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

In 7-dimensional geometry, the 3₃₁ honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3} 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.

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

The E₇ lattice can also be expressed as a union of the vertices of two A₇ lattices, also called A₇:

= ∪

The E₇ lattice (also called E₇) has double the symmetry, represented by ]. 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 A₇:

∪ = ∪ ∪ ∪ = 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	${\displaystyle {\tilde {E}}_{7}}$=E7	${\displaystyle {\bar {T}}_{8}}$=E7
Coxeter diagram
Symmetry			] =
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. "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

• 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,
• 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	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E	Uniform tiling	0	δ3	hδ3	qδ3	Hexagonal
E	Uniform convex honeycomb	0	δ4	hδ4	qδ4
E	Uniform 4-honeycomb	0	δ5	hδ5	qδ5	24-cell honeycomb
E	Uniform 5-honeycomb	0	δ6	hδ6	qδ6
E	Uniform 6-honeycomb	0	δ7	hδ7	qδ7	222
E	Uniform 7-honeycomb	0	δ8	hδ8	qδ8	133 • 331
E	Uniform 8-honeycomb	0	δ9	hδ9	qδ9	152 • 251 • 521
E	Uniform 9-honeycomb	0	δ10	hδ10	qδ10
E	Uniform 10-honeycomb	0	δ11	hδ11	qδ11
E	Uniform (n-1)-honeycomb	0	δn	hδn	qδn	1k2 • 2k1 • k21

• 8-polytopes