Quarter 7-cubic honeycomb

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quarter 7-cubic honeycomb
(No image)
TypeUniform 7-honeycomb
FamilyQuarter hypercubic honeycomb
Schläfli symbolq{4,3,3,3,3,3,4}
Coxeter diagram =
6-face typeh{4,35},
h5{4,35},
{31,1,1}×{3,3} duoprism
Vertex figure
Coxeter group{{\tilde  {D}}}_{7}×2 = [[31,1,3,3,3,31,1]]
Dual
Propertiesvertex-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.

Related honeycombs

This honeycomb is one of 77 uniform honycombs constructed by the {{\tilde  {D}}}_{7} 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 {{\tilde  {B}}}_{7} and {{\tilde  {C}}}_{7} constructions:

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

Regular and uniform honeycombs in 7-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, [Math. Zeit. 200 (1988) 3-45] See p318
  • Richard Klitzing, 7D, Euclidean tesselations#7D
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