In geometry , there are a sequence of regular polytopes and honeycombs, {4,3,p}, with cubic cells . The first is the finite tesseract in 4-dimensional space. The second is the cubic honeycomb that tessellates Euclidean 3-space. The next two tessellate hyperbolic 3-space.
{4,3,p} regular honeycombs
Space
S3
E3
H3
Form
Finite
Affine
Compact
Paracompact
Noncompact
NameFile:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel p.png File:CDel node.png
{4,3,3} File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
{4,3,4} File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel labelinfin.png File:CDel branch 10.png File:CDel 2.png File:CDel labelinfin.png File:CDel branch 10.png File:CDel 2.png File:CDel labelinfin.png File:CDel branch 10.png
{4,3,5} File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png
{4,3,6} File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel node 1.png File:CDel ultra.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel uaub.png File:CDel nodes 11.png
{4,3,7} File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 7.png File:CDel node.png
{4,3,8} File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel label4.png
... {4,3,∞} File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel infin.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel labelinfin.png
Image
60px
60px
80px
80px
60px
60px
60px
Vertex figure File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel p.png File:CDel node.png
60px {3,3} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
60px {3,4} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel split1.png File:CDel nodes.png
60px {3,5} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png
60px {3,6} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel split1.png File:CDel branch.png
61px {3,7} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 7.png File:CDel node.png
60px {3,8} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel node 1.png File:CDel split1.png File:CDel branch.png File:CDel label4.png
60px {3,∞} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel infin.png File:CDel node.png File:CDel node 1.png File:CDel split1.png File:CDel branch.png File:CDel labelinfin.png
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)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN Expression error: Unrecognized punctuation character "[". 99-Expression error: Unrecognized punctuation character "[". Template:Only in print , ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space ) Table III
N. W. Johnson, R. Kellerhals , J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups , (2002) H3 : p130. [1]