Regular polytopes

A regular polytope in n dimensions is a polytope whose facets are all regular polytopes in n-1 dimensions and equivalent, and whose vertices all have regular figures and are equivalent under symmetry. Regular polytopes are particularly interesting since they characterize special rotational symmetries in n dimensions.

In 0 and 1 dimensions, the point and the line segment are regular.

In 2 dimensions, polygons are regular if all of the sides are equal and all of the angles are equal. There are a denumerable number of them, starting from three sides, and characterizes cyclic and dihedral groups.

In 3 dimensions, the regular polyhedra are precisely the Platonic polyhedra and the Kepler-Poinsot polyhedra. They characterize exceptional 3D point groups which do not fall on an infinite family.

In 4 dimensions, the regular polychora include the six regular figures built from tetrahedra, cubes, octahedra, or dodecahedra, as well as the ten Schlafli-Hess polyhedra.

In 5 or more dimensions, there are only three regular figures, which are analogous to the tetrahedron, cube, and octahedron.