilton's icosian game: the Herschel graph describes the smallest convex polyhedron for which this game has no solution. However, Herschel's paper described solutions for the Icosian game only on the graphs of the regular tetrahedron and regular icosahedron; it did not describe the Herschel graph.. The name "the Herschel graph" makes an early appearance in a graph theory textbook by John Adrian Bondy and U. S. R. Murty, published in 1976. However, the graph itself was described earlier, for instance by H. S. M. Coxeter.
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In 2007, Kiessling was inducted as a member of the United States National Academy of Sciences, and in 2017, she received the Tetrahedron Prize for Creativity in Organic Chemistry. Kiessling is a Fellow of the American Association for the Advancement of Sciences and the American Chemical Society, as well as an elected member of the American Academy of Arts and Sciences, American Philosophical Society, and American Academy of Microbiology. Since 2005, she has served as the founding editor-in-chief of ACS Chemical Biology. Kiessling is also the cofounder of Quintessence Biosciences, a company that is working to translate her technological advances into cures for various diseases.
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Inspired by the work of Tukey, Gunther explored ways to help systems analyst visualize performance in a manner similar to that already available in scientific visualization and information visualization. In 1991, he developed a tool called Barry, which employs barycentric coordinates to visualize sampled CPU usage data on large-scale multiprocessor systems. More recently, he has applied the same 2-simplex barycentric coordinates to visualizing the Apdex application performance metric, which is based on categorical response time data. A barycentric 3-simplex (a tetrahedron), that can be swivelled on the computer screen using a mouse, has been found useful for visualizing packet network performance data.
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In addition to his contributions to the reports of the Ordnance Survey, Clarke also published a small number of papers in the learned journals. The papers listed in the Bibliography are almost all related to some aspect of geodesy, the only exceptions being a student publication entitled Propositions on the Tetrahedron and a short paper on Just Intonation which reflects his interest in music. Clarke also contributed two important articles to the ninth edition of the Encyclopædia Britannica (1878), one on Geodesy and a second on the Figure of the Earth (with the title of the latter being Earth, Figure of the). These articles appear unchanged in the tenth edition (1903).
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A polyhedron P is said to have the Rupert property if a polyhedron of the same or larger size and the same shape as P can pass through a hole in P. All five Platonic solids: the cube, the regular tetrahedron, regular octahedron, regular dodecahedron, and regular icosahedron, have the Rupert property. It has been conjectured that all 3-dimensional convex polyhedra have this property. For n greater than 2, the n-dimensional hypercube also has the Rupert property. Of the 13 Archimedean solids, it is known that these nine have the Rupert property: the cuboctahedron, truncated octahedron, truncated cube, rhombicuboctahedron, icosidodecahedron, truncated cuboctahedron, truncated icosahedron, truncated dodecahedron.
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Let's begin by considering the 3rd line of Pascal's triangle, with values 1, 3, 3, 1. A 2-dimensional triangle has one 2-dimensional element (itself), three 1-dimensional elements (lines, or edges), and three 0-dimensional elements (vertices, or corners). The meaning of the final number (1) is more difficult to explain (but see below). Continuing with our example, a tetrahedron has one 3-dimensional element (itself), four 2-dimensional elements (faces), six 1-dimensional elements (edges), and four 0-dimensional elements (vertices). Adding the final 1 again, these values correspond to the 4th row of the triangle (1, 4, 6, 4, 1).
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Each carbon atom in a diamond is covalently bonded to four other carbons in a tetrahedron. These tetrahedrons together form a 3-dimensional network of six-membered carbon rings (similar to cyclohexane), in the chair conformation, allowing for zero bond angle strain. This stable network of covalent bonds and hexagonal rings is the reason that diamond is so strong. Although graphite is the most stable allotrope of carbon under standard laboratory conditions (273 or 298 K, 1 atm), a recent computational study indicated that under idealized conditions (T = 0, p = 0), diamond is the most stable allotrope by 1.1 kJ/mol compared to graphite.
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The crystal and molecular structure was elucidated by X-ray crystallography. The X-ray diffraction study confirmed the tricyclic ring structure and gave insight toward the geometry of the complex. With a tin-nitrogen distance of 2.624 Å, the formal bond order was calculated to be about 0.46. The presence of the tin-nitrogen interaction, albeit weaker than anticipated, led to a few key discoveries: (1) the distortion from ideal trigonal bipyramidal toward monocapped tetrahedron geometry; (2) the lengthening of the apical tin-methyl bond by ~ 0.1 Å (largest known value for any existing tetraorganotin compounds); (3) the observation of unusual hybridization at the apical tin-methyl bond.
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Heilbronner, E.; Plattner, P. A.; Wieland, K. Rearrangement of Azulene to Naphthalene. Experientia 1947, 3, 70–71. Since then, many other isomerizations have been recorded, however the rearrangement of azulene to naphthalene has received the most attention. Likewise, since the characterization of the automerization of naphthalene by Scott in 1977, similar atom scramblings of other aromatic hydrocarbons such as pyrene,Scott, L. T.; Kirms, M. A.; Berg, A.; Hansen, P. E. Automerization of Pyrene a Test for the Mechanism of Naphthalene Automerization. Tetrahedron Letters 1982, 23 (18), 1859–1862. DOI: 10.1016/S0040-4039(00)87204-4 azulene, benz[a]anthraceneScott, L. T.; Tsang, T.-H.
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Ball-and-stick models of the two rotamers of ethane Ethane forms the simplest case for studying the conformation of alkanes, as there is only one C–C bond. If one looks down the axis of the C–C bond, one will see the so-called Newman projection. The hydrogen atoms on both the front and rear carbon atoms have an angle of 120° between them, resulting from the projection of the base of the tetrahedron onto a flat plane. However, the torsion angle between a given hydrogen atom attached to the front carbon and a given hydrogen atom attached to the rear carbon can vary freely between 0° and 360°.
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C and P. Carbon and phosphorus represent an example of a less-well known diagonal relationship, especially in organic chemistry. "Spectacular" evidence of this relationship was provided in 1987 with the synthesis of a ferrocene-like molecule in which six of the carbon atoms were replaced by phosphorus atoms.Rayner-Canham 2011, p. 126 Further illustrating the theme is the "extraordinary" similarity between low coordinate phosphorus compounds and unsaturated carbon compounds, and related research into organophosphorus chemistry.Dillon, Mathey & Nixon 1998 In 2020, the first compound containing three carbon atoms and one phosphorus arranged in a tetrahedron, tri-tert-butyl phosphatetrahedrane, (PC3)(C4H9)3 was synthesised.
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It was first published by Dider Villemin in 1980 describing the synthesis of an Exaltolide precursor,Villemin, D. (1980). “Synthese de Macrolides par Metathese”. Tetrahedron Lett. 21 (18): 1715-1718. . and later become popularized by Robert H. Grubbs and Richard R. Schrock, who shared the Nobel Prize in Chemistry, along with Yves Chauvin, in 2005 for their combined work in olefin metathesis.Grubbs, R. H. (2006). “Olefin-Metathesis Catalysts for the Preparation of Molecules and Materials (Nobel Lecture)”. Angew. Chem. Int. Ed. 45 (23): 3760–3765. .Schrock, R. R. (2006). “Multiple Metal–Carbon Bonds for Catalytic Metathesis Reactions (Nobel Lecture)”. Angew. Chem. Int. Ed. 45 (23), 3748-3759. .
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Supporting this theory is the fact that the three artifacts of the Guardian, first seen in Ultima VII: The Black Gate, are the sphere, the cube, and the tetrahedron, which together are an unmistakable representation of the EA logo during the early 1990s when EA's acquisition of Origin was complete and the games were developed. Furthermore, two members of the Fellowship in The Black Gate, Elizabeth and Abraham (E&A;) provide dubious guidance for the Avatar throughout the game and are ultimately unmasked as evildoers. The Guardian is described as the "Destroyer of Worlds", a possible reference to Origin's tagline which read "We create worlds".
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A 3-simplex, with barycentric subdivisions of 1-faces (edges) 2-faces (triangles) and 3-faces (body). In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or barycenter) of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex.
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A tea bag is being removed from a mug of hot tea to complete the brewing process Tetrahedron-shaped tea bags made of polylactide (PLA), a bioplastic, shown here containing dried peppermint leaves Three different teas in tea bags A tea bag is a small, porous, sealed bag or packet, typically containing tea leaves, which is immersed in water to steep and make an infusion. Originally used only for tea (Camellia sinensis), they are now made with other tisanes as well. Tea bags are commonly made of filter paper or food-grade plastic, or occasionally of silk cotton or silk. The tea bag performs the same function as a tea infuser.
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Heat of vapourisation at the boiling point is 3.85 kcal/mol. The F3NO molecule has C3V symmetry, with all the N-F bonds being equivalent. The shape is almost a tetrahedron as N-O bond is similar to the N-F bonds in nature. The nuclear magnetic resonance (NMR) spectrum of 19F has a triplet line around −363 ppm. JNF is 136 Hz. The infra red spectrum N-O stretch at 1687 cm−1, N-F stretch at 743 cm−1, unsymmetrical N-F stretch 887 cm−1 ∠ONF bend 528 cm−1, wither other bands at 558, 528, 801, 929, 1055, 1410, 1622, 1772, 2435, and 3345 cm−1.
