| Trapezo-rhombic dodecahedron | |
|---|---|
| Type | Johnson solid dual |
| Faces | 6 rhombus, 6 trapezoid |
| Edges | 24 |
| Vertices | 14 |
| Vertex configuration | (2) 4.4.4 (6) 4.4.4.4 (6) 4.4.4 |
| Symmetry group | Dihedral (D3h) |
| Dual polyhedron | Triangular orthobicupola |
| Properties | convex |
The trapezo-rhombic dodecahedron is a convex polyhedron with 6 rhombic and 6 trapezoidal faces.
This shape could be constructed by taking a tall uniform hexagonal prism, and making 3 angled cuts on the top and bottom. The trapezoids represent what remains of the original prism sides, and the 6 rhombi a result of the top and bottom cuts.
A space-filling tessellation (or honeycomb) can be made by translated copies of this cell. Each "layer" is a hexagonal tiling, and alternate layers are connected by shifting their centers and rotating each polyhedron so the rhombic faces match up.
In the special case that the long sides of the trapezoids equals twice the length of the short sides, the solid now represents the 3D Voronoi cell of a sphere in a Hexagonal Close Packing (HCP), next to Face-Centered-Cubic, an ideal way to stack spheres. It is therefore similar to the rhombic dodecahedron, which can be represented by turning the lower half of the picture at right over an angle of 60 degrees. The rhombic dodecahedron is a Voronoi cell of the other ideal way to stack spheres.
It is the polyhedral dual of the triangular orthobicupola.