The isolated icosahedra are not stable, due to the nonuniformity of the honeycomb; thus boron is not a molecular solid, but the icosahedra in it are connected by strong covalent bonds.
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This phase also exists without silicon. Figure 19a shows the network of boron icosahedra in the boron framework of ScB19+xSiy. In this network, 4 icosahedra form a supertetrahedron (figure 18b); its one edge is parallel to the a-axis, and the icosahedra on this edge make up a chain along the a-axis.
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The I4 icosahedra link two networks along the c-axis and therefore form an infinite chain of icosahedra along this axis as shown in figure 10. The unusually short distances (0.4733 and 0.4788 nm) between the neighboring icosahedra in this direction result in the relatively small c-axis lattice constant of 0.95110(7) nm in this compound – other borides with a similar icosahedral chain have this value larger than 1.0 nm. However, the bonding distances between the apex B atoms (0.1619 and 0.1674 nm) of neighboring I4 icosahedra are usual for the considered metal borides.
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Nitrogen atoms strengthen the bonding in the c-plane by bridging three icosahedra, like C atoms in the C-B-C chain. Figure 13 depicts the c-plane network revealing the alternate bridging of the boron icosahedra by N and C atoms. Decreasing the number of the B6 octahedra diminishes the role of nitrogen because the C-B-C chains start bridging the icosahedra. On the other hand, in MgB9N the B6 octahedron layer and the B12 icosahedron layer stack alternatively and there is no C-B-C chains; thus only N atoms bridge the B12 icosahedra.
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Convex regular icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes meaning "twenty" and meaning "seat". The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non-similar shapes of icosahedra, some of them being more symmetrical than others.
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A network of boron icosahedra lying in the (001) plane. Black, blue and red spheres correspond to C, Si and Y atoms, respectively. The crystal has layered structure. Figure 15 shows a network of boron icosahedra that spreads parallel to the (001) plane, connecting with four neighbors through B1–B1 bonds.
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This property originates from electron transfer from metal atoms to the boron icosahedra and is favorable for thermoelectric applications.
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Hemicube is derived from a cube by equating opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces. Grünbaum also discovered the 11-cell, a four-dimensional self-dual object whose facets are not icosahedra, but are "hemi-icosahedra" -- that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the same face . The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12.
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Unit cell of B4C. The green sphere and icosahedra consist of boron atoms, and black spheres are carbon atoms. Fragment of the B4C crystal structure. Boron carbide has a complex crystal structure typical of icosahedron-based borides. There, B12 icosahedra form a rhombohedral lattice unit (space group: Rm (No. 166), lattice constants: a = 0.56 nm and c = 1.212 nm) surrounding a C-B-C chain that resides at the center of the unit cell, and both carbon atoms bridge the neighboring three icosahedra. This structure is layered: the B12 icosahedra and bridging carbons form a network plane that spreads parallel to the c-plane and stacks along the c-axis. The lattice has two basic structure units – the B12 icosahedron and the B6 octahedron.
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Of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them. Other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are often referred to as such.
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This is in contrast to bulk boron allotropes, which are semiconducting and marked by an atomic structure based on B12 icosahedra.
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The higher boride LnB66 is known for all lanthanide metals. The composition is approximate as the compounds are non-stoichiometric. They all have similar complex structure with over 1600 atoms in the unit cell. The boron cubic sub lattice contains super icosahedra made up of a central B12 icosahedra surrounded by 12 others, B12(B12)12.
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This crystalline form has the same stoichiometry as B13C3, which consists of boron icosahedra connected by boron and carbon atoms. Boron suboxide (B6O) has a hardness of about 35 GPa. Its structure contains eight B12 icosahedra units, which are sitting at the vertices of a rhombohedral unit cell. There are two oxygen atoms located along the (111) rhombohedral direction.
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He was also known for startling the travelling public by carrying around a large string bag filled with garishly coloured stellated icosahedra.
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If the twinned icosahedra were independent without twinning then B23 would be a bridge site linking three icosahedra. However, because of twinning, B23 shifts closer to the twinned icosahedra than another icosahedron; thus B23 is currently treated as a member of the twinned icosahedra. In ScB19+xSiy, the two B24 sites which correspond to the vacant sites in the B20 unit are partially occupied; thus, the unit should be referred to as a B22 cluster which is occupied by about 20.6 boron atoms. Scandium atoms occupy 3 of 5 Al sites of α-AlB12, that is Sc1, Sc2 and Sc3 correspond to Al4, Al1 and Al2 sites of α-AlB12, respectively. The Al3 and Al5 sites are empty for ScB19+xSiy, and the Si site links two B22 units.
