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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Examples include " Circoporus octahedrus ", " Circogonia icosahedra ", " Lithocubus geometricus " and " Circorrhegma dodecahedra "; the shapes of these creatures are indicated by their names.
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The great icosahedron, as a uniform " retrosnub tetrahedron " 50px, is similar to these snub-pair compounds : compound of two icosahedra, compound of two snub cubes and compound of two snub dodecahedra.
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The regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge; their dihedral angles thus are ? / 2 and 2? / 5, both of which are less than that of a Euclidean dodecahedron.
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The small and great stellated dodecahedra, sometimes called the "'Kepler polyhedra "', were first recognized as regular by Johannes Kepler in 1619 . He obtained them by convex, as the traditional Platonic solids were.
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One related non-wythoffian form is constructed from the { 3, 5, 3 } vertex figure with 4 ( tetrahedrally arranged ) vertices removed, creating pentagonal antiprisms and dodecahedra filling in the gaps, called a tetrahedrally diminished dodecahedron.
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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 }.
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The vertices in rhombic dodecahedral packing appear to be the centres of tetrahedron or octahedron ( taking the centres of surrounding ) rhombic dodecahedra as vertices-is there a name for this type of ( octahedron / tetrahedron ) / ( rhombic dodecahedron ) dual relationship
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The substance can be seen with empirical formula B 12 C 3 ( i . e ., with B 12 dodecahedra being a motif ), but with less carbon, as the suggested C 3 units are replaced with C-B-C chains, and some smaller ( B 6 ) octahedra are present as well ( see the boron carbide article for structural analysis ).
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There are 6, 384, 634 topologically distinct " convex " dodecahedra, excluding mirror images, having at least 8 vertices . ( Two polyhedra are " topologically distinct " if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces .)
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