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Stereocontrol for cyclohexane rings is well established in organic chemistry, in large part due to the axial/equatorial preferential positioning of substituents on the ring. Macrocyclic stereocontrol models the substitution and reactions of medium and large rings in organic chemistry, with remote stereogenic elements providing enough conformational influence to direct the outcome of a reaction. Early assumptions towards macrocycles in synthetic chemistry considered them far too floppy to provide any degree of stereochemical or regiochemical control in a reaction. The experiments of W. Clark Still in the late 1970s and 1980s challenged this assumption,Still, W. C.; Galynker, I. Tetrahedron 1981, 37, 3981-3996.
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The whale eventually devours the Planet Express ship and swallows the crew alive (with the exception of Zoidberg, who again returns to Earth in an escape pod). Inside the whale, Leela encounters the captain of the first crew, Lando Tucker, fused to the inside of the whale's belly. Lando explains that the whale feeds on obsession, luring obsessive space captains into the Tetrahedron to feed itself. Leela is fused to the whale's belly like Lando, but her overwhelming obsession with completing her delivery grants her control over the whale, allowing her to bring it to the Planet Express building in time for the memorial.
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Tetrahedron with insphere in red (also midsphere in green, circumsphere in blue) In his 1597 book Mysterium Cosmographicum, Kepler modelled of the solar system with its then known six planets' orbits by nested platonic solids, each circumscribed and inscribed by a sphere. In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual to the dual polyhedron's circumsphere. The radius of the sphere inscribed in a polyhedron P is called the inradius of P.
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Removing certain triples of vertices from the triakis tetrahedron separates the remaining vertices into multiple connected components. When no such three- vertex separation exists, a polyhedron is said to be 4-connected. Every 4-connected polyhedron has a representation as an ideal polyhedron; for instance this is true of the tetrakis hexahedron, another Catalan solid. Truncating a single vertex from a cube produces a simple polyhedron (one with three edges per vertex) that cannot be realized as an ideal polyhedron: by Miquel's six circles theorem, if seven of the eight vertices of a cube are ideal, the eighth vertex is also ideal, and so the vertices created by truncating it cannot be ideal.
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Zwittermicin A is an antibiotic that has been identified from the bacterium Bacillus cereus UW85.Haiyin, He (April, 1994) "Zwittermicin A, an Antifungal and Plant Protection Agent from Bacillus cereus", Tetrahedron Letters 35 (16) 2499-2502 doi=10.1016/S0040-4039(00)77154-1 It is a molecule of interest to agricultural industry because it has the potential to suppress plant disease due to its broad spectrum activity against certain gram positive and gram negative prokaryotic micro-organisms. The molecule is also of interest from a metabolic perspective because it represents a new structural class of antibiotic and suggests a crossover between polyketide and non-ribosomal peptide biosynthetic pathways. Zwittermicin A is linear aminopolyol.
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Researchers have synthesized many three-dimensional DNA complexes that each have the connectivity of a polyhedron, such as a cube or octahedron, meaning that the DNA duplexes trace the edges of a polyhedron with a DNA junction at each vertex.Overview: The earliest demonstrations of DNA polyhedra were very work-intensive, requiring multiple ligations and solid-phase synthesis steps to create catenated polyhedra.DNA polyhedra: Subsequent work yielded polyhedra whose synthesis was much easier. These include a DNA octahedron made from a long single strand designed to fold into the correct conformation,DNA polyhedra: and a tetrahedron that can be produced from four DNA strands in one step, pictured at the top of this article.
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In a more general description the RC reaction is any coupling of one active alkene / latent enolate to a second Michael acceptor, creating a new C–C bond between the alpha-position of one activated alkene and the beta-position of a second alkene under the influence of a nucleophilic catalyst.The Rauhut–Currier reaction: a history and its synthetic application Carrie E. Aroyan, Alpay Dermenci and Scott J. Miller Tetrahedron Volume 65, Issue 21, 23 May 2009, pp. 4069–84 The reaction mechanism is essentially that of the related and better known Baylis–Hillman reaction (DABCO not phosphine, carbonyl not enone) but the Rauhut–Currier reaction actually predates it by several years.
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The Nazarov cyclization reaction is a named electrocyclic reaction converting divinylketones to cyclopentenones. A classic example is the thermal ring-opening reaction of 3,4-dimethylcyclobutene. The cis isomer exclusively yields cis,trans-hexa-2,4-diene whereas the trans isomer gives the trans,trans diene:The preparation and isomerization of - and -3,4-dimethylcyclobutene. Tetrahedron Letters, Volume 6, Issue 17, 1965, Pages 1207-1212 Rudolph Ernst K. Winter Dimethylcyclobutene isomerization This reaction course can be explained in a simple analysis through the frontier- orbital method: the sigma bond in the reactant will open in such a way that the resulting p-orbitals will have the same symmetry as the HOMO of the product (a hexadiene).
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Using a regular triangle and the factor 1/2 will result in the Sierpinski triangle, while creating the proper arrangement with four points and a factor 1/2 will create a display of a "Sierpinski Tetrahedron", the three-dimensional analogue of the Sierpinski triangle. As the number of points is increased to a number N, the arrangement forms a corresponding (N-1)-dimensional Sierpinski Simplex. The term has been generalized to refer to a method of generating the attractor, or the fixed point, of any iterated function system (IFS). Starting with any point x0, successive iterations are formed as xk+1 = fr(xk), where fr is a member of the given IFS randomly selected for each iteration.
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PG(3,2) can be represented as a tetrahedron. The 15 points correspond to the 4 vertices + 6 edge-midpoints + 4 face-centers + 1 body-center. The 35 lines correspond to the 6 edges + 12 face-medians + 4 face-incircles + 4 altitudes from a face to the opposite vertex + 3 lines connecting the midpoints of opposite edges + 6 ellipses connecting each edge midpoint with its two non-neighboring face centers. The 15 planes consist of the 4 faces + the 6 "medial" planes connecting each edge to the midpoint of the opposite edge + 4 "cones" connecting each vertex to the incircle of the opposite face + one "sphere" with the 6 edge centers and the body center.
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His original work is lost and his solids come down to us through Pappus. ;China Cubical gaming dice in China have been dated back as early as 600 B.C. By 236 AD, Liu Hui was describing the dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids, using assemblages of these solids as the basis for calculating volumes of earth to be moved during engineering excavations. ;Islamic civilisation After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see Mathematics in medieval Islam). The 9th century scholar Thabit ibn Qurra gave formulae for calculating the volumes of polyhedra such as truncated pyramids.
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The edges of the truncated tetrahedron form a vertex- transitive graph (also a Cayley graph) which is not symmetric. Finite vertex- transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected cycles) are also vertex- transitive, as are the vertices and edges of the Archimedean solids (though only two of these are symmetric). Potočnik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices.. Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs.
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A lens-shaped Δ-biangle rotating in an equilateral triangle Δ curves (delta curves), the simplest of which is the circle, are curves which can be rotated in the equilateral triangle, have many similar properties to curves of constant width. Delta curves of height h all share the same perimeter, . The generalization of the definition of bodies of constant width to convex bodies in R³ and their boundaries leads to the concept of surface of constant width (in the case of a Reuleaux triangle, this does not lead to a Reuleaux tetrahedron, but to Meissner bodies). There is also the concept of space curves of constant width, whose widths are defined by tangent planes.
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An angle consists of a point O (the vertex) and two non- collinear rays out from O (the sides). A triangle is given by three non- collinear points (called vertices) and their three segments AB, BC, and CA. If three points A, B, and C are non-collinear, then a plane ABC is the set of all points collinear with pairs of points on one or two of the sides of triangle ABC. If four points A, B, C, and D are non-coplanar, then a space (3-space) ABCD is the set of all points collinear with pairs of points selected from any of the four faces (planar regions) of the tetrahedron ABCD.
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A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles. The line graph of the complete graph Kn is also known as the triangular graph, the Johnson graph J(n, 2), or the complement of the Kneser graph KGn,2. Triangular graphs are characterized by their spectra, except for n = 8.. See in particular Proposition 8, p. 262. They may also be characterized (again with the exception of K8) as the strongly regular graphs with parameters srg(n(n − 1)/2, 2(n − 2), n − 2, 4).
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In most commonly encountered silicates, including almost all silicate minerals found in the Earth's crust, each silicon atom occupies the center of an idealized tetrahedron whose corners are four oxygen atoms, connected to it by single covalent bonds according to the octet rule. These tetrahedra may occur as isolated orthosilicate anions , but two or more silicon atoms can be joined with oxygen atoms in various ways, to form more complex anions, such as pyrosilicate or the metasilicate ring hexamer . Polymeric silicate anions of arbitrarily large sizes can have chain, double chain, sheet, or three- dimensional structures. Typically, each oxygen atom that does not contribute a negative charge to the anion is a bridge between two silicon atoms.
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Although the tetrahedron is the common coordination geometry for silicon compounds, silicon may also occur with higher coordination numbers. For example, in the anion hexafluorosilicate , the silicon atom is surrounded by six fluorine atoms in an octahedral arrangement. This structure is also seen in the hexahydroxysilicate anion that occurs in thaumasite, a mineral found rarely in nature but sometimes observed among other calcium silicate hydrates artificially formed in cement and concrete submitted to a severe sulfate attack. At very high pressure, even SiO2 adopts the six-coordinated octahedral geometry in the mineral stishovite, a dense polymorph of silica found in the lower mantle of the Earth and also formed by shock during meteorite impacts.
