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Cyclotruncated 6-simplex honeycomb

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Cyclotruncated 6-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Cyclotruncated simplectic honeycomb
Schläfli symbol t0,1{3[7]}
Coxeter diagram
6-face types {35}
t{35}
2t{35}
3t{35}
Vertex figure Elongated 5-simplex antiprism
Symmetry ×2, [[3[7]]]
Properties vertex-transitive

In six-dimensional Euclidean geometry, the cyclotruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.

Structure

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It can be constructed by seven sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-simplex honeycomb divisions on each hyperplane.

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This honeycomb is one of 17 unique uniform honeycombs[1] constructed by the Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

A6 honeycombs
Heptagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
a1 [3[7]]

i2 [[3[7]]] ×2

1

2

r14 [7[3[7]]] ×14

3

See also

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Regular and uniform honeycombs in 6-space:

Notes

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  1. ^ * Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 18-1 cases, skipping one with zero marks

References

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  • 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 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
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