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Quarter 5-cubic honeycomb

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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{4,3},
h4{4,3},
Vertex figure
Rectified 5-cell antiprism
or Stretched birectified 5-simplex
Coxeter group D ~ 5 {\displaystyle {\tilde {D}}_{5}} ×2 = ]
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. Its facets are 5-demicubes and runcinated 5-demicubes.

Related honeycombs

This honeycomb is one of 20 uniform honeycombs constructed by the D ~ 5 {\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
D ~ 5 {\displaystyle {\tilde {D}}_{5}}
<>
↔
↔ 		D ~ 5 {\displaystyle {\tilde {D}}_{5}} ×21 = B ~ 5 {\displaystyle {\tilde {B}}_{5}} , , ,

, , ,

] D ~ 5 {\displaystyle {\tilde {D}}_{5}} ×22 ,
<2>
↔
↔ 		D ~ 5 {\displaystyle {\tilde {D}}_{5}} ×41 = C ~ 5 {\displaystyle {\tilde {C}}_{5}} , , , , ,
>]
↔ ]
↔ 		D ~ 5 {\displaystyle {\tilde {D}}_{5}} ×8 = C ~ 5 {\displaystyle {\tilde {C}}_{5}} ×2 , ,

See also

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
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, See p318
  • Klitzing, Richard. "5D Euclidean tesselations#5D". x3o3o x3o3o *b3*e - spaquinoh
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}}
E Uniform tiling 0 δ3 hδ3 qδ3 Hexagonal
E Uniform convex honeycomb 0 δ4 hδ4 qδ4
E Uniform 4-honeycomb 0 δ5 hδ5 qδ5 24-cell honeycomb
E Uniform 5-honeycomb 0 δ6 hδ6 qδ6
E Uniform 6-honeycomb 0 δ7 hδ7 qδ7 222
E Uniform 7-honeycomb 0 δ8 hδ8 qδ8 133 • 331
E Uniform 8-honeycomb 0 δ9 hδ9 qδ9 152 • 251 • 521
E Uniform 9-honeycomb 0 δ10 hδ10 qδ10
E Uniform 10-honeycomb 0 δ11 hδ11 qδ11
E Uniform (n-1)-honeycomb 0 δn hδn qδn 1k2 • 2k1 • k21
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