3 31 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|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}

Removing the node on the short branch leaves the 6-simplex facet:

: {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

Removing the node on the end of the 3-length branch leaves the 3₂₁ facet:

: {{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 2₃₁ polytope.

: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}

The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (1₃₁).

: {{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea}}

The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (0₃₁).

: {{CDD|branch_10|3a|nodea|3a|nodea|3a|nodea}}

The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes tetrahedral prism {}×{3,3}.

: {{CDD|node_1|2|node_1|3|node|3|node}}

Each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 dimensions; its kissing number is 126, represented by the vertices of its vertex figure 2₃₁.

The 3₃₁ honeycomb's vertex arrangement is called the E₇ lattice.{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E7.html|title = The Lattice E7}}

$\{\backslash tilde\{E\}\}\_7$ contains $\{\backslash tilde\{A\}\}\_7$ as a subgroup of index 144.N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p 177 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 can also be expressed as a union of the vertices of two A₇ lattices, also called A₇²:

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

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}}.

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.

{{3_k1_polytopes}}

• 8-polytope
• 1₃₃ 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] [https://books.google.com/books?id=fUm5Mwfx8rAC&dq=Coxeter&pg=PP1 GoogleBook]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• R. T. Worley, The Voronoi Region of E7*. SIAM J. Discrete Math., 1.1 (1988), 134-141.
• {{Cite book | first = John H. | last = Conway | authorlink = John Horton Conway | author2 = Sloane, Neil J. A. | year = 1998 | title = Sphere Packings, Lattices and Groups | edition = (3rd ed.) | publisher = Springer-Verlag | location = New York | isbn = 0-387-98585-9 | url-access = registration | url = https://archive.org/details/spherepackingsla0000conw_b8u0 | author2-link = Neil Sloane }} p124-125, 8.2 The 7-dimensinoal lattices: E7 and E7*
• {{KlitzingPolytopes|flat.htm#7D|7D Heptacombs|x3o3o3o3o3o3o *d3o - naquoh}}

{{Honeycombs}}

Category:8-polytopes