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 25px {36} 25px
6-face types	22125px {35}25px
5-face types	21125px {34}25px
4-face type	{33}25px
Cell type	{32}25px
Face type	{3}25px
Face figure	031 25px
Edge figure	131 25px
Vertex figure	231 25px
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

The 3₃₁ honeycomb's vertex arrangement is called the 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 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. Template:3 k1 polytopes

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. Discrete Math., 1.1 (1988), 134-141.
• Conway, John H.; Sloane, Neil J. A. (1998). Sphere Packings, Lattices and Groups (3rd ed.). New York: Springer-Verlag. ISBN 0-387-98585-9. https://archive.org/details/spherepackingsla0000conw_b8u0.  p124-125, 8.2 The 7-dimensional lattices: E7 and E7*
• Klitzing, Richard. "7D Heptacombs x3o3o3o3o3o3o *d3o - naquoh". https://bendwavy.org/klitzing/dimensions/flat.htm#7D. 

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	{3[3]}	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	{3[4]}	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	{3[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	{3[6]}	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	{3[7]}	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	{3[8]}	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δ10	hδ10	qδ10
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21