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    Cyclotruncated 6-simplex honeycomb

    From Handwiki - Reading time: 2 min
    Cyclotruncated 6-simplex honeycomb
    (No image)
    Type Uniform honeycomb
    Family Cyclotruncated simplectic honeycomb
    Schläfli symbol t0,1{3[7]}
    Coxeter diagram CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
    6-face types {35} 6-simplex t0.svg
    t{35} 30px
    2t{35} 30px
    3t{35} 6-simplex t23.svg
    Vertex figure Elongated 5-simplex antiprism
    Symmetry [math]\displaystyle{ {\tilde{A}}_6 }[/math]×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

    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.

    Related polytopes and honeycombs

    This honeycomb is one of 17 unique uniform honeycombs[1] constructed by the [math]\displaystyle{ {\tilde{A}}_6 }[/math] Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

    See also

    Regular and uniform honeycombs in 6-space:

    Notes

    1. ↑ * Weisstein, Eric W.. "Necklace". http://mathworld.wolfram.com/Necklace.html. , OEIS sequence A000029 18-1 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 [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]
    Fundamental convex regular and uniform honeycombs in dimensions 2-9
    Space Family [math]\displaystyle{ {\tilde{A}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{C}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{B}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{D}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
    E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
    E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
    E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
    E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
    E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
    E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
    E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
    E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
    En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21



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