Order-4 dodecahedral honeycomb
In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) of hyperbolic 3-space. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.
| Order-4 dodecahedral honeycomb | |
|---|---|
| Type | Hyperbolic regular honeycomb Uniform hyperbolic honeycomb |
| Schläfli symbol | {5,3,4} {5,31,1} |
| Coxeter diagram | ↔ |
| Cells | {5,3} (dodecahedron) |
| Faces | {5} (pentagon) |
| Edge figure | {4} (square) |
| Vertex figure | octahedron |
| Dual | Order-5 cubic honeycomb |
| Coxeter group | BH3, [4,3,5] DH3, [5,31,1] |
| Properties | Regular, Quasiregular honeycomb |
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.