Trapezo-rhombic dodecahedron
Polyhedron
In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has D3h symmetry. A concave form can be constructed with an identical net, seen as excavating trigonal trapezohedra from the top and bottom. It is also called the trapezoidal dodecahedron.
Description
The trapezo-rhombic dodecahedron is a polyhedron with six trapezoidal and six rhombic faces. This polyhedron can be obtained from a rhombic dodecahedron by cutting it in half through the hexagonal cross-section and rotating the halves 60° with respect to each other. It is the dual polyhedron of a triangular orthobicupola, a Johnson solid whose faces are squares and equilateral triangles.
The trapezo-rhombic dodecahedron is a plesiohedron, a special kind of polyhedron that can tile a space with its copy, representing the three-dimensional Voronoi cell of a sphere in a hexagonal close packing; this and the face-centered cubic packing are the densest ways to pack spheres. It is therefore related to the rhombic dodecahedron, as suggested from the construction, which is a Voronoi cell of the other optimal way to pack spheres. The two shapes differ in their combinatorial structure as well as in their geometry: in the rhombic dodecahedron, every edge connects a degree-three vertex to a degree-four vertex, whereas the trapezo-rhombic dodecahedron has six edges that connect vertices of equal degrees.
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This entry incorporates text from “Trapezo-rhombic dodecahedron” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.