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Order-3 icosahedral honeycomb
From Wikipedia, the free encyclopedia
| Order-3 icosahedral honeycomb | |
|---|---|
 |
|
| 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 {5,3} |
| Cells/edge | {3,5}3 |
| Cells/vertex | {3,5}12 |
| Euler characteristic | 0 |
| Dual | Self-dual |
| Coxeter group | [3,5,3] |
| Properties | Regular |
The order-3 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 dodecahedral pattern.
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 nicely around an edge.
The bitruncated form, t1,2{3,5,3}, , of this honeycomb has all truncated dodecahedron cells.
- Seifert-Weber space
- List of regular polytopes
- 11-cell - An abstract regular polytope which shares the {3,5,3} Schläfli symbol.
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294-296)
- Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
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