7-simplex honeycomb

Affine A7 E7 relations

7-simplex honeycomb

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Type

Family

Schläfli symbol

Coxeter diagram

6-face types

6-face types

5-face types

4-face types

Cell types

Face types

Vertex figure

Symmetry

Properties

In seven-dimensional Euclidean geometry, the 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, rectified 7-simplex, birectified 7-simplex, and trirectified 7-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.

A7 lattice

This vertex arrangement is called the A7 lattice or 7-simplex lattice. The 56 vertices of the expanded 7-simplex vertex figure represent the 56 roots of the Ã7 Coxeter group. It is the 7-dimensional case of a simplectic honeycomb. Around each vertex figure are 254 facets: 8+8 7-simplex, 28+28 rectified 7-simplex, 56+56 birectified 7-simplex, 70 trirectified 7-simplex, with the count distribution from the 9th row of Pascal's triangle.

Ẽ7 contains Ã7 as a subgroup of index 144. Both Ẽ7 and Ã7 can be seen as affine extensions from A7 from different nodes:

The A lattice can be constructed as the union of two A7 lattices, and is identical to the E7 lattice.

∪ = .

The A lattice is the union of four A7 lattices, which is identical to the E7* lattice (or E).

∪ ∪ ∪ = + = dual of .

The A lattice (also called A) is the union of eight A7 lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.

∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .

Projection by folding

The 7-simplex honeycomb can be projected into the 4-dimensional tesseractic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

Ã7

C̃4

See also

Regular and uniform honeycombs in 7-space:

  • 7-cubic honeycomb
  • 7-demicubic honeycomb
  • Truncated 7-simplex honeycomb
  • Omnitruncated 7-simplex honeycomb
  • E7 honeycomb

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, 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]