Configuration (polytope)

In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration.

These polytope configurations may be more accurately called incidence matrices, where like elements are collected together in rows and columns.

They all have the same incidence matrix, but symmetry allows vertices and edges to be collected together and counted.

Quadrilaterals exist with dual pairs which will have the same matrix, rotated 180 degrees, with vertices and edges reversed.

Squares and parallelograms, and general quadrilaterals are self-dual by class so their matrices are unchanged when rotated 180 degrees.

Writing Nj for the number of j-spaces present, a given configuration may be represented by the matrix Tetrahedra have matrices that can also be grouped by their symmetry, with a general tetrahedron having 14 rows and columns for the 4 vertices, 6 edges, and 4 faces.

Tetrahedra are self-dual, and rotating the matices 180 degrees (swapping vertices and faces) will leave it unchanged.