3₃₁ honeycomb

3₃₁ honeycomb
(no image)
Type	Uniform tessellation
Schläfli symbol	{3,3,3,3^3,1}
Coxeter symbol	3₃₁
Coxeter-Dynkin diagram	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
7-face types	3₂₁ File:E7 graph.svg {3⁶} File:7-simplex t0.svg
6-face types	2₂₁File:E6 graph.svg {3⁵}File:6-simplex t0.svg
5-face types	2₁₁File:Cross graph 5.svg {3⁴}File:5-simplex t0.svg
4-face type	{3³}File:4-simplex t0.svg
Cell type	{3²}File:3-simplex t0.svg
Face type	{3}File:2-simplex t0.svg
Face figure	0₃₁ File:5-simplex t1.svg
Edge figure	1₃₁ File:6-demicube.svg
Vertex figure	2₃₁ File:Gosset 2 31 polytope.svg
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.

File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

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

File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

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

File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

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

File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

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

File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

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

File:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

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}.

File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png

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: 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:

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:

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	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.png	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
Symmetry	[3^−1,3,1]	[3^0,3,1]	[[3^1,3,1]] = [4,3,3,3,3]	[3^2,3,1]	[3^3,3,1]	[3^4,3,1]
Order	48	720	46,080	2,903,040	∞
Graph		File:5-simplex t0.svg	File:6-cube t5.svg	File:Up2 3 21 t0 E7.svg	-	-
Name	3_1,-1	3₁₀	3₁₁	3₂₁	3₃₁	3₄₁

References

↑ "The Lattice E7".
↑ N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p 177
↑ "The Lattice E7".
↑ The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, 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 [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. Discrete 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*
Klitzing, Richard. "7D Heptacombs x3o3o3o3o3o3o *d3o - naquoh".

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde{A}}_{n - 1}$	${\tilde{C}}_{n - 1}$	${\tilde{B}}_{n - 1}$	${\tilde{D}}_{n - 1}$	${\tilde{G}}_{2}$ / ${\tilde{F}}_{4}$ / ${\tilde{E}}_{n - 1}$
E²	Uniform tiling	0_[3]	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	0_[4]	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	0_[5]	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	0_[6]	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	0_[7]	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	0_[8]	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	0_[9]	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	0_[10]	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	0_[11]	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	0_[n]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

[1] "The Lattice E7".

[2] N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p 177

[3] "The Lattice E7".

[4] The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, Edward Pervin

[1]

[2]

[3]

[4]

3₃₁ honeycomb

Contents

Construction

Kissing number

E7 lattice

Related honeycombs

See also

References

Navigation menu

Page actions

Page actions

Personal tools

Navigation

Search

Tools

In other projects

In other languages

3 31 honeycomb

Construction

Kissing number

E7 lattice

Related honeycombs

See also

References

Navigation menu

Search

3₃₁ honeycomb