3 31 honeycomb
In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the short branch leaves the 6-simplex facet:
Removing the node on the end of the 3-length branch leaves the 321 facet:
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 231 polytope.
The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (131).
The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (031).
The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes tetrahedral prism {}×{3,3}.
Each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 dimensions; its kissing number is 126, represented by the vertices of its vertex figure 231.
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