User:MWinter4/Vertex-transitive polytopes

In geometry, a polytope (e.g. a polygon or polyhedron) is vertex-transitive or isogonal if all its vertices are identical under the polytope's symmetry group. This implies that each vertex is surrounded by the same kinds of face.

More precisely, one requires that for any two vertices there exists a symmetry of the polytope mapping the first vertex onto the second vertex. Other ways of saying this are that the group of automorphisms of the polytope acts transitively on its vertices, or that the vertices lie within a single orbit of the symmetry group.

The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory.

The pseudorhombicuboctahedron – which is not isogonal – demonstrates that simply asserting that "all vertices look the same" is not as restrictive as the definition used here, which involves the group of isometries preserving the polyhedron or tiling.

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.