Omnitruncated 6-simplex honeycomb

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Omnitruncated 6-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Omnitruncated simplectic honeycomb
Schläfli symbol {3[8]}
Coxeter–Dynkin diagrams CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png
Facets 6-simplex t012345.svg
t0,1,2,3,4,5{3,3,3,3,3}
Vertex figure Omnitruncated 6-simplex honeycomb verf.png
Irr. 6-simplex
Symmetry ×14, [7[3[7]]]
Properties vertex-transitive

In six-dimensional Euclidean geometry, the omnitruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 6-simplex facets.

The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

A*
6
lattice
[edit]

The A*
6
lattice (also called A7
6
) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes 01lr.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01lr.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 01l.png = dual of CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

Related polytopes and honeycombs[edit]

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:

Projection by folding[edit]

The omnitruncated 6-simplex honeycomb can be projected into the 4-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png

See also[edit]

Regular and uniform honeycombs in 6-space:

Notes[edit]

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

References[edit]

  • 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 / /
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21