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Chemie Int. Ed. 2006, 45, 3484. This increases the rate of productive reaction and thus subdues unwanted decomposition pathways such as protodeboronation. Cross- coupling reactions have also been accelerated with metal additives such as silverChen, J.; Cammers-Goodwin, A. "2-(Fluorophenyl)pyridines by the Suzuki- Miyaura method: Ag2O accelerates coupling over undesired ipso substitution (SNAr) of fluorine" Tetrahedron Lett. 2003, 44, 1503.Nishihara, Y.; Onodera, H.; Osakada, K. "Synthesis and structural characterization of the first unsymmetrical from transmetallation between 2,4,6-trifluorophenylboronic acid and trans-Pd(C6F5)I(PEt3)2" Chem. Commun. 2004, 192. Korenaga, T.; Kosaki, T.; Fukumura, R.; Ema, T.; Sakai, T. "Suzuki-Miyaura coupling reaction using pentafluorophenylboronic acid". Org. Lett. 2005, 7, 4915.
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The spin ice model is only one subdivision of frustrated systems. The word frustration was initially introduced to describe a system's inability to simultaneously minimize the competing interaction energy between its components. In general frustration is caused either by competing interactions due to site disorder (see also the Villain model or by lattice structure such as in the triangular, face-centered cubic (fcc), hexagonal-close-packed, tetrahedron, pyrochlore and kagome lattices with antiferromagnetic interaction. So frustration is divided into two categories: the first corresponds to the spin glass, which has both disorder in structure and frustration in spin; the second is the geometrical frustration with an ordered lattice structure and frustration of spin.
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Before long, the human colony police come after Nicole's family in "New York" and they flee to the octospider city. After the human colony leader starts bombing the octospider city under a made-up pretext and the octospiders retaliate, the situation becomes dire enough that Rama's controlling intelligence intervenes to end the conflict caused by the humans aboard, by sending everybody into hibernation until the end of the journey, except for the surviving myrmicats which are allowed to lead their daily life. The Rama spacecraft rendezvous with another Node, an enormous tetrahedron near the star Tau Ceti, designed to research any intelligent life capable of spaceflight. The humans are divided into two groups based mainly on the degree of xenophobia they had exhibited during the journey.
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The mineral structure appears to be very similar to that of afghanite, except for the presents of strong H2O bonds. The two minerals contains the same tetrahedron structure, and alloriite appears to follow the typical ABABACAC sequencing layer chain that is found in cancrinite grouped minerals. The A is Na, the B is H2 O, where in afghanite its A is Ca and the B is Cl. Alloriite's crystal structure was found using a single crystal X-ray diffraction, which allowed mineralogists to classify it as a P31c for its point group, making it trigonal crystal structure. In relation to afghanite's and alloriite's physical properties, they both have a trigonal crystal structure, P31c space group, and then have very similar formulas.
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The (half) hypercone may be constructed in a manner analogous to the construction of a 3D cone. A 3D cone may be thought of as the result of stacking progressively smaller discs on top of each other until they taper to a point. Alternatively, a 3D cone may be regarded as the volume swept out by an upright isosceles triangle as it rotates about its base. A 4D hypercone may be constructed analogously: by stacking progressively smaller balls on top of each other in the 4th direction until they taper to a point, or taking the hypervolume swept out by a tetrahedron standing upright in the 4th direction as it rotates freely about its base in the 3D hyperplane on which it rests.
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The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells, and the fact that the tetrahedron has no opposing faces or vertices. One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron, which with some effort can be seen in most of the below perspective projections. A three-dimensional model of the 600-cell, in the collection of the Institut Henri Poincaré, was photographed in 1934–1935 by Man Ray, and formed part of two of his later "Shakesperean Equation" paintings.. See in particular mathematical object mo-6.2, p.
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His accolades include: Fellow of the American Academy of Arts and Sciences (1966); Member of the National Academy of Sciences (1983); Honorary Fellow, Balliol College, Oxford (1974); Tilden Medal of the Chemical Society (1954); Harrison Howe Award of the American Chemical Society (1961); Robert Robinson Medal, Chemical Society (1974); G.W. Wheland Medal of the University of Chicago (1976); Evans Award, The Ohio State University (1977); Southwest Regional Award of the American Chemical Society (1978); Davy Medal (1982); James Flack Norris Award of the American Chemical Society (1984); William H. Nichols Award of the American Chemical Society (1986); Auburn-G. M. Kosolapoff Award of the American Chemical Society (1988); Tetrahedron Prize for Creativity in Organic Chemistry (1989); WATOC Medal (World Association of Theoretical Organic Chemists Meda), (1990).
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It is also possible to define geodesics on some surfaces that are not smooth everywhere, such as convex polyhedra. The surface of a convex polyhedron has a metric that is locally Euclidean except at the vertices of the polyhedron, and a curve that avoids the vertices is a geodesic if it follows straight line segments within each face of the polyhedron and stays straight across each polyhedron edge that it crosses. Although some polyhedra have simple closed geodesics (for instance, the regular tetrahedron and disphenoids have infinitely many closed geodesics, all simple).. others do not. In particular, a simple closed geodesic of a convex polyhedron would necessarily bisect the total angular defect of the vertices, and almost all polyhedra do not have such bisectors.
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In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v1 and v2 of G, there is some automorphism :f\colon G \to G\ such that :f(v_1) = v_2.\ In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices.. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph).
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Pauling noted that the number of configurations conforming to this "two-near, two-far" ice rule grows exponentially with the system size, and, therefore, that the zero-temperature entropy of ice was expected to be extensive. Pauling's findings were confirmed by specific heat measurements, though pure crystals of water ice are particularly hard to create. left Spin ices are materials that consist of regular corner-linked tetrahedra of magnetic ions, each of which has a non- zero magnetic moment, often abridged to "spin", which must satisfy in their low-energy state a "two-in, two-out" rule on each tetrahedron making the crystalline structure (see figure 2). This is highly analogous to the two- near, two far rule in water ice (see figure 1).
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The ligand consists of two 2,5-alkyl-substituted phospholane rings connected by a 1,2-phenyl bridge. The alkyl group can be methyl, ethyl, propyl, or isopropyl. In the closely related bis(dimethylphospholano)ethane or BPE ligand New electron-rich chiral phosphines for asymmetric catalysis Mark J. Burk, John E. Feaster, Richard L. Harlow Organometallics, 1990, 9 (10), pp 2653–2655 New chiral phospholanes; Synthesis, characterization, and use in asymmetric hydrogenation reactions Tetrahedron: Asymmetry, Volume 2, Issue 7, 1991, Pages 569-592 Mark J. Burk, John E. Feaster, Richard L. Harlow the o-phenylene bridge is replaced by a 1,2-ethylene bridge. Both compounds can be obtained from the corresponding chiral diol through conversion to the cyclic sulfate and reaction with lithiated phenylbisphosphine.
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Glotzer made fundamental contributions to the field of the glass transition, for which the molecular dynamics simulation of Lennard-Jones particles exhibiting dynamical heterogeneity in the form of string-like motion in a 3D-liquid is of particular significance. In addition, her paper together with Michael J. Solomon on anisotropy dimensions of patchy particles has become a classic work, inspiring research directions of groups around the world. Glotzer and collaborators also hold the record for the densest tetrahedron packing and discovered that hard tetrahedrons can self-assemble into a dodecagonal quasicrystal. Glotzer and collaborators coined the term ‘Directional Entropic Forces’ in 2011 to denote the effective interaction that drives anisotropic hard particles to align their facets during self-assembly and/or crystallization.
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E.J. Corey has received more than 40 major awards including the Linus Pauling Award (1973), Franklin Medal (1978), Tetrahedron Prize (1983), Wolf Prize in Chemistry (1986), National Medal of Science (1988), Japan Prize (1989), Nobel Prize in Chemistry (1990), Golden Plate Award of the American Academy of Achievement (1991), Roger Adams Award (1993), and the Priestley Medal (2004).See the E.J. Corey, About E.J. Corey, Major Awards tab He was inducted into the Alpha Chi Sigma Hall of Fame in 1998. As of 2008, he has been awarded 19 honorary degrees from universities around the world including Oxford University (UK), Cambridge University (UK), and National Chung Cheng University.See the E.J. Corey, About E.J. Corey, Honorary Degrees tab In 2013, the E.J. Corey Institute of Biomedical Research (CIBR) opened in Jiangyin, Jiangsu Province, China.
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A cycle basis is called weakly fundamental if its cycles can be placed into a linear ordering such that each cycle includes at least one edge that is not included in any earlier cycle. A fundamental cycle basis is automatically weakly fundamental (for any edge ordering).. If every cycle basis of a graph is weakly fundamental, the same is true for every minor of the graph. Based on this property, the class of graphs (and multigraphs) for which every cycle basis is weakly fundamental can be characterized by five forbidden minors: the graph of the square pyramid, the multigraph formed by doubling all edges of a four-vertex cycle, two multigraphs formed by doubling two edges of a tetrahedron, and the multigraph formed by tripling the edges of a triangle..
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Meissner and Schilling showed how to modify the Reuleaux tetrahedron to form a surface of constant width, by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. According to which three edge arcs are replaced (three that have a common vertex or three that form a triangle) there result two noncongruent shapes that are sometimes called Meissner bodies or Meissner tetrahedra. Bonnesen and Fenchel conjectured that Meissner tetrahedra are the minimum-volume three- dimensional shapes of constant width, a conjecture which is still open. In connection with this problem, Campi, Colesanti and Gronchi showed that the minimum volume surface of revolution with constant width is the surface of revolution of a Reuleaux triangle through one of its symmetry axes.
