Order-5 cubic honeycomb

Order-5 cubic honeycomb
Order-5 cubic honeycomb
Hyperb gcubic hc constr.pngHyperb gcubic hc.png
Poincaré disk models
Type Hyperbolic regular honeycomb
Schläfli symbol {4,3,5}
Coxeter-Dynkin diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Cells cube {4,3}
Faces square {4}
Edge figure pentagon {5}
Vertex figure Order-5 cubic honeycomb verf.png
icosahedron
Cells/edge {4,3}5
Cells/vertex {4,3}20
Euler characteristic 0
Coxeter group BH3, [5,3,4]
Dual Order-4 dodecahedral honeycomb
Properties Regular

The order-5 cubic honeycomb is one of four regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. Five cubes exist on each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

Related honeycombs

It is related to the regular (order-4) cubic honeycomb of Euclidean 3-space, which has 4 cubes per edge, and also the tesseract of Euclidean 4-space with 3 cubes per edge.

It has an alternated form, represented by CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png, having icosahedron and tetrahedron cells.

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including this regular form.

See also

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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