Order-4 hexagonal tiling honeycomb

Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps.

[1] The order-4 hexagonal tiling honeycomb has three reflective simplex symmetry constructions.

The half-symmetry uniform construction {6,31,1} has two types (colors) of hexagonal tilings, with Coxeter diagram ↔ .

The latter has a cubic fundamental domain, and an octahedral Coxeter diagram with three axial infinite branches: .

The omnitruncated order-4 hexagonal tiling honeycomb, t0,1,2,3{6,3,4}, has truncated cuboctahedron, truncated trihexagonal tiling, dodecagonal prism, and octagonal prism cells, with an irregular tetrahedron vertex figure.

The runcicantic order-4 hexagonal tiling honeycomb, ↔ , is composed of trihexagonal tiling, truncated cuboctahedron, truncated cube, and triangular prism cells, with a rectangular pyramid vertex figure.