Uniform honeycombs in hyperbolic space

In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.

Four compact regular hyperbolic honeycombs

Order-4 dodecahedral honeycomb {5,3,4}	Order-5 dodecahedral honeycomb {5,3,5}
Order-5 cubic honeycomb {4,3,5}	Icosahedral honeycomb {3,5,3}
Poincaré ball model projections