It has a half symmetry construction, {5,31,1}, with two types (colors) of dodecahedra in the Wythoff construction. ↔ .
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During the Hellenistic era, small, hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe. Their purpose is not certain. In 20th-century art, dodecahedra appear in the work of M. C. Escher, such as his lithographs Reptiles (1943) and Gravitation (1952). In Salvador Dalí's painting The Sacrament of the Last Supper (1955), the room is a hollow regular dodecahedron.
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The dodecahedral prism consists of two dodecahedra connected to each other via 12 pentagonal prisms. The pentagonal prisms are joined to each other via their square faces.
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There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,5,3}, , also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.
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There are four regular compact honeycombs in 3D hyperbolic space: There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb, {5,3,4}, which has only four dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively curved space (the surface of a 4-dimensional sphere), with three dodecahedra on each edge, {5,3,3}. Lastly the dodecahedral ditope, {5,3,2} exists on a 3-sphere, with 2 hemispherical cells. There are nine uniform honeycombs in the [5,3,5] Coxeter group family, including this regular form.
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This family has diploid hexacosichoric symmetry, [5,3,3], of order 120×120=24×600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra. There is one small index subgroups [5,3,3]+, all order 7200.
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The rhombic dodecahedral honeycomb can be dissected into a trigonal trapezohedral honeycomb with each rhombic dodecahedron dissected into 4 trigonal trapezohedrons. Each rhombic dodecahedra can also be dissected with a center point into 12 rhombic pyramids of the rhombic pyramidal honeycomb.
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Other forms include dodecahedra and (rarely) cubes. There is evidence that nitrogen impurities play an important role in the formation of well-shaped euhedral crystals. The largest diamonds found, such as the Cullinan Diamond, were shapeless. These diamonds are pure (i.e.
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Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold. It is a quotient space of the order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.
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It includes new material on knotted polyhedra and on rings of regular octahedra and regular dodecahedra; as the ring of dodecahedra forms the outline of a golden rhombus, it can be extended to make skeletal pentagon-faced versions of the convex polyhedra formed from the golden rhombus, including the Bilinski dodecahedron, rhombic icosahedron, and rhombic triacontahedron. The second edition also includes the Császár polyhedron and Szilassi polyhedron, toroidal polyhedra with non-regular faces but with pairwise adjacent vertices and faces respectively, and constructions by Alaeglu and Giese of polyhedra with irregular but congruent faces and with the same numbers of edges at every vertex.
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Coffinite is isostructural with the orthosilicates zircon (ZrSiO4) and thorite (ThSiO4). Stieff et al. analyzed coffinite using the x-ray powder diffraction technique and determined that it has a tetragonal structure. Occurring naturally with U4+ cations, the UO8 triangular dodecahedra coordinate with edge-sharing, alternating SiO4 tetrahedra in chains along the c-axis.
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Dover Publications. ISBN. The closed ends of the honeycomb cells are also an example of geometric efficiency, though three-dimensional. The ends are trihedral (i.e., composed of three planes) sections of rhombic dodecahedra, with the dihedral angles of all adjacent surfaces measuring 120°, the angle that minimizes surface area for a given volume.
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For the polar reciprocals of the regular and uniform polytopes, the dual facets will be polar reciprocals of the original's vertex figure. For example, in four dimensions, the vertex figure of the 600-cell is the icosahedron; the dual of the 600-cell is the 120-cell, whose facets are dodecahedra, which are the dual of the icosahedron.
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Fine rhombic dodecahedra occur in the schistose rocks of the Zillertal, in Tyrol, and are sometimes cut and polished. An almandine in which the ferrous oxide is replaced partly by magnesia is found at Luisenfeld in German East Africa. In the United States there are many localities which yield almandine. Fine crystals of almandine embedded in mica-schist occur near Wrangell in Alaska.
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Some diamonds have opaque fibers. They are referred to as opaque if the fibers grow from a clear substrate or fibrous if they occupy the entire crystal. Their colors range from yellow to green or gray, sometimes with cloud-like white to gray impurities. Their most common shape is cuboidal, but they can also form octahedra, dodecahedra, macles or combined shapes.
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Bixbyite is a manganese iron oxide mineral with chemical formula: (Mn,Fe)2O3. The iron/manganese ratio is quite variable and many specimens have almost no iron. It is a metallic dark black with a Mohs hardness of 6.0 - 6.5. It is a somewhat rare mineral sought after by collectors as it typically forms euhedral isometric crystals exhibiting various cubes, octahedra, and dodecahedra.
