Note that there are two distinct orbits of this 7-simplex under the honeycomb's full 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
, [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 Weyl group) acts transitively on the k-faces for k ≤ 6. All of the k-faces for k ≤ 7 are simplices.
^Coxeter, 1973, Chapter 5: The Kaleidoscope
^Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48.
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