Polytope groups

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Polytopes can be grouped in several different ways, usually based off of the arrangement of the elements of the polytopes.

Element groupings[edit | edit source]

Polytopes can be put into different vertex groups, edge groups, face groups etc. based on the positions of the corresponding elements within the polytope. These groups have leaders, which are the "most convex" members of the group, which the group is usually named after. These are called by special names as seen on the table.

Element Frame Leader
Vertex Army General
Edge Regiment Colonel
Face Company Captain
n-element n-regiment n-colonel

A leader is chosen based on its element figures. A general is a member of the army which is convex, a colonel is a member of the regiment with a convex vertex figure, a captain is a member of the company with a convex edge figure etc. (Note that in some cases, it is not possible to find a member of a regiment/company/etc. with a convex element figure, in which case the "most convex" representative is chosen. One example of this type of situation is with the icosicosahedron, which lacks edges that a convex-vertex-figured colonel would require.)