 Icosahedral honeycomb

Icosahedral honeycomb
Poincaré disk modelType regular hyperbolic honeycomb Schläfli symbol {3,5,3} CoxeterDynkin diagram Cells icosahedron {3,5} Faces triangle {3} Edge figure triangle {3} Vertex figure
dodecahedronCells/edge {3,5}^{3} Cells/vertex {3,5}^{12} Dual Selfdual Coxeter group J_{3}, [3,5,3] Properties Regular The icosahedral honeycomb is one of four regular spacefilling tessellations (or honeycombs) in hyperbolic 3space.
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 3space. However in hyperbolic space, properly scaled icosahedra can have dihedral angles exactly 120 degrees, so three of these fit around an edge.
Related honeycombs
There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t_{1,2}{3,5,3}, , also called truncated dodecahedral honeycomb, each of whose cells is a truncated dodecahedron.
See also
 Seifert–Weber space
 List of regular polytopes
 Convex uniform honeycombs in hyperbolic space
 11cell  An abstract regular polychoron which shares the {3,5,3} Schläfli symbol.
References
 Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0486614808. (Tables I and II: Regular polytopes and honeycombs, pp. 294296)
 Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0486409198 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212213)
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