6-simplex honeycomb

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

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 ${\displaystyle {\tilde {A}}_{6}}$ Coxeter group. 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

Related polytopes and honeycombs

This honeycomb is one of 17 unique uniform honeycombs constructed by the ${\displaystyle {\tilde {A}}_{6}}$ Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

Projection by folding

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

Regular and uniform honeycombs in 6-space:

• 6-cubic honeycomb
• 6-demicubic honeycomb
• Truncated 6-simplex honeycomb
• Omnitruncated 6-simplex honeycomb
• 2₂₂ honeycomb

Notes

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]