Icosahedral honeycomb

Icosahedral honeycomb
Icosahedral honeycomb
Hyperb icosahedral hc.png
Poincaré disk model
Type regular hyperbolic honeycomb
Schläfli symbol {3,5,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells icosahedron {3,5}
Faces triangle {3}
Edge figure triangle {3}
Vertex figure Order-3 icosahedral honeycomb verf.png
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.

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, t1,2{3,5,3}, CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png, also called truncated dodecahedral honeycomb, each of whose cells is a truncated dodecahedron.

See also


  • 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)