| Icosahedral honeycomb | |
|---|---|
Poincaré disk model |
|
| Type | regular hyperbolic honeycomb |
| Schläfli symbol | {3,5,3} |
| Coxeter-Dynkin diagram | |
| Cells | icosahedron {3,5} |
| Faces | triangle {3} |
| Edge figure | triangle {3} |
| Vertex figure | dodecahedron |
| Cells/edge | {3,5}3 |
| Cells/vertex | {3,5}12 |
| Dual | Self-dual |
| Coxeter group | J3, [3,5,3] |
| Properties | Regular |
The icosahedral honeycomb is one of four regular space-filling tessellations (or honeycombs) in hyperbolic 3-space.
Three icosahedra surround each edge, and 12 icosahedra surround each vertex, in a regular dodecahedral vertex figure.
The dihedral angle of a Euclidean icosahedron is 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However in hyperbolic space, properly scaled icosahedra can have dihedral angles exactly 120 degrees, so three of these fit around an edge.
There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,5,3}, , also called truncated dodecahedral honeycomb, each of whose cells is a truncated dodecahedron.