5 21 honeycomb
This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure (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-simplices.
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.
The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 121 polytope.
The E8 lattice can also be constructed as a union of the vertices of two 8-demicube honeycombs (called a D82 or D8+ lattice), as well as the union of the vertices of three 8-simplex honeycombs (called an A83 lattice):
The 521 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.