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The features of the cube and its dual octahedron correspond one-for-one with dimensions reversed. There are many distinct but interrelated dualities in which geometric or topological objects correspond to other objects of the same type, but with a reversal of the dimensions of the features of the objects. A classical example of this is the duality of the platonic solids, in which the cube and the octahedron form a dual pair, the dodecahedron and the icosahedron form a dual pair, and the tetrahedron is self-dual. The dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the vertices of the dual correspond one-for-one with the faces of the primal.
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The structure floats with its arms open wide. They can be pulled shut by adding a fourth strand of DNA (D) "programmed" to stick to both of the dangling, unpaired sections of strands B and C. The closing of the tweezers was proven by tagging strand A at either end with light-emitting molecules that do not emit light when they are close together. To re-open the tweezers add a further strand (E) with the right sequence to pair up with strand D. Once paired up, they have no connection to the machine BAC, so float away. The DNA machine can be opened and closed repeatedly by cycling between strands D and E. These tweezers can be used for removing drugs from inside fullerenes as well as from a self assembled DNA tetrahedron.
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Bubbles can be effectively used to teach and explore a wide variety of concepts to even young children. Flexibility, colour formation, reflective or mirrored surfaces, concave and convex surfaces, transparency, a variety of shapes (circle, square, triangle, sphere, cube, tetrahedron, hexagon), elastic properties, and comparative sizing, as well as the more esoteric properties of bubbles listed on this page. Bubbles are useful in teaching concepts starting from 2 years old and into college years. A Swiss university professor, Dr. Natalie Hartzell, has theorized that usage of artificial bubbles for entertainment purposes of young children has shown a positive effect in the region of the child's brain that controls motor skills and is responsible for coordination with children exposed to bubbles at a young age showing measurably better motion skills than those who were not.
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The molecules are made by an expanding supersonic jet of neon gas. The neon dimer has an average distance of 3.3 Å between atoms. The neon trimer is shaped approximately like an equilateral triangle with sides 3.3 Å long. However the shape is floppy and isosceles triangle shapes are also common. The first excited state of the neon trimer is 2 meV above the ground state. The neon tetramer takes the form of a tetrahedron with sides around 3.2 Å. Van der Waals molecules with metals include LiNe. More Van der Waals molecules include CF4Ne and CCl4Ne, Ne2Cl2, Ne3Cl2, I2Ne, I2Ne2, I2Ne3, I2Ne4, I2NexHey (x=1-5, y=1-4). Van der Waals molecules formed with organic molecules in gas include aniline, dimethyl ether, 1,1-difluoroethylene, pyrimidine, chlorobenzene, cyclopentanone, cyanocyclobutane, and cyclopentadienyl.
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The Tsuji–Trost reaction (also called the Trost allylic alkylation or allylic alkylation) is a palladium-catalysed substitution reaction involving a substrate that contains a leaving group in an allylic position. The palladium catalyst first coordinates with the allyl group and then undergoes oxidative addition, forming the -allyl complex. This allyl complex can then be attacked by a nucleophile, resulting in the substituted product.Strategic Applications of Named Reactions in Organic Synthesis (Paperback) by Laszlo Kurti, Barbara Czako Tsuji–Trost allylation This work was first pioneered by Jiro Tsuji in 1965Organic syntheses by means of noble metal compounds XVII. Reaction of -allylpalladium chloride with nucleophiles Tetrahedron Letters, Volume 6, Issue 49, 1965, Pages 4387-4388 Jiro Tsuji, Hidetaka Takahashi, Masanobu Morikawa and, later, adapted by Barry Trost in 1973 with the introduction of phosphine ligands.
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The Reye configuration can be realized in three-dimensional projective space by taking the lines to be the 12 edges and four long diagonals of a cube, and the points as the eight vertices of the cube, its center, and the three points where groups of four parallel cube edges meet the plane at infinity. Two regular tetrahedra may be inscribed within a cube, forming a stella octangula; these two tetrahedra are perspective figures to each other in four different ways, and the other four points of the configuration are their centers of perspectivity. These two tetrahedra together with the tetrahedron of the remaining 4 points form a desmic system of three tetrahedra. Any two disjoint spheres in three dimensional space, with different radii, have two bitangent double cones, the apexes of which are called the centers of similitude.
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The cube and regular octahedron are dual graphs of each other According to Steinitz's theorem, every polyhedral graph (the graph formed by the vertices and edges of a three- dimensional convex polyhedron) must be planar and 3-vertex-connected, and every 3-vertex-connected planar graph comes from a convex polyhedron in this way. Every three-dimensional convex polyhedron has a dual polyhedron; the dual polyhedron has a vertex for every face of the original polyhedron, with two dual vertices adjacent whenever the corresponding two faces share an edge. Whenever two polyhedra are dual, their graphs are also dual. For instance the Platonic solids come in dual pairs, with the octahedron dual to the cube, the dodecahedron dual to the icosahedron, and the tetrahedron dual to itself.. Polyhedron duality can also be extended to duality of higher dimensional polytopes,.
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Structural models prior to this work had been two-dimensional, and van 't Hoff and Le Bel theorized that the arrangement of groups around this tetrahedron could dictate the optical activity of the resulting compound through what became known as the Le Bel–van 't Hoff rule. Marckwald's brucine- catalyzed enantioselective decarboxylation of 2-ethyl-2-methylmalonic acid, resulting in a slight excess of the levorotary form of the 2-methylbutyric acid product. In 1894 Hermann Emil Fischer outlined the concept of asymmetric induction; in which he correctly ascribed selective the formation of D-glucose by plants to be due to the influence of optically active substances within chlorophyll. Fischer also successfully performed what would now be regarded as the first example of enantioselective synthesis, by enantioselectively elongating sugars via a process which would eventually become the Kiliani–Fischer synthesis.
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Hibarimicinone is an organic atropisomeric small molecule, derived from hibarimicins, which are isolated from the microbial culture strain of Microbispora rosea from the Hibari region of Japan.Hori, H.; Higashi, K.; Ishiyama, T.; Uramoto, M.; Uehara, Y.; Oki, T. Tetrahedron Lett. 1996 37, 2785-2788Kajiura, T.; Furumai, T.; Igarashi, Y.; Hori, H.; Higashi, K.; Ishiyama, T.; Uramoto, M.; Uehara, Y.; Oki, T. J. Antibiot. 1998, 51, 394-401Hori, H.; Kajiura, T.; Igarashi, Y.; Hori, H.; Higashi, K; Ishiyama, T.; Uramoto, M; Uehara, Y.; Oki, T. J. Antibiot. 2002, 55, 53-60Hori, H.; Kajiura, T.; Igarashi, Y.; Furumai, T; Higashi, K.; Ishiyama, T; Uramoto, M; Uehara, Y; Oki, T. J. Antibiot., 2002, 55, 46-52 Analysis of the bacteria identified a new class of molecule containing a dimeric-tetracyclic polyketide backbone, which are now known as the hibarimicins.
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Aristotle added a fifth element, aithēr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.Wildberg (1988): Wildberg discusses the correspondence of the Platonic solids with elements in Timaeus but notes that this correspondence appears to have been forgotten in Epinomis, which he calls "a long step towards Aristotle's theory", and he points out that Aristotle's ether is above the other four elements rather than on an equal footing with them, making the correspondence less apposite. Euclid completely mathematically described the Platonic solids in the Elements, the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order.
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The first examples of the HDDA reaction were reported independently in 1997 by the groups of Ueda and Johnson.Bradley, A. Z.; Johnson, R. P. J. Am. Chem. Soc., 1997, 119, 9917 Cahill, K. J.; Ajaz, A.; Johnson, R. P. Aust. J. Chem., 2010, 63, 1007 Miyawaki, K.; Suzuki, R.; Kawano, T.; Ueda, I. Tetrahedron Lett., 1997, 38, 3943 Johnson and co-workers observed the cyclization of 1,3,8-nonatriyne under flash vacuum thermolysis (600 °C, 10−2 torr) to form two products, indane and the dehydrogenation product indene, in 95% combined yield. Deuterium labeling studies suggested that the product was formed by a [4+2] cycloaddition to a benzyne intermediate, followed by in-situ reduction to form the observed products. Ueda and co-workers observed that acyclic tetraynes cyclized at room temperature to form 5H-fluorenol derivatives.
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In VSEPR theory the electron pairs on the oxygen atom in water form the vertices of a tetrahedron with the lone pairs on two of the four vertices. The H–O–H bond angle is 104.5°, less than the 109° predicted for a tetrahedral angle, and this can be explained by a repulsive interaction between the lone pairs.Organic Chemistry Marye Anne Fox, James K. Whitesell 2nd Edition 2001Organic chemistry John McMurry 5th edition 2000Concise Inorganic Chemistry J.D. Lee 4th Edition 1991 Various computational criteria for the presence of lone pairs have been proposed. While electron density ρ(r) itself generally does not provide useful guidance in this regard, the laplacian of the electron density is revealing, and one criterion for the location of the lone pair is where L(r) = –∇2ρ(r) is a local maximum.
