1 33 honeycomb

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram.

: {{CDD|node_1|3|node|split1|nodes|3ab|nodes|3ab|nodes}}

Removing a node on the end of one of the 3-length branch leaves the 1₃₂, its only facet type.

: {{CDD|node_1|3|node|split1|nodes|3ab|nodes|3a|nodea}}

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 0₃₃.

: {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes}}

The edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, {3,3}×{3,3}.

: {{CDD|nodes_11|3ab|nodes|3ab|nodes}}

Each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known for 7 dimensions (the kissing number) is 126.

The $\{\backslash tilde\{E\}\}\_7$ group is related to the $\{\backslash tilde\{F\}\}\_4$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.

$\{\backslash tilde\{E\}\}\_7$ 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 extension from $A\_7$ from different nodes: File:Affine_A7_E7_relations.png

The E₇^* lattice (also called E₇²){{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html|title=The Lattice E7}} has double the symmetry, represented by 3,3^3,3. The Voronoi cell of the E₇^* lattice is the 1₃₂ polytope, and voronoi tessellation the 1₃₃ honeycomb.[http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices] {{Webarchive|url=https://web.archive.org/web/20160130193811/http://home.digital.net/~pervin/publications/vermont.html |date=2016-01-30 }}, Edward Pervin The E₇^* lattice is constructed by 2 copies of the E₇ lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A₇^* lattices, also called A₇⁴:

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

The 1₃₃ is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3k series. The final is a noncompact hyperbolic honeycomb, 1₃₄.

{{1 3k polytopes}}

The rectified 1₃₃ or 0₃₃₁, Coxeter diagram {{CDD|node|3|node_1|split1|nodes|3ab|nodes|3ab|nodes}} has facets {{CDD|node_1|3|node|split1|nodes|3ab|nodes|3b|nodeb}} and {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes}}, and vertex figure {{CDD|node_1|2|nodes_11|3ab|nodes|3ab|nodes}}.

• 8-polytope
• 3₃₁ honeycomb

{{reflist}}

• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, {{ISBN|978-0-486-40919-1}} (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• 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 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• {{KlitzingPolytopes|flat.htm#7D|7D Heptacombs|o3o3o3o3o3o3o *d3x - linoh}}
• {{KlitzingPolytopes|flat.htm#7D|7D Heptacombs|o3o3o3x3o3o3o *d3o - rolinoh}}

{{Honeycombs}}

Category:8-polytopes