Tesseractic honeycomb honeycomb

In the geometry of hyperbolic 5-space, the tesseractic honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,3,4,3}, it has three tesseractic honeycombs around each cell. It is dual to the order-4 24-cell honeycomb honeycomb.

Tesseractic honeycomb honeycomb
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TypeHyperbolic regular honeycomb
Schläfli symbol{4,3,3,4,3}
{4,3,31,1,1}
Coxeter diagram
↔
5-faces {4,3,3,4}
4-faces {4,3,3}
Cells {4,3}
Faces {4}
Cell figure {3}
Face figure {4,3}
Edge figure {3,4,3}
Vertex figure {3,3,4,3}
DualOrder-4 24-cell honeycomb honeycomb
Coxeter groupR5, [3,4,3,3,4]
PropertiesRegular

It is related to the regular Euclidean 4-space tesseractic honeycomb, {4,3,3,4}.

It is analogous to the paracompact cubic honeycomb honeycomb, {4,3,4,3}, in 4-dimensional hyperbolic space, square tiling honeycomb, {4,4,3}, in 3-dimensional hyperbolic space, and the order-3 apeirogonal tiling, {∞,3} of 2-dimensional hyperbolic space, each with hypercube honeycomb facets.

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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)
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