Cubic honeycomb honeycomb

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Cubic honeycomb honeycomb
(No image)
Type Hyperbolic regular honeycomb
Schläfli symbol {4,3,4,3}
{4,31,1,1}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel splitsplit1.pngCDel branch3.pngCDel node.png ↔ CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1-43.pngCDel nodes.png
CDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node 1.png ↔ CDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
4-faces Cubic honeycomb.png {4,3,4}
Cells Hexahedron.png {4,3}
Faces Regular polygon 4 annotated.svg {4}
Face figure Regular polygon 3 annotated.svg {3}
Edge figure Hexahedron.png {4,3}
Vertex figure Schlegel wireframe 24-cell.png {3,4,3}
Dual Order-4 24-cell honeycomb
Coxeter group R4, [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 {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. 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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