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Crystal structure of YB25. Black and green spheres indicate Y and B atoms, respectively. The structure of yttrium borides with B/Y ratio of 25 and above consists of a network of B12 icosahedra. The boron framework of YB25 is one of the simplest among icosahedron-based borides – it consists of only one kind of icosahedra and one bridging boron site.
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The unit cell is orthorhombic and its most salient feature is four boron-containing icosahedra. Each icosahedron contains 12 boron atoms. Eight more boron atoms connect the icosahedra to the other elements in the unit cell. The occupancy of metal sites in the lattice is lower than one, and thus, while the material is usually identified with the formula AlMgB14, its chemical composition is closer to Al0.75Mg0.75B14.
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It has 2 icosahedra connected by 6 tetrahedra and 8 octahedra, with 24 irregular tetrahedra in the alternated gaps. In total it has 40 cells, 112 triangular faces, 96 edges, and 24 vertices. It has [4,(3,2)+] symmetry, order 48, and also [3,3,2]+ symmetry, order 24. A construction exists with two regular icosahedra in snub positions with two edge lengths in a ratio of around 0.831 : 1.
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In the 1960s Branko Grünbaum issued a call to the geometric community to consider generalizations of the concept of regular polytopes that he called polystromata. He developed a theory of polystromata, showing examples of new objects including the 11-cell. The 11-cell is a self- dual 4-polytope whose facets are not icosahedra, but are "hemi-icosahedra" -- that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the same face (Grünbaum, 1977). A few years after Grünbaum's discovery of the 11-cell, H.S.M. Coxeter discovered a similar polytope, the 57-cell (Coxeter 1982, 1984), and then independently rediscovered the 11-cell.
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The C3 and Si3 site atoms strengthen the network by bridging the boron icosahedra. Contrary to other boron-rich icosahedral compounds, the boron icosahedra from different layers are not directly bonded. The icosahedra within one layer are linked through Si8 ethane-like clusters with (B12)3≡Si-C≡(B12)3 bonds, as shown in figures 16a and b. There are eight atomic sites in the unit cell: one yttrium Y, four boron B1–B4, one carbon C3 and three silicon sites Si1–Si3. Atomic coordinates, site occupancy and isotropic displacement factors are listed in table Va; 68% of the Y sites are randomly occupied and remaining Y sites are vacant.
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Both the Y atoms and B12 icosahedra form zigzags along the x-axis. The bridging boron atoms connect three equatorial boron atoms of three icosahedra and those icosahedra make up a network parallel to the (101) crystal plane (x-z plane in the figure). The bonding distance between the bridging boron and the equatorial boron atoms is 0.1755 nm, which is typical for the strong covalent B-B bond (bond length 0.17–0.18 nm); thus, the bridging boron atoms strengthen the individual network planes. On the other hand, the large distance between the boron atoms within the bridge (0.2041 nm) reveals a weaker interaction, and thus the bridging sites contribute little to the bonding between the network planes.
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Both the Y atoms and B12 icosahedra form zigzags along the x-axis. The bridging boron atoms connect three equatorial boron atoms of three icosahedra and those icosahedra make up a network parallel to the (101) crystal plane (x-z plane in the figure). The bonding distance between the bridging boron and the equatorial boron atoms is 0.1755 nm, which is typical for the strong covalent B-B bond (bond length 0.17–0.18 nm); thus, the bridging boron atoms strengthen the individual network planes. On the other hand, the large distance between the boron atoms within the bridge (0.2041 nm) suggests weaker interaction, and thus the bridging sites contribute little to the bonding between the network planes.
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Other complex higher borides LnB50 (Tb, Dy, Ho Er Tm Lu) and LnB25 are known (Gd, Tb, Dy, Ho, Er) and these contain boron icosahedra in the boron framework.
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There are 58 independent atomic sites in the unit cell. Three of them are occupied by either B or Si atoms (mixed-occupancy sites), one is a Si bridge site and one is Y site. From the remaining 53 boron sites, 48 form icosahedra and 5 are bridging sites. Atomic coordinates and site occupancies are summarized in table III. The boron framework of YB41Si1.2 consists of five B12 icosahedra (I1–I5) and a B12Si3 polyhedron shown in figure 8a.
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Atomic coordinates, site occupancies and isotropic displacement factors are listed in table IX. More than 500 atoms are available in the unit cell. In the crystal structure, there are six structurally independent icosahedra I1–I6, which are constructed from B1–B12, B13–B24, B25–B32, B33–B40, B41–B44 and B45–B56 sites, respectively; B57–B62 sites form a B8 polyhedron. The Sc4.5–xB57–y+zC3.5–z crystal structure is layered, as shown in figure 26. This structure has been described in terms of two kinds of boron icosahedron layers, L1 and L2. L1 consists of the icosahedra I3, I4 and I5 and the C65 "dimer", and L2 consists of the icosahedra I2 and I6. I1 is sandwiched by L1 and L2 and the B8 polyhedron is sandwiched by L2.
