# Polytope groups

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

## Element groupings

### Faceting based

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.

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.)

One may also group polytopes into brigades, which are the unions of all regiments in an army using the same edge length. Within a brigade, one may also define cohorts, which are subsets of brigades closed under blending.

### Stellation based

The group of polytopes with the same number of facets that occupy the same hyperplanes is called a navy, a dual structure to an army. Likewise, they can be subdivided into k-navies such that all elements from dimension n-1 to n-k occupy the same (n-p)-space.