5 21 honeycomb

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.