Order-5 dodecahedral honeycomb

Order-5 dodecahedral honeycomb
(No image)
Type	Hyperbolic regular honeycomb
Schläfli symbol	{5,3,5}
Coxeter-Dynkin diagram
Cells	dodecahedron {5,3}
Faces	pentagon {5}
Edge figure	pentagon {5}
Vertex figure	{3,5}
Cells/edge	{5,3}5
Cells/vertex	{5,3}20
Euler characteristic	0
Dual	Self-dual
Coxeter group	K3, [5,3,5]
Properties	Regular

The order-5 dodecahedral honeycomb is one of four regular space-filling tessellations (or honeycombs) in hyperbolic 3-space.

Five dodecahedra exist on each edge, and 20 dodecahedra around each vertex.

The dihedral angle of a dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In curved space, however, the dihedral angle depends on the size of the figure; a hyperbolic regular dodecahedron can be made with dihedral angles as small as one-sixth of a circle (if its vertices are at infinity).

Related honeycombs

There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb which has only 4 dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively-curved space (the surface of a 4 dimensional sphere), with 3 dodecahedra on each edge.

There are nine uniform honeycombs in the [5,3,5] Coxeter group family, including this regular form. Also the bitruncated form, t_1,2{5,3,5}, , of this honeycomb has all truncated icosahedron cells.

See also

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes
• 57-cell - An abstract regular polychoron which shared the {5,3,5} symbol.

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)

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