Order-3 icosahedral honeycomb

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Order-3 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 {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, t_1,2{3,5,3}, , of this honeycomb has all truncated dodecahedron cells.

[edit] See also

Seifert-Weber space
List of regular polytopes
11-cell - An abstract regular polychoron which shares the {3,5,3} Schläfli symbol.

[edit] References

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)