Cubic honeycomb honeycomb explained

bgcolor=#e7dcc3 colspan=2	Cubic honeycomb honeycomb
bgcolor=#ffffff align=center colspan=2	(No image)
Type	Hyperbolic regular honeycomb
Schläfli symbol
Coxeter diagram	↔ ↔
4-faces
Cells
Faces
Face figure
Edge figure
Vertex figure
Dual	Order-4 24-cell honeycomb
Coxeter group	4, [4,3,4,3]
Properties	Regular

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, it has three cubic honeycombs around each face, and with a 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,, which also has a 24-cell vertex figure.

It is analogous to the paracompact tesseractic honeycomb honeycomb,, in 5-dimensional hyperbolic space, square tiling honeycomb,, in 3-dimensional hyperbolic space, and the order-3 apeirogonal tiling, 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)