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Light Up is 20 feet and 6 inches in height and most noted for it being a bright shade of yellow; a departure from the sculptor's more numerous black works. It combines a tetrahedron and octahedron into what Smith described as a "continuous space grid" which "may be seen as interruptions in an otherwise unbroken flow of space." In its public settings it has been seen to be a stark comparison to the surrounding colors and shapes of its neighboring architecture, with one writer describing it in comparison to an adjacent Ludwig Mies van der Rohe skyscraper in its temporary New York retrospective setting as being "a jazzy counterpoint to its staid geometry." For this reason some posit Smith's intention, for its original setting, was to lighten up its surroundings and perhaps the attitudes of passers-by.
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Seshadri, who was a member of the scientific advisory committee to the Government of India and UNESCO, served as an expert advisor to several government agencies such as the Council of Scientific and Industrial Research, Indian Council of Medical Research, Indian Council of Agricultural Research and Department of Atomic Energy and chaired many expert committees related to education, health and science. He was the president of the Indian National Science Academy (INSA) (1967–68) and the Indian Academy of Sciences (IAS) and was a member of the editorial boards of the Indian Journal of Chemistry and two international journals, Tetrahedron and Phytochemistry. He also served as the president of organizations such as the Indian Chemical Society, Indian Pharmaceutical Association, Oil Technology Association, the North India chapter of the Royal Institute of Chemistry, London, Indian Pharmaceutical Congress and Indian Science Congress Association (1966–67).
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Professor Farnsworth sends the Planet Express crew to collect a monumental statue of his first crew for a memorial marking the 50th anniversary of their disappearance. To save time on the return to Earth after forcing the statue to be recarved to fix a grammar error, Leela travels through the Bermuda Tetrahedron where they find a graveyard of lost spaceships, including the first crew's Planet Express ship. While the crew investigates the ship, a four-dimensional space whale appears and devours the old ship and statue; Zoidberg, the only member of the first crew who returned to Earth, identifies the whale as the one responsible for the first crew's disappearance. Leela becomes obsessed with killing the whale to take revenge for eating the statue and delaying their return to Earth in time for the memorial, and grows increasingly insane with each failed effort.
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357; . If a polyhedron with vertices is formed by repeating the Kleetope construction some number of times, starting from a tetrahedron, then its longest path has length ; that is, the shortness exponent of these graphs is , approximately 0.630930. The same technique shows that in any higher dimension , there exist simplicial polytopes with shortness exponent .. Similarly, used the Kleetope construction to provide an infinite family of examples of simplicial polyhedra with an even number of vertices that have no perfect matching. Kleetopes also have some extreme properties related to their vertex degrees: if each edge in a planar graph is incident to at least seven other edges, then there must exist a vertex of degree at most five all but one of whose neighbors have degree 20 or more, and the Kleetope of the Kleetope of the icosahedron provides an example in which the high-degree vertices have degree exactly 20..
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HNLs and the action they mediate is a key target for study of protein engineering, as the formation of mandelonitrile is a key step in a wide variety of organic syntheses with medical and therapeutic potential. The step mediated by these enzymes is essential to the synthesis of stereospecific bond formation in (R)-Salbutamol bronchodilators, (S)-amphetamines, (1R, 2S)-(-)-ephedrine bronchodilators,Jackson WR., Jacob HA., Matthew BR., Jayatilake GS., Watson KG. Stereoselctive syntheses of ephedrine and related 2-aminoalcohols of high optical purity from protected cyanohydrins. Tetrahedron Lett. 1990; 31:1447-1450 in addition to many others, including Lipitor,Maureen AR. Biocatalysis buzz, deals underscore interest in biotechnology-based methods to improve chemical processes. Chem Eng News 2002; 80:86 Thalidomide,Ziegler T., Horsch B., Effenberger F., A convenient route to (R)-α-hydroxy carboxylic acids and (2R)-1-amin-2-alkanols from (R)-cyanohydrins.
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248px A corollary of the usual law of sines is that in a tetrahedron with vertices O, A, B, C, we have :\sin\angle OAB\cdot\sin\angle OBC\cdot\sin\angle OCA = \sin\angle OAC\cdot\sin\angle OCB\cdot\sin\angle OBA.\, One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface. Putting any of the four vertices in the role of O yields four such identities, but at most three of them are independent: If the "clockwise" sides of three of them are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity. Three angles are the angles of some triangle if and only if their sum is 180° (π radians).
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Dutch chemist Jacobus Henricus van 't Hoff is generally considered to be one of the founders of the field of stereochemistry. In 1874, Van 't Hoff built on the work on isomers of German chemist Johannes Wislicenus, and showed that the four valencies of the carbon atom were probably directed in space toward the four corners of a regular tetrahedron, a model which explained how optical activity could be associated with an asymmetric carbon atom. He shares credit for this with the French chemist Joseph Le Bel, who independently came up with the same idea. Three months before his doctoral degree was awarded Van 't Hoff published this theory, which today is regarded as the foundation of stereochemistry, first in a Dutch pamphlet in the fall of 1874, and then in the following May in a small French book entitled La chimie dans l'espace.
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Tetrahedral packing: The dihedral angle of a tetrahedron is not commensurable with 2; consequently, a hole remains between two faces of a packing of five tetrahedra with a common edge. A packing of twenty tetrahedra with a common vertex in such a way that the twelve outer vertices form an irregular icosahedron The stability of metals is a longstanding question of solid state physics, which can only be understood in the quantum mechanical framework by properly taking into account the interaction between the positively charged ions and the valence and conduction electrons. It is nevertheless possible to use a very simplified picture of metallic bonding and only keeps an isotropic type of interactions, leading to structures which can be represented as densely packed spheres. And indeed the crystalline simple metal structures are often either close packed face-centered cubic (fcc) or hexagonal close packing (hcp) lattices.
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H-Cube adopted by the top 20 Pharma Since then a series of modular components have been developed by the company that expand the possibilities provided by the H-Cube, as well as developing a newer version of the initial product (called the H-Cube Pro). The use of ThalesNano's equipment for flow chemistry is supported by a number of scientific publications. Heterogeneous hydrogenation reactions using a continuous flow high pressure device, Desai, B. and Kappe, O.,J.Comb. Chem.; 2005; 7; 5; 641-643 Enantiomer selective acylation of racemic alcohols by lipases in continuous-flow bioreactors, Csajagi, Cs., Szatzker, G., Toke, E.R., Urge, L., Darvas, F., Poppe, L., Tetrahedron: Asymmetry, 2008, 19(2), 237-246 Construction and Validation of an Automated Flow Hydrogenation Instrument for Application in High-Throughput Organic Chemistry, Clapham, B., Wilson, N.S., Mischmerhuizen, M.J., Blanchard, D.P., Dingle, D.M., Nemcek, T.A., Pan, J.Y., Sauer, D.R., J. Comb. Chem.
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Each convex set containing X must (by the assumption that it is convex) contain all convex combinations of points in X, so the set of all convex combinations is contained in the intersection of all convex sets containing X. Conversely, the set of all convex combinations is itself a convex set containing X, so it also contains the intersection of all convex sets containing X, and therefore the second and third definitions are equivalent., p. 12; , p. 17. In fact, according to Carathéodory's theorem, if X is a subset of a d-dimensional Euclidean space, every convex combination of finitely many points from X is also a convex combination of at most d+1 points in X. The set of convex combinations of a (d+1)-tuple of points is a simplex; in the plane it is a triangle and in three-dimensional space it is a tetrahedron.
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Roman caltrop at the Westphalian Museum of Archeology (German: Westfälisches Museum für Archäologie), Herne, North Rhine-Westphalia, Germany A caltrop (also known as caltrap, galtrop, cheval trap, galthrap, galtrap, calthrop, jackrock or crow's footBattle of Alesia (Caesar's conquest of Gaul in 52 BC)), Battlefield Detectives program, (2006), rebroadcast: 2008-09-08 on History Channel International (13;00-14:00 hrs EDST); Note: No mention of name caltrop at all, but illustrated and given as battle key to defend Roman lines of circumvallation per recent digs evidence.) is an area denial weapon made up of two or more sharp nails or spines arranged in such a manner that one of them always points upward from a stable base (for example, a tetrahedron). Historically, caltrops were part of defences that served to slow the advance of troops, especially horses, chariots, and war elephants, and were particularly effective against the soft feet of camels. In modern times, caltrops are effective when used against wheeled vehicles with pneumatic tires.
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Structural motif found in α-quartz, but also found in almost all forms of silicon dioxide Relationship between refractive index and density for some SiO2 forms In the majority of silicates, the silicon atom shows tetrahedral coordination, with four oxygen atoms surrounding a central Si atom. The most common example is seen in the quartz polymorphs. It is a 3 dimensional network solid in which each silicon atom is covalently bonded in a tetrahedral manner to 4 oxygen atoms. For example, in the unit cell of α-quartz, the central tetrahedron shares all four of its corner O atoms, the two face-centered tetrahedra share two of their corner O atoms, and the four edge-centered tetrahedra share just one of their O atoms with other SiO4 tetrahedra. This leaves a net average of 12 out of 24 total vertices for that portion of the seven SiO4 tetrahedra that are considered to be a part of the unit cell for silica (see 3-D Unit Cell).
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The airbags were made of four inter-connected multi-layer vectran bags that surrounded the tetrahedron lander. They were designed and tested to accommodate grazing angle impacts as high as 28 m/s. However, as the airbags were designed for no more than about 15 m/s vertical impacts, three solid retrorockets were mounted above the lander in the backshell. These were fired at 98 m above the ground. The lander's on-board computer estimated the best time to fire the rockets and cut the bridle so that the lander velocity would be reduced to about 0 m/s between 15 and 25 m above the ground. After 2.3 seconds, while the rockets were still firing, the lander cut the bridle loose about 21.5 m above the ground and fell to the ground. The rockets flew up and away with the backshell and parachute (they have since been sighted by orbital images).
