Trapezo-rhombic dodecahedron

Template:Short description Template:Infobox polyhedron File:Trapezo-rhombic dodecahedron.stl

In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has Template:Math 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.Template:R

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.Template:R It is the dual polyhedron of a triangular orthobicupola, a Johnson solid whose faces are squares and equilateral triangles.Template:R

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.Template:R 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.Template:R

A trapezo-rhombic dodecahedron has two different edges lengths, $\frac{2}{3} a$ and $\frac{4}{3} a$ . Its surface area $A$ and volume $V$ are given byTemplate:R $A = 8 \sqrt{2} a^{2} \approx 11.314 a^{2}, V = \frac{16}{9} \sqrt{3} a^{3} \approx 3.079 a^{3} .$

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 Template:Math 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.Template:Cn