6-simplex honeycomb

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6-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[7]}
Coxeter–Dynkin diagrams CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
6-face types {35} 6-simplex t0.svg, t1{35} 6-simplex t1.svg
t2{35} 6-simplex t2.svg
5-face types {34} 5-simplex t0.svg, t1{34} 5-simplex t1.svg
t2{34} 5-simplex t2.svg
4-face types {33} 4-simplex t0.svg, t1{33} 4-simplex t1.svg
Cell types {3,3} 3-simplex t0.svg, t1{3,3} 3-simplex t1.svg
Face types {3} 2-simplex t0.svg
Vertex figure t0,5{35} 6-simplex t05.svg
Symmetry {\tilde{A}}_6×2, [[3[7]]]
Properties vertex-transitive

In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.

A6 lattice[edit]

This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the {\tilde{A}}_6 Coxeter group.[1] It is the 6-dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6-simplex, 21+21 rectified 6-simplex, 35+35 birectified 6-simplex, with the count distribution from the 8th row of Pascal's triangle.

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 10lr.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[2] constructed by the {\tilde{A}}_6 Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

Heptagon
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycombs
a1 [3[7]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png ×1

CDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.png CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.png

i2 [[3[7]]] CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel 3ab.pngCDel nodeab c3.pngCDel 3ab.pngCDel branch c4.png ×2

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png1 CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png2 CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

r14 [7[3[7]]] CDel node c1.pngCDel split1.pngCDel nodeab c1.pngCDel 3ab.pngCDel nodeab c1.pngCDel 3ab.pngCDel branch c1.png ×14

CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png3

Projection by folding[edit]

The 6-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

{\tilde{A}}_6 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
{\tilde{C}}_3 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

See also[edit]

Regular and uniform honeycombs in 6-space:

Notes[edit]

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]