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Boron framework of YAlB14 is one of the simplest among icosahedron-based borides – it consists of only one kind of icosahedra and one bridging boron site. The bridging boron site is tetrahedrally coordinated by four boron atoms. Those atoms are another boron atom in the counter bridge site and three equatorial boron atoms of one of three B12 icosahedra. Aluminium atoms are separated by 0.2911 nm and are arranged in lines parallel to the x-axis, whereas yttrium atoms are separated by 0.3405 nm.
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The virus is short, rod-shaped, and non- enveloped. The nucleocapsid has been measured at 38 nm long and 15-22 nm in diameter. Each nucleocapsid includes 22 capsomeres. Particles are two incomplete icosahedra joined together.
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J. W. Mellor, A Comprehensive Treatise on Inorganic and Theoretical Chemistry Vol. 5, Longmans & Co. (1924) p. 27. The silicon borides may be grown from boron-saturated silicon in either the solid or liquid state. The SiB6 crystal structure contains interconnected icosahedra (polyhedra with 20 faces), icosihexahedra (polyhedra with 26 faces), as well as isolated silicon and boron atoms. Due to the size mismatch between the silicon and boron atoms, silicon can be substituted for boron in the B12 icosahedra up to a limiting stoichiometry corresponding to SiB2.89.
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Truly icosahedral crystals may be formed by quasicrystalline materials which are very rare in nature but can be produced in a laboratory.. A more recent discovery is of a series of new types of carbon molecule, known as the fullerenes (see Curl, 1991). Although C60, the most easily produced fullerene, looks more or less spherical, some of the larger varieties (such as C240, C480 and C960) are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across. Circogonia icosahedra, a species of Radiolaria. Polyhedra appear in biology as well.
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Virus particles are non-enveloped. The nucleocapsid is 38 nanometers (nm) long and 15–22 nm in diameter. While particles have basic icosahedral symmetry, they consist of two incomplete icosahedra—missing one vertex—joined together. There are 22 capsomeres per nucleocapsid.
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B54–B59 sites form the irregularly shaped B16 polyhedron in which only 10.7 boron atoms are available because most of sites are too close to each other to be occupied simultaneously. Ten bridging sites C60–B69 interconnect polyhedron units or other bridging sites to form a 3D boron framework structure. One description of the crystal structure uses three pillar-like units that extend along the c-axis that however results in undesired overlaps between those three pillar- like units. An alternative is to define two pillar-like structure units. Figure 29 shows the boron framework structure of Sc3.67–xB41.4–y–zC0.67+zSi0.33–w viewed along the c-axis, where the pillar- like units P1 and P2 are colored in dark green and light green respectively and are bridged by yellow icosahedra I4 and I7. These pillar-like units P1 and P2 are shown in figures 30a and b, respectively. P1 consists of icosahedra I1 and I3, an irregularly shaped B16 polyhedron and other bridge site atoms where two supericosahedra can be seen above and below the B16 polyhedron. Each supericosahedron is formed by three icosahedra I1 and three icosahedra I3 and is the same as the supericosahedron O(1) shown in figure 24a.
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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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An unusual linkage is depicted in figure 8b, where two B12-I5 icosahedra connect via two B atoms of each icosahedron forming an imperfect square. The boron framework of YB41Si1.2 can be described as a layered structure where two boron networks (figures 9a,b) stack along the z-axis. One boron network consists of 3 icosahedra I1, I2 and I3 and is located in the z = 0 plane; another network consists of the icosahedron I5 and the B12Si3 polyhedron and lies at z = 0.5. The icosahedron I4 bridges these networks, and thus its height along the z-axis is 0.25.
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The opposite edge of the supertetrahedron is parallel to the b-axis and the icosahedra on this edge form a chain along the b-axis. As shown in figure 19, there are wide tunnels surrounded by the icosahedron arrangement along the a- and b-axes. The tunnels are filled by the B22 units which strongly bond to the surrounding icosahedra; the connection of the B22 units is helical and it runs along the c-axis as shown in figure 19b. Scandium atoms occupy the voids in the boron network as shown in figure 19c, and the Si atoms bridge the B22 units.
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The truncated 600-cell consists of 600 truncated tetrahedra and 120 icosahedra. The truncated tetrahedral cells are joined to each other via their hexagonal faces, and to the icosahedral cells via their triangular faces. Each icosahedron is surrounded by 20 truncated tetrahedra.
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Thus, REB22C2N and REB28.5C4 have rather large c-lattice constants. Because of the small size of the B6 octahedra, they cannot interconnect. Instead, they bond to the B12 icosahedra in the neighboring layer, and this decreases bonding strength in the c-plane.
