Order-6 dodecahedral honeycomb are on the ideal surface. It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures. There are 15 uniform honeycombs in the [5,3,6] Coxeter group family, including this regular form and its regular dual, order-5 hexagonal tiling honeycomb, {6,3,5}. This honeycomb is a part of a sequence of polychora and honeycombs with triangular tiling vertex figures: This honeycomb is a part of a sequence of regular polytopes and honeycombs with dodecahedral cells: The rectified order-6 dodecahedral honeycomb, t{5,3,6} has icosidodecahedron and triangular tiling cells connected in
Honeycomb (geometry) The tetrahedral-octahedral honeycomb and gyrated tetrahedral-octahedral honeycombs are generated by 3 or 2 positions of slab layer of cells, each alternating tetrahedra and octahedra. An infinite number of unique honeycombs can be created by higher order of patterns of repeating these slab layers. A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric. In the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a "space-filling polyhedron". A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, ruling out any of