This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.

This table shows a summary of regular polytope counts by rank.

There are no Euclidean regular star tessellations in any number of dimensions.

There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol { }, or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a dion and gives it the Schläfli symbol { }.

Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon.

The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol {p}.

Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed.

The Schläfli symbol {p} represents a regular p-gon.