Order-4 dodecahedral honeycomb

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Order-4 dodecahedral honeycomb
Type	Hyperbolic regular honeycomb
Schläfli symbol	{5,3,4}
Coxeter-Dynkin diagram
Cells	dodecahedron {5,3}
Faces	pentagon {5}
Edge figure	square {4}
Vertex figure	octahedron {3,4}
Cells/edge	{5,3}4
Cells/vertex	{5,3}8
Euler characteristic	0
Dual	Order-5 cubic honeycomb
Coxeter group	[5,3,4]
Properties	Regular

The order-4 dodecahedral honeycomb is one of four regular space-filling tessellation (or honeycombs) in hyperbolic 3-space.

Four dodecahedra exist on each edge, and 8 dodecahedra around each vertex. Its vertices are constructed from 3 orthogonal axes, just like the cubic honeycomb of Euclidean 3-space.

There is another regular honeycomb in hyperbolic 3-space called the order-5 dodecahedral honeycomb which has 5 dodecahedra per edge.

This honeycomb is also related to the 120-cell which has 120 dodecahedra in 4-dimensional space, with 3 dodecahedra on each edge.

The dihedral angle of a dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly scaled dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge.

[edit] See also

• Poincaré sphere Poincaré dodecahedral space
• Seifert-Weber space Seifert-Weber dodecahedral space
• List of regular polytopes

[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)
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II) [1]