5_{21} honeycomb  

Type  Uniform honeycomb 
Family  k_{21} polytope 
Schläfli symbol  {3,3,3,3,3,3^{2,1}} 
Coxeter symbol  5_{21} 
CoxeterDynkin diagram  
8faces  5_{11} {3^{7}} 
7faces  {3^{6}} Note that there are two distinct orbits of this 7simplex under the honeycomb's full automorphism group.

6faces  {3^{5}} 
5faces  {3^{4}} 
4faces  {3^{3}} 
Cells  {3^{2}} 
Faces  {3} 
Cell figure  1_{21} 
Face figure  2_{21} 
Edge figure  3_{21} 
Vertex figure  4_{21} 
Symmetry group  , [3^{5,2,1}] 
In geometry, the 5_{21} honeycomb is a uniform tessellation of 8dimensional Euclidean space. The symbol 5_{21} is from Coxeter, named for the length of the 3 branches of its CoxeterDynkin diagram.^{[1]}
By putting spheres at its vertices one obtains the densestpossible 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 9ic semiregular figure^{[2]} (Gosset regarded honeycombs in n dimensions as degenerate n+1 polytopes).
Each vertex of the 5_{21} honeycomb is surrounded by 2160 8orthoplexes and 17280 8simplicies.
The vertex figure of Gosset's honeycomb is the semiregular 4_{21} polytope. It is the final figure in the k_{21} family.
This honeycomb is highly regular in the sense that its symmetry group (the affine Weyl group) acts transitively on the kfaces for k ≤ 6. All of the kfaces for k ≤ 7 are simplices.
It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8dimensional space.
The facet information can be extracted from its CoxeterDynkin diagram.
Removing the node on the end of the 2length branch leaves the 8orthoplex, 6_{11}.
Removing the node on the end of the 1length branch leaves the 8simplex.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 4_{21} polytope.
The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 3_{21} polytope.
The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 2_{21} polytope.
The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 1_{21} polytope.
Each vertex of this tessellation is the center of a 7sphere in the densest packing in 8 dimensions; its kissing number is 240, represented by the vertices of its vertex figure 4_{21}.
contains as a subgroup of index 5760.^{[3]} Both and can be seen as affine extensions of from different nodes:
contains as a subgroup of index 270.^{[4]} Both and can be seen as affine extensions of from different nodes:
The vertex arrangement of 5_{21} is called the E8 lattice.^{[5]}
The E8 lattice can also be constructed as a union of the vertices of two 8demicube honeycombs (called a D_{8}^{2} or D_{8}^{+} lattice), as well as the union of the vertices of three 8simplex honeycombs (called an A_{8}^{3} lattice):^{[6]}
Using a complex number coordinate system, it can also be constructed as a 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 3edges, 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]}
The 5_{21} 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.
k_{21} figures in n dimensions  

Space  Finite  Euclidean  Hyperbolic  
E_{n}  3  4  5  6  7  8  9  10  
Coxeter group 
E_{3}=A_{2}A_{1}  E_{4}=A_{4}  E_{5}=D_{5}  E_{6}  E_{7}  E_{8}  E_{9} = = E_{8}^{+}  E_{10} = = E_{8}^{++}  
Coxeter diagram 

Symmetry  [3^{−1,2,1}]  [3^{0,2,1}]  [3^{1,2,1}]  [3^{2,2,1}]  [3^{3,2,1}]  [3^{4,2,1}]  [3^{5,2,1}]  [3^{6,2,1}]  
Order  12  120  1,920  51,840  2,903,040  696,729,600  ∞  
Graph      
Name  −1_{21}  0_{21}  1_{21}  2_{21}  3_{21}  4_{21}  5_{21}  6_{21} 
Space  Family  / /  

E^{2}  Uniform tiling  0_{[3]}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  0_{[4]}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  0_{[5]}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  0_{[6]}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  0_{[7]}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  0_{[8]}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  0_{[9]}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  0_{[10]}  δ_{10}  hδ_{10}  qδ_{10}  
E^{10}  Uniform 10honeycomb  0_{[11]}  δ_{11}  hδ_{11}  qδ_{11}  
E^{n1}  Uniform (n1)honeycomb  0_{[n]}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 