User:MWinter4/Symmetries of polytopes

The symmetries of a polytope are geometric or combinatorial transformations that keep aspects of the polytope unchanged.

Isometries
Usually the symmetries of a polytope are the isometries (or distance-preserving mappings) that map a polytope onto itself. Under this notion of symmetry group, a vertex-transitive polytope is inscribed, that is, all its vertices lie on a common sphere.

Other geometric symmetries
One can also consider more general geometric symmetry groups of a polytope, such as its affine or projective symmetries. This leads to considering polytopes as vertex-transitive that are usually not seen as such, e.g. any triangle (for affine symmetries) or any quadrilateral (for projective symmetries). However, these polytopes that are vertex-transitive under these more general symmetries are not richer in the combinatorial sense. Any polytope that is affinely or projectively vertex-transitive can be transformed using an affine or projective transformation into a polytope that is vertex-transitive via isometries.

Combinatorial symmetries
The most general notion of vertex-transitivity for a polytope is defined via combinatial symmetries, that is, symmetries of the polytope's face lattice. It is not know whether every polytope that is combinatorially vertex-transitive also has a geometrically vertex-transitive realization.