8-simplex honeycomb

This vertex arrangement is called the A8 lattice or 8-simplex lattice. The 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the $\{\backslash tilde\{A\}\}\_8$ Coxeter group.{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A8.html|title = The Lattice A8}} It is the 8-dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9 8-simplex, 36+36 rectified 8-simplex, 84+84 birectified 8-simplex, 126+126 trirectified 8-simplex, with the count distribution from the 10th row of Pascal's triangle.

$\{\backslash tilde\{E\}\}\_8$ contains $\{\backslash tilde\{A\}\}\_8$ as a subgroup of index 5760.N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p.294 Both $\{\backslash tilde\{E\}\}\_8$ and $\{\backslash tilde\{A\}\}\_8$ can be seen as affine extensions of $A\_8$ from different nodes: File:Affine A8 E8 relations.png

The A{{sup sub|3|8}} lattice is the union of three A₈ lattices, and also identical to the E8 lattice.Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)

: {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}} ∪ {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes_10lr|3ab|branch}} ∪ {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes_01lr|3ab|branch}} = {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.

The A{{sup sub|*|8}} lattice (also called A{{sup sub|9|8}}) is the union of nine A₈ lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex

:

{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}} ∪

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

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

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

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

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

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

{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch_10l}} ∪

{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch_01l}} = dual of {{CDD|node_1|split1|nodes_11|3ab|nodes_11|3ab|nodes_11|3ab|branch_11}}.

{{8-simplex honeycomb family}}

The 8-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 8-space:
• 8-cubic honeycomb
• 8-demicubic honeycomb
• Truncated 8-simplex honeycomb
• 5₂₁ honeycomb
• 2₅₁ honeycomb
• 1₅₂ 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:9-polytopes