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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[6]}
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 ×22, [[3[6]]]
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

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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.

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This honeycomb is one of 12 unique uniform honeycombs[1] constructed by the Coxeter group. The extended symmetry of the hexagonal diagram of the 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 [3[6]]
d2 <[3[6]]> ×21 1, , , ,
p2 [[3[6]]] ×22 2,
i4 [<[3[6]]>] ×21×22 ,
d6 <3[3[6]]> ×61
r12 [6[3[6]]] ×12 3

See also

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Regular and uniform honeycombs in 5-space:

Notes

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