24-cell honeycomb honeycomb

In the geometry of hyperbolic 5-space, the 24-cell honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because it has infinite facets, whose vertices exist on 4-horospheres and converge to a single ideal point at infinity. With Schläfli symbol {3,4,3,3,3}, it has three 24-cell honeycombs around each cell. It is dual to the 5-orthoplex honeycomb.

24-cell honeycomb honeycomb
(No image)
Type	Hyperbolic regular honeycomb
Schläfli symbol	{3,4,3,3,3}
Coxeter diagram	=
5-faces	{3,4,3,3}
4-faces	{3,4,3}
Cells	{3,4}
Faces	{3}
Cell figure	{3}
Face figure	{3,3}
Edge figure	{3,3,3}
Vertex figure	{4,3,3,3}
Dual	5-orthoplex honeycomb
Coxeter group	U5, [3,3,3,4,3]
Properties	Regular

Related honeycombs

It is related to the regular Euclidean 4-space 24-cell honeycomb, {3,4,3,3}, and the hyperbolic 5-space order-4 24-cell honeycomb honeycomb.

See also

• List of regular polytopes

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)