Quarter 7-cubic honeycomb

Summary

quarter 7-cubic honeycomb
(No image)
Type Uniform 7-honeycomb
Family Quarter hypercubic honeycomb
Schläfli symbol q{4,3,3,3,3,3,4}
Coxeter diagram =
6-face type h{4,35},
h5{4,35},
{31,1,1}×{3,3} duoprism
Vertex figure
Coxeter group ×2 = [[31,1,3,3,3,31,1]]
Dual
Properties vertex-transitive

In seven-dimensional Euclidean geometry, the quarter 7-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 7-demicubic honeycomb, and a quarter of the vertices of a 7-cube honeycomb.[1] Its facets are 7-demicubes, pentellated 7-demicubes, and {31,1,1}×{3,3} duoprisms.

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This honeycomb is one of 77 uniform honeycombs constructed by the   Coxeter group, all but 10 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 77 permutations are listed with its highest extended symmetry, and related   and   constructions:

D7 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[31,1,3,3,3,31,1]             ×1            ,            ,            ,            ,            ,            ,
[[31,1,3,3,3,31,1]]             ×2            ,            ,            ,            
<[31,1,3,3,3,31,1]>
↔ [31,1,3,3,3,3,4]
           
             
×2 ...
<<[31,1,3,3,3,31,1]>>
↔ [4,3,3,3,3,3,4]
           
               
×4 ...
[<<[31,1,3,3,3,31,1]>>]
↔ [[4,3,3,3,3,3,4]]
           
               
×8 ...

See also

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Regular and uniform honeycombs in 7-space:

Notes

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  1. ^ Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318

References

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  • Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318 [2]
  • Klitzing, Richard. "7D Euclidean tesselations#7D".
Space Family           /   /  
E2 Uniform tiling 0[3] δ3 3 3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 4 4
E4 Uniform 4-honeycomb 0[5] δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 6 6
E6 Uniform 6-honeycomb 0[7] δ7 7 7 222
E7 Uniform 7-honeycomb 0[8] δ8 8 8 133331
E8 Uniform 8-honeycomb 0[9] δ9 9 9 152251521
E9 Uniform 9-honeycomb 0[10] δ10 10 10
E10 Uniform 10-honeycomb 0[11] δ11 11 11
En-1 Uniform (n-1)-honeycomb 0[n] δn n n 1k22k1k21