521 honeycomb
Type	Uniform honeycomb
Family	k21 polytope
Schläfli symbol	{3,3,3,3,3,32,1}
Coxeter symbol	521
Coxeter-Dynkin diagram
8-faces	511 {37}
7-faces	{36} Note that there are two distinct orbits of this 7-simplex under the honeycomb's full ${\displaystyle {\tilde {E}}_{8}}$ automorphism group.
6-faces	{35}
5-faces	{34}
4-faces	{33}
Cells	{32}
Faces	{3}
Cell figure	121
Face figure	221
Edge figure	321
Vertex figure	421
Symmetry group	${\displaystyle {\tilde {E}}_{8}}$, [35,2,1]

In geometry, the 5₂₁ honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5₂₁ is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.^[1]

By putting spheres at its vertices one obtains the densest-possible packing of spheres in 8 dimensions. This was proven by Maryna Viazovska in 2016 using the theory of modular forms. Viazovska was awarded the Fields Medal for this work in 2022.

This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure^[2] (Gosset regarded honeycombs in n dimensions as degenerate n+1 polytopes).

Each vertex of the 5₂₁ honeycomb is surrounded by 2160 8-orthoplexes and 17280 8-simplicies.

The vertex figure of Gosset's honeycomb is the semiregular 4₂₁ polytope. It is the final figure in the k₂₁ family.

This honeycomb is highly regular in the sense that its symmetry group (the affine ${\displaystyle {\tilde {E}}_{8}}$ Weyl group) acts transitively on the k-faces for k ≤ 6. All of the k-faces for k ≤ 7 are simplices.

Construction

It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram.

Removing the node on the end of the 2-length branch leaves the 8-orthoplex, 6₁₁.

Removing the node on the end of the 1-length branch leaves the 8-simplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 4₂₁ polytope.

The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 3₂₁ polytope.

The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 2₂₁ polytope.

The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 1₂₁ polytope.

Kissing number

Each vertex of this tessellation is the center of a 7-sphere in the densest packing in 8 dimensions; its kissing number is 240, represented by the vertices of its vertex figure 4₂₁.

E8 lattice

${\displaystyle {\tilde {E}}_{8}}$ contains ${\displaystyle {\tilde {A}}_{8}}$ as a subgroup of index 5760.^[3] Both ${\displaystyle {\tilde {E}}_{8}}$ and ${\displaystyle {\tilde {A}}_{8}}$ can be seen as affine extensions of ${\displaystyle A_{8}}$ from different nodes:

${\displaystyle {\tilde {E}}_{8}}$ contains ${\displaystyle {\tilde {D}}_{8}}$ as a subgroup of index 270.^[4] Both ${\displaystyle {\tilde {E}}_{8}}$ and ${\displaystyle {\tilde {D}}_{8}}$ can be seen as affine extensions of ${\displaystyle D_{8}}$ from different nodes:

The vertex arrangement of 5₂₁ is called the E8 lattice.^[5]

The E8 lattice can also be constructed as a union of the vertices of two 8-demicube honeycombs (called a D₈² or D₈⁺ lattice), as well as the union of the vertices of three 8-simplex honeycombs (called an A₈³ lattice):^[6]

= ∪ = ∪ ∪

Regular complex honeycomb

Using a complex number coordinate system, it can also be constructed as a ${\displaystyle \mathbb {C} ^{4}}$ regular complex polytope, given the symbol 3{3}3{3}3{3}3{3}3, and Coxeter diagram . Its elements are in relative proportion as 1 vertex, 80 3-edges, 270 ₃{3}₃ faces, 80 ₃{3}₃{3}₃ cells and 1 ₃{3}₃{3}₃{3}₃ Witting polytope cells.^[7]

Related polytopes and honeycombs

The 5₂₁ is seventh in a dimensional series of semiregular polytopes, identified in 1900 by Thorold Gosset. Each member of the sequence has the previous member as its vertex figure. All facets of these polytopes are regular polytopes, namely simplexes and orthoplexes.

k21 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
En	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+	E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter diagram
Symmetry	[3−1,2,1]	[30,2,1]	[31,2,1]	[32,2,1]	[33,2,1]	[34,2,1]	[35,2,1]	[36,2,1]
Order	12	120	1,920	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	−121	021	121	221	321	421	521	621

See also

• E8 lattice
• 1₅₂ honeycomb
• 2₅₁ honeycomb

Notes

References

• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Coxeter, H. S. M. (1973). Regular Polytopes ((3rd ed.) ed.). New York: Dover Publications. ISBN 0-486-61480-8.
• 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
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: Geometries and Transformations, (2015)

Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
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
E10	Uniform 10-honeycomb	{3[11]}	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21