A regular polytope is a polytope whose symmetry group acts transitively on its flags, where a flag contains
one face of each dimension in a maximal chain. Equivalently, any flag can be carried
to any other by a symmetry of the polytope. A regular 
-polytope is described by a Schläfli
symbol 
.
Regular polygons are the two-dimensional examples, regular polyhedra the three-dimensional examples, and regular polychora the four-dimensional examples. There are three families of convex regular polytopes in every dimension at least 5: the simplices, hypercubes, and cross polytopes.
