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Cyclotruncated 8-simplex honeycomb

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Euclidean Geometry, 8-dimensional: Cyclotruncated 8-simplex
Cyclotruncated 8-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Cyclotruncated simplectic honeycomb
Schläfli symbol t0,1{3}
Coxeter diagram
8-face types {3} , t0,1{3}
t1,2{3} , t2,3{3}
t3,4{3}
Vertex figure Elongated 7-simplex antiprism
Symmetry A ~ 8 {\displaystyle {\tilde {A}}_{8}} ×2, ]
Properties vertex-transitive

In eight-dimensional Euclidean geometry, the cyclotruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, truncated 8-simplex, bitruncated 8-simplex, tritruncated 8-simplex, and quadritruncated 8-simplex facets. These facet types occur in proportions of 2:2:2:2:1 respectively in the whole honeycomb.

Structure

It can be constructed by nine sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 7-simplex honeycomb divisions on each hyperplane.

Related polytopes and honeycombs

This honeycomb is one of 45 unique uniform honeycombs constructed by the A ~ 8 {\displaystyle {\tilde {A}}_{8}} Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:

A8 honeycombs
Enneagon
symmetry 		Symmetry Extended
diagram 		Extended
group 		Honeycombs
a1 A ~ 8 {\displaystyle {\tilde {A}}_{8}}

i2 ] A ~ 8 {\displaystyle {\tilde {A}}_{8}} ×2

1 2

i6 ] A ~ 8 {\displaystyle {\tilde {A}}_{8}} ×6
r18 ] A ~ 8 {\displaystyle {\tilde {A}}_{8}} ×18 3

See also

Regular and uniform honeycombs in 8-space:

Notes

  1. * Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 46-1 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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