| Order-5 dodecahedral honeycomb | |
|---|---|
| (No image) | |
| Type | Hyperbolic regular honeycomb |
| Schläfli symbol | {5,3,5} |
| Coxeter-Dynkin diagram | |
| Cells | dodecahedron {5,3} |
| Faces | pentagon {5} |
| Edge figure | pentagon {5} |
| Vertex figure | {3,5} |
| Cells/edge | {5,3}5 |
| Cells/vertex | {5,3}20 |
| Euler characteristic | 0 |
| Dual | Self-dual |
| Coxeter group | K3, [5,3,5] |
| Properties | Regular |
The order-5 dodecahedral honeycomb is one of four regular space-filling tessellations (or honeycombs) in hyperbolic 3-space.
Five dodecahedra exist on each edge, and 20 dodecahedra around each vertex.
The dihedral angle of a dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In curved space, however, the dihedral angle depends on the size of the figure; a hyperbolic regular dodecahedron can be made with dihedral angles as small as one-sixth of a circle (if its vertices are at infinity).
There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb which has only 4 dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively-curved space (the surface of a 4 dimensional sphere), with 3 dodecahedra on each edge.
There are nine uniform honeycombs in the [5,3,5] Coxeter group family, including this regular form. Also the bitruncated form, t1,2{5,3,5}, , of this honeycomb has all truncated icosahedron cells.