Trigonal trapezohedral honeycombTypeDual uniform honeycombCoxeter-Dynkin diagramsCellTrigonal trapezohedron(1/4 of rhombic dodecahedron)FacesRhombusSpace groupCoxeter group (double)vertex figures DualQuarter cubic honeycombPropertiesCell-transitive, Face-transitive
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.
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 honeycombRhombic dodecahedra dissectionRhombic 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
References