3 31 honeycomb

331 honeycomb
(no image)
Type	Uniform tessellation
Schläfli symbol	{3,3,3,33,1}
Coxeter symbol	331
Coxeter-Dynkin diagram
7-face types	321 {36}
6-face types	221 {35}
5-face types	211 {34}
4-face type	{33}
Cell type	{32}
Face type	{3}
Face figure	031
Edge figure	131
Vertex figure	231
Coxeter group	${\tilde{E}}_7$, [33,3,1]
Properties	vertex-transitive

In 7-dimensional geometry, the 3₃₁ honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3^3,1} and is composed of 3₂₁ and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

Construction

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.

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

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

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

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

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

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}.

Kissing number

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.

E7 lattice

${\tilde{E}}_7$ contains ${\tilde{A}}_7$ as a subgroup of index 144. Both ${\tilde{E}}_7$ and ${\tilde{A}}_7$ can be seen as affine extension from $A_7$ from different nodes: File:Affine A7 E7 relations.png

The vertex arrangement of 3₃₁ is called the E₇ lattice. The E₇ lattice can also be expressed as a union of the vertices of two A₇ lattices, also called A₇²:

= ∪

The E₇^* lattice (also called E₇²) 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. 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:

∪ = ∪ ∪ ∪ = dual of .

Related honeycombs

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 dimensional figures

Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\tilde{E}}_{7}$=E7+	${\bar{T}}_8$=E7++
Coxeter diagram
Symmetry	[3−1,3,1]	[30,3,1]	[[31,3,1]]	[32,3,1]	[33,3,1]	[34,3,1]
Order	48	720	46,080	2,903,040	∞
Graph					-	-
Name	31,-1	310	311	321	331	341

See also

• 8-polytope
• 1₃₃ honeycomb

References

• 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 [1] 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. Disc. Math., 1.1 (1988), 134-141.
• p124-125, 8.2 The 7-dimensinoal lattices: E7 and E7*