Cubic honeycomb honeycomb(No image)TypeHyperbolic regular honeycombSchläfli symbol{4,3,4,3}{4,31,1,1}Coxeter diagram ↔ ↔ 4-faces {4,3,4}Cells {4,3}Faces {4}Face figure {3}Edge figure {4,3}Vertex figure {3,4,3}DualOrder-4 24-cell honeycombCoxeter group4, [4,3,4,3]PropertiesRegular 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. Acronym: chont
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
Notes
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, pp. 212–213) o3o4o3o4x - chont