6-simplex honeycomb

Summary

6-simplex honeycomb
(No image)
Type Uniform 6-honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[7]} = 0[7]
Coxeter diagram
6-face types {35} , t1{35}
t2{35}
5-face types {34} , t1{34}
t2{34}
4-face types {33} , t1{33}
Cell types {3,3} , t1{3,3}
Face types {3}
Vertex figure t0,5{35}
Symmetry ×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

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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   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.

                                                  = dual of        

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This honeycomb is one of 17 unique uniform honeycombs[2] constructed by the   Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

A6 honeycombs
Heptagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
a1 [3[7]]          

               

i2 [[3[7]]]          ×2

       1                                                

                       2                                

r14 [7[3[7]]]          ×14

       3

Projection by folding

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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:

         
         

See also

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Regular and uniform honeycombs in 6-space:

Notes

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  1. ^ "The Lattice A6". Archived from the original on 2012-01-19. Retrieved 2011-05-11.
  2. ^ * Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 18-1 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