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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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Further discovered structures include complex structures with internal spheres/foam cells. Some dry foam structures with interior cells were found to consist of a chain of pentagonal dodecahedra or Kelvin cells in the centre of the tube. For many more arrangements of this type, it was observed that the outside bubble layer is ordered, with each internal layer resembling a different, simpler columnar structure by using X-ray tomography.
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Matila Ghyka, The Geometry of Art and Life (1977), p.68 The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be thought of as the 4-dimensional analog of the regular dodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedra, with 3 around each edge.
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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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The pentagonal-prism-first orthographic projection of the dodecahedral prism into 3D space has a decagonal envelope (see diagram). Two of the pentagonal prisms project to the center of this volume, each surrounded by 5 other pentagonal prisms. They form two sets (each consisting of a central pentagonal prism surrounded by 5 other non-uniform pentagonal prisms) that cover the volume of the decagonal prism twice. The two dodecahedra project onto the decagonal faces of the envelope.
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The substance can be seen with empirical formula B12C3 (i.e., with B12 dodecahedra being a motif), but with less carbon, as the suggested C3 units are replaced with C-B-C chains, and some smaller (B6) octahedra are present as well (see the boron carbide article for structural analysis). The repeating polymer plus semi-crystalline structure of boron carbide gives it great structural strength per weight. It is used in tank armor, bulletproof vests, and numerous other structural applications.
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The dihedral angle of a Euclidean regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°.
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In 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size. The regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge; their dihedral angles thus are π/2 and 2π/5, both of which are less than that of a Euclidean dodecahedron. Apart from this effect, the hyperbolic honeycombs obey the same topological constraints as Euclidean honeycombs and polychora. The 4 compact and 11 paracompact regular hyperbolic honeycombs and many compact and paracompact uniform hyperbolic honeycombs have been enumerated.
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However, a 4-polytope can be considered a tessellation of a 3-dimensional non- Euclidean space, namely, a tessellation of the surface of a four-dimensional sphere (a 4-dimensional spherical tiling). A regular dodecahedral honeycomb, {5,3,4}, of hyperbolic space projected into 3-space. Locally, this space seems like the one we are familiar with, and therefore, a virtual-reality system could, in principle, be programmed to allow exploration of these "tessellations", that is, of the 4-dimensional regular polytopes. The mathematics department at UIUC has a number of pictures of what one would see if embedded in a tessellation of hyperbolic space with dodecahedra.
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360 B.C.), as a personage of Plato's dialogue, associates the other four platonic solids with the four classical elements, adding that there is a fifth solid pattern which, though commonly associated with the regular dodecahedron, is never directly mentioned as such; "this God used in the delineation of the universe."Plato, Timaeus, Jowett translation [line 1317–8]; the Greek word translated as delineation is diazographein, painting in semblance of life. Aristotle also postulated that the heavens were made of a fifth element, which he called aithêr (aether in Latin, ether in American English). Regular dodecahedra have been used as dice and probably also as divinatory devices.
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For, in the Bilinski dodecahedron, the long body diagonal is parallel to the short diagonals of two faces, and to the long diagonals of two other faces. In the rhombic dodecahedron, the corresponding body diagonal is parallel to four short face diagonals, and in any affine transformation of the rhombic dodecahedron this body diagonal would remain parallel to four equal-length face diagonals. Another difference between the two dodecahedra is that, in the rhombic dodecahedron, all the body diagonals connecting opposite degree-4 vertices are parallel to face diagonals, while in the Bilinski dodecahedron the shorter body diagonals of this type have no parallel face diagonals.
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Another method based on the diffusion model is the hex-path algorithm, developed by R. Andrew Russel for underground chemical odor localization with a buried probe controlled by a robotic manipulator. The probe moves at a certain depth along the edges of a closely packed hexagonal grid. At each state junction , there are two paths (left and right) for choosing, and the robot will take the path that leads to higher concentration of the odor based on the previous two junction states odor concentration information , . In the 3D version of the hex-path algorithm, the dodecahedron algorithm, the probe moves in a path that corresponds to a closely packed dodecahedra, so that at each state point there are three possible path choices.
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This shape was called a Siamese dodecahedron in the paper by Hans Freudenthal and B. L. van der Waerden (1947) which first described the set of eight convex deltahedra.. The dodecadeltahedron name was given to the same shape by , referring to the fact that it is a 12-sided deltahedron. There are other simplicial dodecahedra, such as the hexagonal bipyramid, but this is the only one that can be realized with equilateral faces. Bernal was interested in the shapes of holes left in irregular close-packed arrangements of spheres, so he used a restrictive definition of deltahedra, in which a deltahedron is a convex polyhedron with triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres. This restrictive definition disallows the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and icosahedron (because it has interior room for another sphere).
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