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Their crystal structure and chemical bonding depend strongly on the metal element M and on its atomic ratio to boron. When B/M ratio exceeds 12, boron atoms form B12 icosahedra which are linked into a three-dimensional boron framework, and the metal atoms reside in the voids of this framework. Those icosahedra are basic structural units of most allotropes of boron and boron-rich rare-earth borides. In such borides, metal atoms donate electrons to the boron polyhedra, and thus these compounds are regarded as electron- deficient solids. The crystal structures of many boron-rich borides can be attributed to certain types including MgAlB14, YB66, REB41Si1.2, B4C and other, more complex types such as RExB12C0.33Si3.0.
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Because of the small size of the B6 octahedra, they cannot interconnect. Instead, they bond to the B12 icosahedra in the neighboring layer, and this decreases bonding strength in the c-plane. Because of the B12 structural unit, the chemical formula of "ideal" boron carbide is often written not as B4C, but as B12C3, and the carbon deficiency of boron carbide described in terms of a combination of the B12C3 and B12C2 units. Some studies indicate the possibility of incorporation of one or more carbon atoms into the boron icosahedra, giving rise to formulas such as (B11C)CBC = B4C at the carbon-heavy end of the stoichiometry, but formulas such as B12(CBB) = B14C at the boron-rich end.
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Vertex figure for the bialternatosnub 16-cell The bialternatosnub 16-cell or runcic snub rectified 16-cell, constructed by removing alternating long rectangles from the octagons, but is also not uniform. Like the omnisnub tesseract, it has a highest symmetry construction of order 192, with 8 rhombicuboctahedra (with Th symmetry), 16 icosahedra (with T symmetry), 24 rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry), 32 triangular prisms, with 96 triangular prisms (as Cs-symmetry wedges) filling the gaps. A variant with regular icosahedra and uniform triangular prisms has two edge lengths in the ratio of 1 : 2, and occurs as a vertex-faceting of the scaliform runcic snub 24-cell.
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The Fifty-Nine Icosahedra. 59 is one of the factors that divides the smallest composite Euclid number. In this case 59 divides the Euclid number 13# + 1 = 2 × 3 × 5 × 7 × 11 × 13 + 1 = 59 × 509 = 30031. 59 is the highest integer a single symbol may represent in the Sexagesimal system.
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Its dihedral angle is approximately 138.19°. Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex polytope in n dimensions, at least three facets must meet at a peak and leave a positive defect for folding in n-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the snub 24-cell), just as hexagons can be used as faces in semi-regular polyhedra (for example the truncated icosahedron). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the icosahedral 120-cell, one of the ten non-convex regular polychora.
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John Flinders Petrie was a lifelong friend of Coxeter and had a remarkable ability to visualise four-dimensional geometry. He and Coxeter had worked together on many mathematical problems. His direct contribution to the fifty nine icosahedra was the exquisite set of three-dimensional drawings which provide much of the fascination of the published work.
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The icosahedral shell encapsulating enzymes and labile intermediates are built of different types of proteins with BMC domains. In 1904, Ernst Haeckel described a number of species of Radiolaria, including Circogonia icosahedra, whose skeleton is shaped like a regular icosahedron. A copy of Haeckel's illustration for this radiolarian appears in the article on regular polyhedra.
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The alternating cell/edge path above maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex (six triangular bipyramids) in the 600-cell. This latter path corresponds to a ring of six icosahedra meeting face to face in the snub 24-cell (or icosahedral pyramids in the 600-cell).
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The 11-cell, discovered independently by H. S. M. Coxeter and Branko Grünbaum, is an abstract 4-polytope. Its facets are hemi-icosahedra. Since its facets are, topologically, projective planes instead of spheres, the 11-cell is not a tessellation of any manifold in the usual sense. Instead, the 11-cell is a locally projective polytope.
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The bridging boron site is tetrahedrally coordinated by four boron atoms. Those atoms are another boron atom in the counter bridge site and three equatorial boron atoms of one of three B12 icosahedra. The yttrium sites have partial occupancies of ca. 60–70%, and the YB25 formula merely reflects the average atomic ratio [B]/[Y] = 25.
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Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron. It is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron.
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For the Third Edition, Kate and David Crennell reset the text and redrew the diagrams. They also added a reference section containing tables, diagrams, and photographs of some of the Cambridge models (which at that time were all thought to be Flather's). Corrections to this edition have been published online.K. and D. Crennell; The Fifty-Nine Icosahedra, Fortran Friends, (retrieved 14 September 2017).
