In the geometry of hyperbolic 4-space, the cubic honeycomb honeycomb is one of two paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because it has infinite facets, whose vertices exist on 3-horospheres and converge to a single ideal point at infinity. With Schläfli symbol { 4,3,4,3}, it has three cubic honeycombs around each face, and with a { 3,4,3} vertex figure. It is dual to the order-4 24-cell honeycomb.
Related honeycombs
It is related to the Euclidean 4-space 16-cell honeycomb, { 3,3,4,3}, which also has a 24-cell vertex figure.
It is analogous to the paracompact tesseractic honeycomb honeycomb, { 4,3,3,4,3}, in 5-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.
See also
List of regular polytopes
References
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
Honeycombs (geometry)