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.
A trapezo-rhombic dodecahedron has two different edges lengths, 2 3 a {\textstyle {\frac {2}{3}}a} and 4 3 a {\textstyle {\frac {4}{3}}a} . Its surface area A {\displaystyle A} and volume V {\displaystyle V} are given by A = 8 2 a 2 ≈ 11.314 a 2 , V = 16 9 3 a 3 ≈ 3.079 a 3 . {\displaystyle A=8{\sqrt {2}}a^{2}\approx 11.314a^{2},\quad V={\frac {16}{9}}{\sqrt {3}}a^{3}\approx 3.079a^{3}.}
Variations The trapezo-rhombic dodecahedron can be seen as an elongation of another dodecahedron, which can be called a rhombo-triangular dodecahedron, with 6 rhombi (or squares) and 6 triangles. It also has D3h symmetry and is space-filling. It has 21 edges and 11 vertices. With square faces, it can be seen as a cube split across the 3-fold axis, separated with the two halves rotated 180°, and filling the gaps with triangles. When used as a space-filler, connecting dodecahedra on their triangles leaves two cubical step surfaces on the top and bottom, which can connect with complementary steps.
See also Elongated dodecahedron Hexagonal prismatic honeycomb
References
External links Weisstein, Eric W. "Trapezo-Rhombic Dodecahedron". MathWorld. VRML model [1]
