# Trigonal trapezohedral honeycomb

The trigonal trapezohedral honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. Cells are identical trigonal trapezohedron or rhombohedra. John Horton Conway calls it an oblate cubille.

Trigonal trapezohedral honeycomb
(No image)
TypeDual uniform honeycomb
Coxeter-Dynkin diagrams
Cell
Trigonal trapezohedron
(1/4 of rhombic dodecahedron)
Faces Rhombus
Space groupFd3m (227)
Coxeter group${\displaystyle {\tilde {A}}_{3}}$ ×2, [[3[4]]] (double)
vertex figures
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DualQuarter cubic honeycomb
PropertiesCell-transitive, Face-transitive

This honeycomb can be seen as a rhombic dodecahedral honeycomb, with the rhombic dodecahedra dissected with its center into 4 trigonal trapezohedra or rhombohedra.

 rhombic dodecahedral honeycomb Rhombic dodecahedra dissection Rhombic net

It is analogous to the regular hexagonal being dissectable into 3 rhombi and tiling the plane as a rhombille. The rhombille tiling is actually an orthogonal projection of the trigonal trapezohedral honeycomb. A different orthogonal projection produces the quadrille where the rhombi are distorted into squares.

## Dual tiling

It is dual to the quarter cubic honeycomb with tetrahedral and truncated tetrahedral cells: