Gosset Polytope

A Gosset polytope is one of the semiregular polytopes discovered by Gosset (1900). The finite higher-dimensional Gosset polytopes in Coxeter's series are , , , and . The Gosset graph is the skeleton of the Gosset polytope , while the Gosset polytope is also called the root polytope.

The figure above shows a two-dimensional projection of the skeleton of the Gosset polytope in the style of a Coxeter plane projection, meaning that it is chosen to display a large cyclic symmetry of the polytope rather than to preserve ordinary 8-dimensional distances. This graph has 240 vertices corresponding to the 240 roots of the root system and 6720 edges; equivalently, each vertex is adjacent to its 56 nearest neighbors in the 8-dimensional root configuration. The displayed embedding places the vertices on eight concentric rings of 30 vertices. Its ring radii have been normalized for clarity, so the planar distances in the drawing are not the Euclidean distances in the underlying 8-dimensional configuration. The skeleton of the Gosset polytope is implemented in Wolfram Language as GraphData["421PolytopeGraph"].