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Cyclotruncated 5-simplex honeycomb

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Cyclotruncated 5-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Cyclotruncated simplectic honeycomb
Schläfli symbol t0,1{3}
Coxeter diagram or
5-face types {3,3,3,3}
t{3,3,3,3}
2t{3,3,3,3}
4-face types {3,3,3}
t{3,3,3}
Cell types {3,3}
t{3,3}
Face types {3}
t{3}
Vertex figure
Elongated 5-cell antiprism
Coxeter groups A ~ 5 {\displaystyle {\tilde {A}}_{5}} ×22, ]
Properties vertex-transitive

In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1:1:1.

Structure

Its vertex figure is an elongated 5-cell antiprism, two parallel 5-cells in dual configurations, connected by 10 tetrahedral pyramids (elongated 5-cells) from the cell of one side to a point on the other. The vertex figure has 8 vertices and 12 5-cells.

It can be constructed as six sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-cell honeycomb divisions on each hyperplane.

Related polytopes and honeycombs

This honeycomb is one of 12 unique uniform honeycombs constructed by the A ~ 5 {\displaystyle {\tilde {A}}_{5}} Coxeter group. The extended symmetry of the hexagonal diagram of the A ~ 5 {\displaystyle {\tilde {A}}_{5}} Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:

A5 honeycombs
Hexagon
symmetry 		Extended
symmetry 		Extended
diagram 		Extended
group 		Honeycomb diagrams
a1 A ~ 5 {\displaystyle {\tilde {A}}_{5}}
d2 <> A ~ 5 {\displaystyle {\tilde {A}}_{5}} ×21 1, , , ,
p2 ] A ~ 5 {\displaystyle {\tilde {A}}_{5}} ×22 2,
i4 >] A ~ 5 {\displaystyle {\tilde {A}}_{5}} ×21×22 ,
d6 <3> A ~ 5 {\displaystyle {\tilde {A}}_{5}} ×61
r12 ] A ~ 5 {\displaystyle {\tilde {A}}_{5}} ×12 3

See also

Regular and uniform honeycombs in 5-space:

Notes

  1. mathworld: Necklace, OEIS sequence A000029 13-1 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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