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Omnitruncated 7-simplex honeycomb

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Omnitruncated 7-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Omnitruncated simplectic honeycomb
Schläfli symbol {3}
Coxeter–Dynkin diagrams
6-face types t0123456{3,3,3,3,3,3}
Vertex figure
Irr. 7-simplex
Symmetry A ~ 8 {\displaystyle {\tilde {A}}_{8}} ×16, ]
Properties vertex-transitive

In seven-dimensional Euclidean geometry, the omnitruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 7-simplex facets.

The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

A7 lattice

The A
7 lattice (also called A
7) is the union of eight A7 lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.

∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .

Related polytopes and honeycombs

This honeycomb is one of 29 unique uniform honeycombs constructed by the A ~ 7 {\displaystyle {\tilde {A}}_{7}} Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:

A7 honeycombs
Octagon
symmetry 		Extended
symmetry 		Extended
diagram 		Extended
group 		Honeycombs
a1 A ~ 7 {\displaystyle {\tilde {A}}_{7}}

d2 <> A ~ 7 {\displaystyle {\tilde {A}}_{7}} ×21

1

p2 ] A ~ 7 {\displaystyle {\tilde {A}}_{7}} ×22

2

d4 <2> A ~ 7 {\displaystyle {\tilde {A}}_{7}} ×41

p4 ] A ~ 7 {\displaystyle {\tilde {A}}_{7}} ×42

d8 ] A ~ 7 {\displaystyle {\tilde {A}}_{7}} ×8
r16 ] A ~ 7 {\displaystyle {\tilde {A}}_{7}} ×16 3

See also

Regular and uniform honeycombs in 7-space:

Notes

  1. Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 30-1 cases, skipping one with zero marks

References

  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, (1.9 Uniform space-fillings)
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
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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