# 5 21 honeycomb

In geometry, the **5 _{21} honeycomb** is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5

_{21}is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

^{[1]}

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 5_{21} honeycomb is surrounded by 2160 8-orthoplexes and 17280 8-simplices.

The vertex figure of Gosset's honeycomb is the semiregular 4_{21} polytope. It is the final figure in the k_{21} family.

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, 6_{11}.

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 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 7-sphere in the densest known packing in 8 dimensions; its kissing number is 240, represented by the vertices of its vertex figure 4_{21}.

The E8 lattice can also be constructed as a union of the vertices of two 8-demicube honeycombs (called a D_{8}^{2} or D_{8}^{+} lattice), as well as the union of the vertices of three 8-simplex honeycombs (called an A_{8}^{3} lattice):^{[6]}

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.