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The HRTEM lattice image of figure 22a reproduces well the (a, b) plane of the crystal structure shown in figure 21a, with the clearly visible rings membered by icosahedra I1 and I2 and centered by the "tube". Figure 22b proves that ScB17C0.25 does not have layered character but its c-axis direction is built up by the ring-like structure and tubular structures.
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The superoctahedron O(1) consists of 6 icosahedra I(3) and bridge sites B, C18, C1 and Si1; here Si1 and C1 exhibit a tetrahedral arrangement at the center of O(1). The B10 polyhedra also arrange octahedrally, without the central atom, as shown in figure 24c where the B and C19 atoms bridge the B10 polyhedra to form the octahedral supercluster of the B10 polyhedra. Fig. 25. Boron framework structure of Sc0.83–xB10.0–yC0.17+ySi0.083–z depicted by supertetrahedra T(1) and T(2), superoctahedron O(1) and the superoctahedron based on B10 polyhedron. Vertexes of each superpolyhedron are adjusted to the center of the constituent icosahedra, thus the real volumes of these superpolyhedra are larger than appear in the picture. Using these large polyhedra, the crystal structure of Sc0.83–xB10.0–yC0.17+ySi0.083–z can be described as shown in figure 25.
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These became known as Miller's rules. The 1938 book on the fifty-nine icosahedra resulted, written by Coxeter and Patrick du Val.Stellation and facetting - a brief history In the 1930s, Coxeter and Miller found 12 new uniform polyhedra, a step in the process of their complete classification in the 1950s.Peter R. Cromwell, Polyhedra: "One of the Most Charming Chapters of Geometry" (1999), p. 178.
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In the early 20th century, Ernst Haeckel described a number of species of Radiolaria, some of whose skeletons are shaped like various regular polyhedra (Haeckel, 1904). Examples include Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus and Circorrhegma dodecahedra; the shapes of these creatures are indicated by their names. The outer protein shells of many viruses form regular polyhedra. For example, HIV is enclosed in a regular icosahedron.
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Collectively they are called the Kepler-Poinsot polyhedra. The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by H.S.M. Coxeter and others in 1938, with the now famous paper The 59 icosahedra.. The reciprocal process to stellation is called facetting (or faceting).
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The icosahedra I1 and I2 form a ring centered by the "tube" shown in figure 21b, which probably governs the properties of the ScB17C0.25 crystal. B/C15 and B/C16 mixed-occupancy sites interconnect the rings. A structural similarity can be seen between ScB17C0.25 and BeB3. Figures 22a and b present HRTEM lattice images and electron diffraction patterns taken along the [0001] and [110] crystalline directions, respectively.
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When the composition ratio exceeds 12, boron forms B12 icosahedra (Fig. 1c) which are linked into a three-dimensional boron framework, and the metal atoms reside in the voids of this framework. This complex bonding behavior originates from the fact that boron has only three valence electrons; this hinders tetrahedral bonding as in diamond or hexagonal bonding as in graphite. Instead, boron atoms form polyhedra.
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Black and blue spheres indicate Y and Al atoms, respectively. Vacancies at the Y and Al sites are ignored. Figure 3 shows the crystal structure of YAlB14 viewed along the x-axis. The large black spheres are Y atoms, the small blue spheres are Al atoms and the small green spheres are the bridging boron sites; B12 clusters are depicted as the green icosahedra.
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They have elongated, geminate capsids with two incomplete T=1 icosahedra joined at the missing vertex. The capsids range in size from 18–20 nm in diameter with a length of about 30 nm. Geminiviruses constitute a large family of phytopathogens (Geminiviridae). The family geminiviridae have been classified into four genera, namely Begomovirus, Curtovirus, Topocuvirus and Mastrevirus, depending on their genomes, mode of transmission and host range.
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Boron carbide Crystal structure of B6O Other hard boron-rich compounds include B4C and B6O. Amorphous a-B4C has a hardness of about 50 GPa, which is in the range of superhardness. It can be looked at as consisting of boron icosahedra-like crystals embedded in an amorphous medium. However, when studying the crystalline form of B4C, the hardness is only about 30 GPa.
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The P2 unit consists of icosahedra I2, I5 and I6, B10 polyhedron and other bridge site atoms. Eight Sc sites with occupancies between 0.49 (Sc8) and 0.98 (Sc1) spread over the boron framework. As described above, this hexagonal phase originates from a cubic phase, and thus one may expect a similar structural element in these phases. There is an obvious relation between the hexagonal ab-plane and the cubic (111) plane.
