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Quarter 5-cubic honeycomb

## Summary

quarter 5-cubic honeycomb
(No image)
Type Uniform 5-honeycomb
Family Quarter hypercubic honeycomb
Schläfli symbol q{4,3,3,3,4}
Coxeter-Dynkin diagram =
5-face type h{4,33},
h4{4,33},
Vertex figure
Rectified 5-cell antiprism
or Stretched birectified 5-simplex
Coxeter group ${\displaystyle {\tilde {D}}_{5}}$×2 = [[31,1,3,31,1]]
Dual
Properties vertex-transitive

In five-dimensional Euclidean geometry, the quarter 5-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 5-demicubic honeycomb, and a quarter of the vertices of a 5-cube honeycomb.[1] Its facets are 5-demicubes and runcinated 5-demicubes.

This honeycomb is one of 20 uniform honeycombs constructed by the ${\displaystyle {\tilde {D}}_{5}}$  Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:

D5 honeycombs
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
[31,1,3,31,1]         ${\displaystyle {\tilde {D}}_{5}}$
<[31,1,3,31,1]>
↔ [31,1,3,3,4]

${\displaystyle {\tilde {D}}_{5}}$ ×21 = ${\displaystyle {\tilde {B}}_{5}}$         ,        ,        ,

,        ,        ,

[[31,1,3,31,1]]         ${\displaystyle {\tilde {D}}_{5}}$ ×22        ,
<2[31,1,3,31,1]>
↔ [4,3,3,3,4]

${\displaystyle {\tilde {D}}_{5}}$ ×41 = ${\displaystyle {\tilde {C}}_{5}}$         ,        ,        ,        ,        ,
[<2[31,1,3,31,1]>]
↔ [[4,3,3,3,4]]

${\displaystyle {\tilde {D}}_{5}}$ ×8 = ${\displaystyle {\tilde {C}}_{5}}$ ×2        ,        ,

Regular and uniform honeycombs in 5-space:

## Notes

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

## References

• 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. "5D Euclidean tesselations#5D". x3o3o x3o3o *b3*e - spaquinoh
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$  ${\displaystyle {\tilde {C}}_{n-1}}$  ${\displaystyle {\tilde {B}}_{n-1}}$  ${\displaystyle {\tilde {D}}_{n-1}}$  ${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
E10 Uniform 10-honeycomb {3[11]} δ11 11 11
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21