Trigonal trapezohedral honeycomb

In geometry, the trigonal trapezohedral honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. Cells are identical trigonal trapezohedra or rhombohedra. Conway, Burgiel, and Goodman-Strauss call it an oblate cubille.^[1]

Trigonal trapezohedral honeycomb
Type	Dual uniform honeycomb
Coxeter-Dynkin diagrams
Cell	Trigonal trapezohedron (1/4 of rhombic dodecahedron)
Faces	Rhombus
Space group	Fd3m (227)
Coxeter group	⁠${\displaystyle {\tilde {A}}_{3}}$⁠×2, 3[4] (double)
vertex figures	|
Dual	Quarter cubic honeycomb
Properties	Cell-transitive, Face-transitive

Related honeycombs and tilings

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:

See also

• Architectonic and catoptric tessellation