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Coxeter explained in 1937 how he and Petrie began to expand the classical subject of regular polyhedra: :One day in 1926, J. F. Petrie told me with much excitement that he had discovered two new regular polyhedral; infinite but free of false vertices. When my incredulity had begun to subside, he described them to me: one consisting of squares, six at each vertex, and one consisting of hexagons, four at each vertex.H.S.M. Coxeter (1937) "Regular skew polyhedral in three and four dimensions and their topological analogues", Proceedings of the London Mathematical Society (2) 43: 33 to 62 In 1938 Petrie collaborated with Coxeter, Patrick du Val, and H.T. Flather to produce The Fifty-Nine Icosahedra for publication.H. S. M. Coxeter, Patrick du Val, H.T. Flather, J.F. Petrie (1938) The Fifty-nine Icosahedra, University of Toronto studies, mathematical series 6: 1–26 Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes.
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The main group metals, lanthanides and actinides form a wide variety of boron- rich borides, with metal:boron ratios up to YB66. The properties of this group vary from one compound to the next, and include examples of compounds that are semi conductors, superconductors, diamagnetic, paramagnetic, ferromagnetic or anti-ferromagnetic. They are mostly stable and refractory. Some metallic dodecaborides contain boron icosahedra, others (for example yttrium, zirconium and uranium) have the boron atoms arranged in cuboctahedra.
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The B80 boron cluster occupies the large space between four supericosahedra as described in the REB66 section. On the other hand, the 2-dimensional supericosahedron networks in the Sc4.5–xB57–y+zC3.5–z crystal structure stack in-phase along the z-axis. Instead of the B80 cluster, a pair of the I2 icosahedra fills the open space staying within the supericosahedron network, as shown in figure 28 where the icosahedron I2 is colored in yellow.
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The diversity of the crystal structures of rare-earth borides results in unusual physical properties and potential applications in thermopower generation. Thermal conductivity of boron icosahedra based compounds is low because of their complex crystal structure; this property is favored for thermoelectric materials. On the other hand, these compounds exhibit very low (variable range hopping type) p-type electrical conductivity. Increasing the conductivity is a key issue for thermoelectric applications of these borides.
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Continuous transformation pausing at the vertex position of Jessen's icosahedron Jessen's icosahedron is one of a continuous series of icosahedra with 8 regular faces and 12 isosceles faces, described by H. S. M. Coxeter in 1948. The shapes in this family range from cuboctahedron to regular octahedron (as limit cases), which can be inscribed in a regular octahedron. The twisting, expansive-contractive transformations between members of this family were named Jitterbug transformations by Buckminster Fuller.
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Vertex figure for the omnisnub tesseract The full snub tesseract or omnisnub tesseract, defined as an alternation of the omnitruncated tesseract, can not be made uniform, but it can be given Coxeter diagram , and symmetry [4,3,3]+, and constructed from 8 snub cubes, 16 icosahedra, 24 square antiprisms, 32 octahedra (as triangular antiprisms), and 192 tetrahedra filling the gaps at the deleted vertices. It has 272 cells, 944 faces, 864 edges, and 192 vertices.
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Self-assembly is also a method to create patchy particles. This method allows formation of complex structures like chains, sheets, rings, icosahedra, square pyramids, tetrahedra, and twisted staircase structures. By coating the surface of particles with highly anisotropic, highly directional, weakly interacting patches, the arrangement of the attractive patches can organize disordered particles into structures. The coating and the arrangement of the attractive patches is what contributes to the size, shape, and structure of the resulting particle.
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These icosahedra arrange in a thirteen-icosahedron unit (B12)12B12 which is called supericosahedron. The icosahedron formed by the B1 site atoms is located at the center of the supericosahedron. The supericosahedron is one of the basic units of the boron framework of YB66. There are two types of supericosahedra: one occupies the cubic face centers and another, which is rotated by 90°, is located at the center of the cell and at the cell edges.
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Blinn used the vertex coordinates of regular icosahedra and dodecahedra to determine the placement of electric field lines radiating away from point charges. Most of the narration was voiced by actor Aaron Fletcher, who also played Galileo Galilei in the historical segments. Some portions, such as explanations of particular technical details, were narrated by Sally Beaty, the show's executive producer. Shorter versions of Mechanical Universe episodes, 10 to 20 minutes in length, were created for use in high schools.
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RExB12C0.33Si3.0 (RE=Y and Gd–Lu) have a unique crystal structure with two units – a cluster of B12 icosahedra and a Si8 ethane-like complex – and one bonding configuration (B12)3≡Si-C≡(B12)3. A representative compound of this group is YxB12C0.33Si3.0 (x=0.68). It has a trigonal crystal structure with space group Rm (No. 166) and lattice constants a = b = 1.00841(4) nm, c = 1.64714(5) nm, α = β = 90° and γ = 120°. Fig. 15.
