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.pngCDel 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.pngCDel 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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