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The regular dodecahedron can be found in many popular cultures: Roman dodecahedron, the children's story, toys, and painting arts. It can also be found in nature and supramolecules, as well as the shape of the universe. The skeleton of a regular dodecahedron can be represented as the graph called the dodecahedral graph, a Platonic graph.
A pyritohedron is a dodecahedron with pyritohedral (T h) symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices (see figure). [3] However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis.
There are 3 symmetry classes of forms: {3+,3} 1,0 for a tetrahedron, {4+,3} 1,0 for a cube, and {5+,3} 1,0 for a dodecahedron. Values for b , c are divided into three classes: Class I (b=0 or c=0): {3, q +} b ,0 or {3, q +} 0, b represent a simple division with original edges being divided into b sub-edges.
The conjugacy classes of full tetrahedral symmetry, T d ≅ S 4, are: identity; 8 × rotation by 120° 3 × rotation by 180° 6 × reflection in a plane through two rotation axes; 6 × rotoreflection by 90° The conjugacy classes of pyritohedral symmetry, T h, include those of T, with the two classes of 4 combined, and each with inversion: identity
The rhombicosidodecahedron shares its vertex arrangement with three nonconvex uniform polyhedra: the small stellated truncated dodecahedron, the small dodecicosidodecahedron (having the triangular and pentagonal faces in common), and the small rhombidodecahedron (having the square faces in common).
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb.
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