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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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Amorphous boron contains B12 regular icosahedra that are randomly bonded to each other without long range order. Pure amorphous boron can be produced by thermal decomposition of diborane at temperatures below 1000 °C. Annealing at 1000 °C converts amorphous boron to β-rhombohedral boron. Amorphous boron nanowires (30–60 nm thick) or fibers can be produced by magnetron sputtering and laser- assisted chemical vapor deposition, respectively; and they also convert to β-rhombohedral boron nanowires upon annealing at 1000 °C.
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The resulting cluster of 19 atoms has two interior atoms (the centers of the two icosahedra) with 17 atoms in the outer shell in the pattern of the Errera graph. The dual graph of the Errera graph is a fullerene with 30 vertices, designated in the chemistry literature as C30(D5h) or F30(D5h) to indicate its symmetry and distinguish it from other 30-vertex fullerenes. This shape also plays a central role in the construction of higher-dimensional fullerenes.
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Compressing boron above 160 GPa produces a boron phase with an as yet unknown structure. Contrary to other phases, which are semiconductors, this phase is a metal and becomes a superconductor with a critical temperature increasing from 4 K at 160 GPa to 11 K at 250 GPa. This structural transformation occurs at pressures at which theory predicts the icosahedra will dissociate. Speculation as to the structure of this phase has included face-centred cubic (analogous to Al); α-Ga, and body-centred tetragonal (analogous to In).
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Nonuniform variants with [4,3,4] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra to produce a nonuniform honeycomb with truncated octahedra and hexagonal prisms (as ditrigonal trapezoprisms). Its vertex figure is a C2v-symmetric triangular bipyramid. This honeycomb can then be alternated to produce another nonuniform honeycomb with pyritohedral icosahedra, octahedra (as triangular antiprisms), and tetrahedra (as sphenoids). Its vertex figure has C2v symmetry and consists of 2 pentagons, 4 rectangles, 4 isosceles triangles (divided into two sets of 2), and 4 scalene triangles.
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Although a very small amount of carbon (less than 2 wt%!) plays an important role in the phase stability, carbon does not have its own sites but shares with boron two interstitial sites B/C15 and B/C16. There are two inequivalent B12 icosahedra, I1 and I2, which are constructed by the B1–B5 and B8–B12 sites, respectively. A "tube" is another characteristic structure unit of ScB17C0.25. It extends along the c-axis and consists of B13, B14, B17 and B18 sites where B13 and B14 form 6-membered rings.
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This unit can be observed in β-tetragonal boron and is a modification of the B20 unit of α-AlB12 (or B19 unit in early reports). The B20 unit is a twinned icosahedron made from B13 to B22 sites with two vacant sites and one B atom (B23) bridging both sides of the unit. The twinned icosahedron is shown in figure 18a. B23 was treated as an isolated atom in the early reports; it is bonded to each twinned icosahedra through B18 and to another icosahedron through B5 site.
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Mathematical model of the tensegrity icosahedron Different shapes of tensegrity icosahedra, depending on the ratio between the lengths of the tendons and the struts. The polyhedron which corresponds directly to the geometry of the tensegrity icosahedron is called the Jessen's icosahedron. Its spherical dynamics were of special interest to Buckminster Fuller, who referred to its expansion-contraction transformations around a stable equilibrium as jitterbug motion. The following is a mathematical model for figures related to the tensegrity icosahedron, explaining why it is a stable construction, albeit with infinitesimal mobility.
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The stellation of a polyhedron extends the faces of a polyhedron into infinite planes and generates a new polyhedron that is bounded by these planes as faces and the intersections of these planes as edges. The Fifty Nine Icosahedra enumerates the stellations of the regular icosahedron, according to a set of rules put forward by J. C. P. Miller, including the complete stellation. The Du Val symbol of the complete stellation is H, because it includes all cells in the stellation diagram up to and including the outermost "h" layer.
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The stellation diagram for the icosahedron with the central triangle marked for the original icosahedron The Fifty-Nine Icosahedra is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put forward by J. C. P. Miller. First published by the University of Toronto in 1938, a Second Edition reprint by Springer-Verlag followed in 1982. Tarquin's 1999 Third Edition included new reference material and photographs by K. and D. Crennell.
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A small amount of Si was added into the floating zone crystal growth and thus this phase is a quaternary compound. Its rare cubic structure has 26 sites in the unit cell: three Sc sites, two Si sites, one C site and 20 B sites; 4 out of 20 B sites are boron-carbon mixed- occupancy sites. Atomic coordinates, site occupancies and isotropic displacement factors are listed in table VIII. In the unit cell, there are three independent icosahedra, I1, I2 and I3, and a B10 polyhedron which are formed by the B1–B4, B5–B8, B9–B13 and B14–B17 sites, respectively.