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The five Platonic solids – the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron – viewed as two-dimensional surfaces, have the property that any flag (a triple of a vertex, edge, and face that all meet each other) can be taken to any other flag by a symmetry of the surface. More generally, a map embedded in a surface with the same property, that any flag can be transformed to any other flag by a symmetry, is called a regular map. If a regular map is used to generate a clean dessin, and the resulting dessin is used to generate a triangulated Riemann surface, then the edges of the triangles lie along lines of symmetry of the surface, and the reflections across those lines generate a symmetry group called a triangle group, for which the triangles form the fundamental domains. For example, the figure shows the set of triangles generated in this way starting from a regular dodecahedron.
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Hydrazone iodination is an organic reaction in which a hydrazone is converted into a vinyl iodide by reaction of iodine and a non-nucleophilic base such as DBUA new reaction of hydrazones Barton, D. H. R. , R. E. O'Brien and S. Sternhell Journal of the Chemical Society,1962, 470 - 476 AbstractStudies on the oxidation of hydrazones with iodine and with phenylselenenyl bromide in the presence of strong organic bases; an improved procedure for the synthesis of vinyl iodides and phenyl-vinyl selenides Barton, D. H. R.; Bashiardes, G.; Fourrey, J.-L. Tetrahedron 1988, 44, 147 Abstract. First published by Derek Barton in 1962 the reaction is sometimes referred to as the Barton reaction (although there are many different Barton reactions) or, more descriptively, as the Barton vinyl iodine procedure. The reaction has earlier roots with the 1911 discovery by Wieland and Roseeu that the reaction of hydrazones with iodine alone (without base) results in the azine dimer (structure 2 in scheme 1).
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Great stellated dodecahedron enclosed by a skeletal icosahedron from Perspectiva corporum regularium The book focuses on the five Platonic solids, with the subtitles of its title page citing Plato's Timaeus and Euclid's Elements for their history. Each of these five shapes has a chapter, whose title page relates the connection of its polyhedron to the classical elements in medieval cosmology: fire for the tetrahedron, earth for the cube, air for the octahedron, and water for the icosahedron, with the dodecahedron representing the heavens, its 12 faces corresponding to the 12 symbols of the zodiac. Each chapter includes four engravings of polyhedra, each showing six variations of the shape including some of their stellations and truncations, for a total of 120 polyhedra. This great amount of variation, some of which obscures the original Platonic form of each polyhedron, demonstrates the theory of the time that all the variation seen in the physical world comes from the combination of these basic elements.
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60px A carbon dioxide fire extinguisher rated for flammable liquids and gasses These are fires whose fuel is flammable or combustible liquid or gas. The US system designates all such fires "Class B". In the European/Australian system, flammable liquids are designated "Class B" having flash point less than 100 °C, while burning gases are separately designated "Class C". These fires follow the same basic fire tetrahedron (heat, fuel, oxygen, chemical reaction) as ordinary combustible fires, except that the fuel in question is a flammable liquid such as gasoline, or gas such as natural gas. A solid stream of water should never be used to extinguish this type because it can cause the fuel to scatter, spreading the flames. The most effective way to extinguish a liquid or gas fueled fire is by inhibiting the chemical chain reaction of the fire, which is done by dry chemical and Halon extinguishing agents, although smothering with CO2 or, for liquids, foam is also effective.
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Tetrahedron Lett. 21 (31): 2955-2958. . Tsuji describes the olefin metathesis reaction as “…potentially useful in organic synthesis” and addresses the need for the development of a more versatile catalyst to tolerate various functional groups. center In 1987, Siegfried Warwel and Hans Kaitker published a synthesis of symmetric macrocycles through a cross-metathesis dimerization of starting cycloolefins to afford C14, C18, and C20 dienes in 58-74% yield, as well as C16 in 30% yield, using Re2O7 on Al2O3 and Me4Sn for catalyst activation.Warwel, S.; Katker, H. (1987). “Eine einfache Synthese makrocyclischer Kohlenwasserstoffe durch Metathese von Cyclooflefinen”. Synthesis. 935-937. center After a decade since its initial discovery, Grubbs and Fu published two influential reports in 1992 detailing the synthesis of O- and N- heterocycles via RCM utilizing Schrock’s molybdenum alkylidene catalysts, which had proven more robust and functional group tolerant than the tungsten chloride catalysts.Fu, G. C.; Grubbs, R. H. (1992). “The Application of Catalytic Ring-Closing Olefin Metathesis to the Synthesis of Unsaturated Oxygen Heterocycles”.
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Working in DMSO solvent does not allow isolation of the bicyclic ketol intermediate, it leads directly to the optically active bicyclic dione.Z. G. Hajos, D. R. Parrish, German Patent DE 2102623 (29 July 1971) and USP 3,975,440 (Aug.17, 1976) Example 21. The reaction is called the Hajos-Parrish reaction or the Hajos- Parrish-Eder-Sauer-Wiechert reaction.1 H-Indene-1,5(6 H)-dione, 2,3,7,7a-tetrahydro-7a-methyl-, (S)- Zoltan G. Hajos and David R. Parrish Organic Syntheses, Coll. Vol. 7, p.363 (1990); Vol. 63, p.26 (1985) Article Identical reaction with 2-methyl-1,3-cyclopentanedione (5 membered ring instead of a 6 membered ring) This reaction has also been performed in a one- step procedure, leading to 49% yield and 76% Enantiomeric excess (ee):A proline-catalyzed asymmetric Robinson annulation reaction Tetrahedron Letters, Volume 41, Issue 36, September 2000, Pages 6951-6954 Tommy Bui and Carlos F. Barbas :Wieland Miescher Barbas 2000 Other proline-based catalysts have been investigated Org. Synth.
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In 1874, Jacobus Henricus van 't Hoff and Joseph Achille Le Bel independently proposed that the phenomenon of optical activity could be explained by assuming that the chemical bonds between carbon atoms and their neighbors were directed towards the corners of a regular tetrahedron. This led to a better understanding of the three-dimensional nature of molecules. Emil Fischer developed the Fischer projection technique for viewing 3-D molecules on a 2-D sheet of paper: 200px In 1898, Ludwig Boltzmann, in his Lectures on Gas Theory, used the theory of valence to explain the phenomenon of gas phase molecular dissociation, and in doing so drew one of the first rudimentary yet detailed atomic orbital overlap drawings. Noting first the known fact that molecular iodine vapor dissociates into atoms at higher temperatures, Boltzmann states that we must explain the existence of molecules composed of two atoms, the "double atom" as Boltzmann calls it, by an attractive force acting between the two atoms.
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In 2016, he also received the Hoffman Medal of the German Chemical SocietyBen Feringa awarded August-Wilhelm-von-Hofmann-Denkmünze - website of the Research Center for Functional Molecular Systems (FMS) and the Tetrahedron Prize awarded by Elsevier.Tetrahedron Prize for Creativity in Organic Chemistry - website of Elsevier In 2017, Feringa received the Centenary Prize of the Royal Society of Chemistry.Centenary Prize 2017 Winner - website of RSC In August 2018, Feringa was awarded the European Gold Medal presented by the European Chemical Society (EuChemS) during the 7th EuChemS Chemistry Congress, held in Liverpool, UK. In 2019, Feringa accepted the Raman Chair of the Indian Academy of Sciences, an honorary position whereby eminent scientists are invited to lecture on their work and interact with the research community in India. In 2008, he was appointed a Knight of the Order of the Netherlands Lion by Queen Beatrix of the Netherlands, and on 23 November 2016 he was promoted to Commander of the same Order by King Willem-Alexander of the Netherlands.
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In organic chemistry, a carbo-mer (often carbo-mer or carbomer) is an expanded molecule obtained by insertion of C2 units into a given molecule.Carbomers. I. A general concept of expanded molecules Remi Chauvin Tetrahedron Letters Volume 36, Issue 3, 16 January 1995, Pages 397-400 Carbo-mers differ from their templates in size but not in symmetry when each C–C single bond is replaced by an alkyne bond C-C≡C-C, each C=C double bond is replaced by an allene bond C=C=C=C, and each C≡C triple bond is replaced by C≡C-C≡C. The size of the carbo-mer continues to increase when more C2 units are inserted, so carbo-mers are also called carbon-molecules, where "n" is the number of acetylene or allene groups in an n-expansion unit. This concept, devised by Rémi Chauvin in 1995, is aimed at introducing new chemical properties for existing chemical motifs.
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The Szilassi polyhedron, a non-convex polyhedral realization of the Heawood graph with the topology of a torus In any dimension higher than three, the algorithmic Steinitz problem (given a lattice, determine whether it is the face lattice of a convex polytope) is complete for the existential theory of the reals by Richter-Gebert's universality theorem. However, because a given graph may correspond to more than one face lattice, it is difficult to extend this completeness result to the problem of recognizing the graphs of 4-polytopes, and this problem's complexity remains open. Researchers have also found graph-theoretic characterizations of the graphs of certain special classes of three-dimensional non-convex polyhedra.. and four-dimensional convex polytopes.... However, in both cases, the general problem remains unsolved. Indeed, even the problem of determining which complete graphs are the graphs of non-convex polyhedra (other than K4 for the tetrahedron and K7 for the Császár polyhedron) remains unsolved.. László Lovász has shown a correspondence between polyhedral representations of graphs and matrices realizing the Colin de Verdière graph invariants of the same graphs..
