7-simplex honeycomb

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 $\{\backslash tilde\{A\}\}\_7$ Coxeter group.{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A7.html|title = The Lattice A7}} 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.

$\{\backslash tilde\{E\}\}\_7$ contains $\{\backslash tilde\{A\}\}\_7$ as a subgroup of index 144.N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294 Both $\{\backslash tilde\{E\}\}\_7$ and $\{\backslash tilde\{A\}\}\_7$ can be seen as affine extensions from $A\_7$ from different nodes: File:Affine_A7_E7_relations.png

The A{{sup sub|2|7}} lattice can be constructed as the union of two A₇ lattices, and is identical to the E7 lattice.

:

{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}} ∪ {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node_1}} = {{CDD|node|3|node|split1|nodes|3ab|nodes|3ab|nodes_10l}}.

The A{{sup sub|4|7}} lattice is the union of four A₇ lattices, which is identical to the E7* lattice (or E{{sup sub|2|7}}).

:

{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}} ∪ {{CDD|node|split1|nodes|3ab|nodes_10lr|3ab|nodes|split2|node}} ∪ {{CDD|node|split1|nodes|3ab|nodes_01lr|3ab|nodes|split2|node}} ∪ {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node_1}} = {{CDD|node|3|node|split1|nodes|3ab|nodes|3ab|nodes_10l}} + {{CDD|node|3|node|split1|nodes|3ab|nodes|3ab|nodes_01l}} = dual of {{CDD|node_1|3|node|split1|nodes|3ab|nodes|3ab|nodes}}.

The A{{sup sub|*|7}} lattice (also called A{{sup sub|8|7}}) 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.

:

{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}} ∪

{{CDD|node|split1|nodes_10lur|3ab|nodes|3ab|nodes|split2|node}} ∪

{{CDD|node|split1|nodes_01lr|3ab|nodes|3ab|nodes|split2|node}} ∪

{{CDD|node|split1|nodes|3ab|nodes_10lr|3ab|nodes|split2|node}} ∪

{{CDD|node|split1|nodes|3ab|nodes_01lr|3ab|nodes|split2|node}} ∪

{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes_10lru|split2|node}} ∪

{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes_01lr|split2|node}} ∪

{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node_1}} = dual of {{CDD|node_1|split1|nodes_11|3ab|nodes_11|3ab|nodes_11|split2|node_1}}.

{{7-simplex honeycomb family}}

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:

Regular and uniform honeycombs in 7-space:

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

{{reflist}}

• 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
• (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]

{{Honeycombs}}

Category:Honeycombs (geometry)

Category:8-polytopes