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1, pp. 701–708. In 1924 he began part-time teaching (in addition to his high school teaching) as an adjunct instructor of geometry, first at Brown University and then at Wellesley College; however, his college-level adjunct teaching ended by the early 1930s. Wheeler and H. S. M. Coxeter planned to be coauthors (with two other mathematicians) of a short book, which was eventually named The Fifty-Nine Icosahedra and became a minor classic of mathematical literature. However, in 1938 Wheeler objected to Coxeter's expository style so that Coxeter replaced Wheeler's name on the book's title page by another author, although Wheeler is briefly mentioned in the text.
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Model of an icosahedron made with metallic spheres and magnetic connectors The 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly subdividing each edge into the golden mean along the direction of its vector. The five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in this way from any given octahedron form a uniform polyhedron compound. Regular icosahedron and its circumscribed sphere.
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If these faces are aligned with a congruent patch of faces on the icosidodecahedron, then the hexagonal face will lie in the plane midway between two opposing triangular faces of the icosidodecahedron. The triangular hebesphenorotunda also has clusters of faces that can be aligned with corresponding faces of the rhombicosidodecahedron: the three lunes, each lune consisting of a square and two antipodal triangles adjacent to the square. The faces around each (33.5) vertex can also be aligned with the corresponding faces of various diminished icosahedra. Johnson uses the prefix hebespheno- to refer to a blunt wedge-like complex formed by three adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides.
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By the early years of the twentieth century, mathematicians had moved on and geometry was little studied. Coxeter's analysis in The Fifty-Nine Icosahedra introduced modern ideas from graph theory and combinatorics into the study of polyhedra, signalling a rebirth of interest in geometry. Coxeter himself went on to enumerate the star uniform polyhedra for the first time, to treat tilings of the plane as polyhedra, to discover the regular skew polyhedra and to develop the theory of complex polyhedra first discovered by Shephard in 1952, as well as making fundamental contributions to many other areas of geometry. In the second part of the twentieth century, Grünbaum published important works in two areas.
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Boron carbide was discovered in the 19th century as a by-product of reactions involving metal borides, but its chemical formula was unknown. It was not until the 1930s that the chemical composition was estimated as B4C.Ridgway, Ramond R "Boron Carbide", European Patent CA339873 (A), publication date: 1934-03-06 Controversy remained as to whether or not the material had this exact 4:1 stoichiometry, as, in practice the material is always slightly carbon- deficient with regard to this formula, and X-ray crystallography shows that its structure is highly complex, with a mixture of C-B-C chains and B12 icosahedra. These features argued against a very simple exact B4C empirical formula.
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This phase is not described in the phase diagram of figure 17 because it is a quaternary compound. Its hexagonal structure is rare and has 79 atomic positions in the unit cell: eight partially occupied Sc sites, 62 B sites, two C sites, two Si sites and six B/C sites. Six B sites and one of the two Si sites have partial occupancies. The associated atomic coordinates, site occupancies and isotropic displacement factors are listed in table X. There are seven structurally independent icosahedra I1–I7 which are formed by B1–B8, B9–B12, B13–B20, B/C21–B24, B/C25–B29, B30–B37 and B/C38–B42 sites, respectively; B43–B46 sites form the B9 polyhedron and B47–B53 sites construct the B10 polyhedron.
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The first condensed polyhedral borane, , is formed by sharing four vertices between two icosahedra. According to Wade's n + 1 rule for n-vertex closo structures, should have a charge of +2 (n + 1 = 20 + 1 = 21 pairs required; 16 BH units provide 16 pairs; four shared boron atoms provide 6 pairs; thus 22 pairs are available). To account for the existence of as a neutral species, and to understand the electronic requirement of condensed polyhedral clusters, a new variable, m, was introduced and corresponds to the number of polyhedra (sub- clusters). In Wade's n + 1 rule, the 1 corresponds to the core bonding molecular orbital (BMO) and the n corresponds to the number of vertices, which in turn is equal to the number of tangential surface BMOs.
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B17 and B18 sites also form 6-membered rings; however, their mutual distances (0.985 Å for B17 and 0.955 Å for B18) are too short for a simultaneous occupation of the neighboring sites. Therefore, boron atoms occupy 2nd neighbor site forming a triangle. The occupancies of B17 and B18 sites should be 50%, but the structure analysis suggests larger values. The crystal structure viewed along the a-axis is shown in figure 20, which suggests that the ScB17C0.25 is a layered material. Two layers, respectively constructed by the icosahedra I1 and I2, alternatively stack along the c-axis. However, the ScB17C0.25 crystal is not layered. For example, during arc-melting, ScB17C0.25 needle crystals violently grow along the c-axis – this never happens in layered compounds. The crystal structure viewed along the c-axis is shown in figure 21a.
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