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.
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).
The vertex figure of Gosset's honeycomb is the semiregular 421 polytope.
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.
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.
from different nodes: The vertex arrangement of 521 is called the E8 lattice.
regular complex polytope, given the symbol 3{3}3{3}3{3}3{3}3, and Coxeter diagram .
[7] The 521 is seventh in a dimensional series of semiregular polytopes, identified in 1900 by Thorold Gosset.