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A Vietoris–Rips complex of a set of 23 points in the Euclidean plane. This complex has sets of up to four points: the points themselves (shown as red circles), pairs of points (black edges), triples of points (pale blue triangles), and quadruples of points (dark blue tetrahedrons). In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming a simplex for every finite set of points that has diameter at most δ. That is, it is a family of finite subsets of M, in which we think of a subset of k points as forming a (k − 1)-dimensional simplex (an edge for two points, a triangle for three points, a tetrahedron for four points, etc.); if a finite set S has the property that the distance between every pair of points in S is at most δ, then we include S as a simplex in the complex.
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Lucent's aerial performers use both traditional aerial apparatus such as trapeze, aerial silk, aerial hoop, span sets, aerial rope or Spanish web, as well as custom-made aerial apparatus unique to the group, including a large aerial moon, mechanical Lotus flower, bird cage, Star tetrahedron, spider web, straps (similar in effect to a straitjacket), and aerial carousel. Fire performers work with different styles of tools including fire staff, fire poi, fire fans, fire hoop, fire sword, fire knives, fire cups, fire rope, fire chains, aerial fire acts, and fire nunchaku. The interactive portions of Lucent's performances include oracle or tarot readings, rose water foot baths, gypsies offering herbal elixirs, a tribal temporary tattoo parlor, a "Music Shack" where songs are custom- written, Post Portal Post Office where lost letters are delivered from other dimensions and performed on the spot for guests, and a "Transformation Station" where audience members can receive fantasy hair and makeup applications. All the performers dress in fantastical costumes and makeup unique to Lucent Dossier.
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In 1995/96 he shared the Wolf Prize in Chemistry with Gilbert Stork of Columbia University for "designing and developing novel chemical reactions which have opened new avenues to the synthesis of complex molecules, particularly polysaccharides and many other biologically and medicinally important compounds".The Wolf Prize in Chemistry He is the recipient of several other awards including the American Chemical Society's Guenther Award and Aldrich Award for Creative Work in Synthetic Organic Chemistry, the F.A. Cotton Medal for Excellence in Chemical Research of the American Chemical Society, the Tetrahedron Prize (1996), the Arthur C. Cope Award (1998), the New York City Mayor's Award for Science and Technology, and the Bristol Myers Squibb Lifetime Achievement Award. In 2006, he was awarded the Benjamin Franklin Medal in Chemistry for his achievements in synthetic organic chemistry, particularly for the development of methods for preparing complex substances found in nature, and their emerging applications in the field of cancer treatment.Samuel Danishefsky, the winner of Benjamin Franklin Medal in Chemistry in 2006 He is a member of the Board of Scientific Governors at The Scripps Research Institute.
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Satterlyite is hydroxyl bearing iron phosphate with a space group P31m. The structure of satterlyite is made up of two pairs of face shared, distorted (Fe,Mg)O6 octahedra, the two faces are linked together by sharing edges to form double chains along the [001] plain. The double chains share ligands with six other double chains to make a 3D network holding three PO4 tetrahedra, linked by the corners to the (Fe,Mg)O6 octahedra (Kolitsch, 2002). The two (Fe,Mg)O6 Octahedra have different occupancies; the Fe to Mg ratio of the M sites are 0.838(2):0.162(2) for the M(1) site and 0.706(2):0.294(2) for the M(2) site (Kolitsch, 2002). The structure contains three H atoms; two are share ligands with two (Fe,Mg)O6 octahedra and the third strongly disordered H atom is bonded to O of the PO4 tetrahedron (Kolitsch, 2002). Holtedahlite, a mineral that was found in Tingelstadtjern quarry in Norway, with the formula (Mg12PO4)5(PO3OH,CO3)(OH,O)6 is isostructural with satterlyite (Raade, 1979).
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The main achievement of the Shield of the Trinity diagram is to transfer a large part of the essential "mystery" or "paradox" of the Christian doctrine of the Trinity from the realm of complex verbal philosophical abstractions and esoteric theological vocabulary to the realm of simple logic, as presented in the relatively easily graspable form of a concrete and conveniently compact visual diagram. It is remarkable as a basically successful attempt, roughly 800 years old, to represent a complex set of abstract concepts in precise graphic form (as opposed to many of the near-contemporary attempts of Joachim of Fiore and Raymund Lull, which were not so successful). Thus it is perhaps one of the oldest widely attested "graphs", in the sense of graph theory (technically, it is a complete graph on 4 vertices, the same as the vertices and edges of a tetrahedron). Of course, if the diagram is interpreted according to ordinary logic, then it contains a number of contradictions (since the set of twelve propositions listed above is mutually contradictory).
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Dr Kagan is a member of the French Academy of Sciences and has won many awards in the field including:Frankin Laureate Citation Silver Medal of the French National Scientific Research Center, Prelog Medal, August-Wihelm-von Hoffman Medal, Nagoya Mel,jnkm,ldal of Organic Chemistry, Tetrahedron Prize for Creativity in Organic Chemistry, Wolf Prize in Chemistry, Grand Prix de la Fondation de la Maison de la Chimie, Chevalier de la Légion d'honneur, JSPS Award for Eminent Scientists, Ryoji Noyori Prize, and the 2005 Benjamin Franklin Medal. In 2001 controversy was caused when Kagan was not given the Nobel prize which had been shared by K. Barry Sharpless of the Scripps Research Institute in La Jolla, California, Ryōji Noyori of Nagoya University, Japan, and William Knowles, formerly of Monsanto Company in St Louis, Missouri, for work on catalytic asymmetric synthesis. It was thought that as Kagan was one of the pioneers of the field he too should have been honoured. However, as the prize can be given to a maximum of three people he was left off.
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The spaceship graveyard in the Bermuda Tetrahedron contains many spaceships from popular culture, including Discovery One from 2001: A Space Odyssey, Oceanic Airlines Flight 815 from the television series Lost, the Satellite of Love from the comic science fiction television series Mystery Science Theater 3000, the Jupiter II from the science fiction television series Lost in Space, the spaceship from the animated television series Josie and the Pussycats in Outer Space, the spaceship commonly seen on albums by Electric Light Orchestra, Journey and Boston, an Apollo Lunar Module with the ascent and descent stages still attached, an Apollo Command/Service Module labeled "Apollo 100", and Skylab. There are also two spaceships named after two popular GPS brands: Garmin and TomTom. The Fourth Doctor from the British science fiction television series Doctor Who, as portrayed by Tom Baker, makes a cameo appearance emerging from the body of the four-dimensional space whale near the end of the episode. The Monolith from 2001: A Space Odyssey also makes an appearance.
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Inscribed circles of various polygons An inscribed triangle of a circle A tetrahedron (red) inscribed in a cube (yellow) which is, in turn, inscribed in a rhombic triacontahedron (grey). (Click here for rotating model) In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon (or a sphere or ellipsoid inscribed in a convex polyhedron) is tangent to every side or face of the outer figure (but see Inscribed sphere for semantic variants). A polygon inscribed in a circle, ellipse, or polygon (or a polyhedron inscribed in a sphere, ellipsoid, or polyhedron) has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure.
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When it exists, a circumscribed sphere need not be the smallest sphere containing the polyhedron; for instance, the tetrahedron formed by a vertex of a cube and its three neighbors has the same circumsphere as the cube itself, but can be contained within a smaller sphere having the three neighboring vertices on its equator. However, the smallest sphere containing a given polyhedron is always the circumsphere of the convex hull of a subset of the vertices of the polyhedron.. In De solidorum elementis (circa 1630), René Descartes observed that, for a polyhedron with a circumscribed sphere, all faces have circumscribed circles, the circles where the plane of the face meets the circumscribed sphere. Descartes suggested that this necessary condition for the existence of a circumscribed sphere is sufficient, but it is not true: some bipyramids, for instance, can have circumscribed circles for their faces (all of which are triangles) but still have no circumscribed sphere for the whole polyhedron. However, whenever a simple polyhedron has a circumscribed circle for each of its faces, it also has a circumscribed sphere.
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He received awards from the US, Japan, Bulgaria, China, the Czech Republic, the Netherlands, Italy, Saudi Arabia, Sweden (the Scheele Award, 1992), Switzerland, Taiwan and the UK. In an unprecedented international alliance, the Nakanishi Prize of the American Chemical Society (ACS) and the Chemical Society of Japan (CSJ) was established in 1995 and is awarded in alternate years in Japan and the U.S. to recognize achievements in chemical and spectroscopic methods to the study of biological phenomena; it is the only CSJ prize with an individual's name. In 1999, he was awarded one of Japan's highest honors, "Person of Cultural Merit" Columbia News ::: Columbia Chemist Koji Nakanishi Wins Prestigious Japanese Prize for his breakthrough research in the organic chemistry of natural products. His many honors include the prestigious King Faisal International Prize in Science, the Welch Award, the Arthur C. Cope Award, the Paul Karrer Gold Medal, the Imperial Prize of the Japan Academy, the National Academy of Sciences Award in Chemical Sciences, the Tetrahedron Prize (2004) and the Japan Academy Prize, as well as honorary doctorates from Williams College, Georgetown University and the University of Uppsala.
