3 31 honeycomb

331 honeycomb
(no image)
Type	Uniform tessellation
Schläfli symbol	{3,3,3,3^3,1}
Coxeter symbol	331
Coxeter-Dynkin diagram
7-face types	321 {3^6}
6-face types	221 {3^5}
5-face types	211 {3^4}
4-face type	{3^3}
Cell type	{3^2}
Face type	{3}
Face figure	031
Edge figure	131
Vertex figure	231
Coxeter group	${\tilde {E}}_{7}$, [3^3,3,1]
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

The 3₃₁ honeycomb's vertex arrangement is called the E₇ lattice.^[1]

${\tilde {E}}_{7}$ contains ${\tilde {A}}_{7}$ as a subgroup of index 144.^[2] Both ${\tilde {E}}_{7}$ and ${\tilde {A}}_{7}$ can be seen as affine extension from $A_{7}$ from different nodes:

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)^[3] 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.^[4] 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]	[3^0,3,1]	[[3<sup>1,3,1</sup>]]	[3^2,3,1]	[3^3,3,1]	[3^4,3,1]
Order	48	720	46,080	2,903,040	∞
Graph					-	-
Name	31,-1	310	311	321	331	341

See also

• 8-polytope
• 1₃₃ honeycomb

References

1. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E7.html
2. N.W. Johnson: Geometries and Transformations, (2015) Chapter 12: Euclidean symmetry groups, p 177
3. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html
4. The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin

• 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 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.
• Conway, John H.; Sloane, Neil J. A. (1998). Sphere Packings, Lattices and Groups ((3rd ed.) ed.). New York: Springer-Verlag. ISBN 0-387-98585-9. p124-125, 8.2 The 7-dimensinoal lattices: E7 and E7*

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
Uniform 5-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ6	hδ6	qδ6
Uniform 7-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
Uniform 8-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
Uniform 9-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
Uniform 10-honeycomb	{3^[10]}	δ10	hδ10	qδ10
Uniform n-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21