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According to historian Greg Mulheirn the Robinson effort was hampered by his micromanagement style of leadership and the Woodward effort was greatly facilitated by his good relationships with chemical industry. Around 1949 steroids like cortisone were produced from natural resources but expensive. Chemical companies Merck & Co. and Monsanto saw commercial opportunities for steroid synthesis and not only funded Woodward but also provided him with large quantities of certain chemical intermediates from pilot plants. Hard work also helped the Woodward effort: one of the intermediate compounds was named Christmasterone as it was synthesized on Christmas Day 1950 by Sondheimer. Other cholesterol schemes have also been developed: racemic cholesterol was synthesized in 1966 by W.S. Johnson,Steroid total synthesis—hydrochrysene approach—XVI: Racemic conessine, progesterone, cholesterol, and some related natural products W. S. Johnson, J. A. Marshall, J. F. W. Keana, R. W. Franck, D. G. Martin and J. V. Bauer Tetrahedron Volume 22, Supplement 8, 1966, Pages 541-601 the enantiomer of natural cholesterol was reported in 1996 by Rychnovsky and Mickus,Synthesis of ent-cholesterol, the unnatural enantiomer Scott D. Rychnovsky, Daniel E. Mickus J. Org. Chem.
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In the reaction mechanism for this organic reaction the base abstracts the acidic α-proton of the peroxide 1 to form the carbanion 4 as a reactive intermediate which rearranges to the ketone 2 with expulsion of the hydroxyl anion 3'. This intermediate gains a proton forming the alcohol 3. :Kornblum-DeLaMare rearrangement Mechanism Deprotonation and rearrangement can also be a concerted reaction without formation of 4. An alternative reaction mechanism involving direct nucleophilic displacement on the peroxide link of the amine followed by an elimination reaction is considered unlikely based on the outcome of this model reaction:The mechanism of the tertiary amine catalysed isomerisation of endoperoxides to hydroxyketones: synthesis and chemistry of the intermediate postulated in the peroxide attack mechanism David R. Kelly, Harjinder Bansal and J. J. Gwynfor Morgan Tetrahedron Letters Volume 43, Issue 51 , 16 December 2002, Pages 9331–9333 :Kornblum-DeLaMare rearrangement alternative mechanism The peroxide 1 converts to the hydroxyketone 2 by action of triethylamine but the alternative route through hydroxylamine 3 by nucleophilic displacement with Lithium diisopropylamide and the ammonium salt 4 (by methylation with methyl trifluoromethanesulfonate) fails.
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Simonkolleite is rhombohedral, space group R _3_ m. There are two crystallographically distinct zinc sites in Simonkolleite, both of which are fully occupied by zinc. The Zn(1) site is coordinated by six hydroxyl (OH) groups in an octahedral geometry [Zn(OH)6]. The Zn(2) site is coordinated by three OH groups, and one Cl atom in a tetrahedral geometry [Zn(OH)3Cl]. The [Zn(OH)6] octahedra form an edge-sharing dioctahedral sheet similar to that observed in dioctahedral micas. On each site of the vacant octahedron, a [Zn(OH)3Cl] tetrahedron is attached to three anions of the sheet and points away from the sheet. Intercalated between adjacent sheets are interstitial water (H2O) groups. The sheets are held together by hydrogen bonding from OH groups of one sheet to Cl anions of adjacent sheets, and to interstitial H2O groups. The [Zn(OH)6] octahedra have four long equatorial bonds (at 2.157 Å) and two short apical bonds (at 2.066 Å). This apical shortening is a result of the bond-valence requirements of the coordinating OH groups and the connectivity of polyhedra in the structure.
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When a slight enantiomeric excess is present, the solubility's of the pure and racemic crystal can be manipulated to generate large ee's of the pure enantiomer. If we define certain solubility's as such: KL= [L] represents the solubility of the pure enantiomer KDL= [D][L] represents the solubility product of the racemic mixture such that [D]= KDL/[L] We can then define the ratio of [L]/[D] = [L]2/KDL When both enantiomers are present, a racemic crystal structure is formed-however it is lower in energy, has a higher melting point, and is less soluble than the enantiomerically pure crystal structure. As a result, when a slight excess of one enantiomer is present, the ee can be amplified by evaporating solvent-causing the racemate to precipitate. Researchers have been able to start with an ee of 1% L, and ultimately end up with 95:5 solution of L: D. The results discussed above (particularly the synchrotron argument) led Breslow to propose that the D amino acids and L sugars could generate life in other parts of the universe.Breslow, R. “The Origin of Homochirality in Amino Acids and Sugars on Prebiotic Earth” Tetrahedron Lett.
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The Hajos 1974 carbinolamine mechanism has had an unwitting support in a more recent paper by Michael Limbach.β-Homoamino acids as catalysts on enantioselective intra- and intermoelcular aldol reactions by Michael Limbach, Tetrahedron Letters 47 (2006) 3843-3847 The triketone starting material 2- methyl-2-(3-oxobutyl)-1,3-cyclopentanedione gave the expected optically active bicyclic ketol (+)-(3aS,7aS)-3a,4,7,7a-tetrahydro-3a-hydroxy-7a-methyl-1,5(6H)-indanedione with (S)-(−)-proline catalyst. On the other hand, the stereochemical outcome is reversed with ee selectivities of up to 83% by using the homologue amino acid catalysts, such as (S)-β-homoproline, [(pyrrolidine-(2S)-yl) acetic acid]. The virtual anomaly can be explained with a top side approach of the bulkier beta amino acids to the above triketone starting material of reflective symmetry. The top side approach results in the formation of an enantiotopic carbinolamine to give the (−)-(3aR,7aR)-3a,4,7,7a-tetrahydro-3a-hydroxy-7a-methyl-1,5(6H)-indanedione bicyclic ketol enantiomer identical to the one obtained with unnatural (R)-(+)-proline. List in 2010 on the other hand is perplexed and surprised that Hajos rejected the enamine mechanism, certainly in light of earlier work by Spencer in 1965 on amine catalysed aldol reactions.
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The Schöllkopf method or Schöllkopf Bis-Lactim Amino Acid Synthesis is a method in organic chemistry for the asymmetric synthesis of chiral amino acids.Leo A. Paquette :Chiral Reagents for Asymmetric Synthesis, S.220-223, 2003, Wiley and Sons, Jan Bülle, Aloys Hüttermann : Das Basiswissen der organischen Chemie: Die wichtigsten organischen Reaktionen im Labor und in der Natur, S.310/311, 2000, Wiley-VCH, The method was established in 1981 by Ulrich Schöllkopf.Enantioselective Syntheses of (R)-Amino Acids Using L-Valine as Chiral Agent Angewandte Chemie International Edition in English Volume 20, Issue 9, Date: September 1981, Pages: 798-799 Ulrich Schöllkopf, Ulrich Groth, Chuanzheng Deng Enantioselective Synthesis of -Methyl--aminocarboxylic Acids by Alkylation of the Lactim Ether of cyclo-(L-Ala-L-Ala) Angewandte Chemie International Edition in English Volume 18, Issue 11, Date: November 1979, Pages: 863-864 Ulrich Schöllkopf, Wolfgang Hartwig, Ulrich Groth Asymmetric synthesis via heterocyclic intermediates - XXXIX1 : Asymmetric synthesis of (enantiomerically and diastereomerically virtually pure) methyl 2-amino-4,5-epoxy-3-hydroxy-alkanoates and methyl 2-amino-3-hydroxy-4,5-methylene-alkanoates by the bislactimether method Tetrahedron, Volume 44, Issue 17, 1988, Pages 5293-5305 Ulrich Schöllkopf, Thomas Tiller, Jürgen Bardenhagen In it glycine is a substrate, valine a chiral auxiliary and the reaction taking place an alkylation.
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Two distinct expansions of benzene can be called carbo-benzene (C18H6): :carbo-benzene One (above right) expands each C-H bond to C-C≡C-H, making hexaethynylbenzene, a substituted benzene derivative.Hexaethynylbenzene Rainer Diercks, James C. Armstrong, Roland Boese, K. Peter C. Vollhardt. Hexaethynylbenzene. Angewandte Chemie International. 1986; 25(3):268-269. One (above left) expands each C=C and C≡C bond of the benzene core, making 1,2,4,5,7,8,10,11,13,14,16,17-dodecadehydro[18]annulene. An analog of this molecule, with the hydrogen atoms replaced by phenyl groups, 3,6,9,12,15,18-hexaphenyl-1,2,4,5,7,8,10,11,13,14,16,17-dodecadehydro[18]annulene, is stable.Synthesis of the first 3,6,9,15,18,18-hexa- substituted-1,2,4,5,7,8,10,11,13,14,16,17-dodecadehydro[18]annulenes with D6h-symmetry Yoshiyuki Kuwatani, Naoto Watanabe and Ikuo Ueda Tetrahedron Letters Volume 36, Issue 1, 2 January 1995, Pages 119-122 Its proton NMR spectrum shows that the phenyl protons are shifted downfield compared to a proton position in benzene itself (chemical shift position for the ortho proton is 9.49 ppm), suggesting the presence of a diamagnetic ring current and thus aromaticity. The final step in its organic synthesis is reaction of the triol with stannous chloride and hydrochloric acid in diethyl ether: :carbo- benzene synthesis With both core and periphery expanded, the total carbo-mer of benzene (C30H6) only exists in silico (computer simulation).
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