Editing Polytope (section)

===Regular polytopes===
{{Main|Regular polytope}}
[[Regular polytope]]s have the highest degree of symmetry of all polytopes. The symmetry group of a regular polytope acts transitively on its [[flag (geometry)|flags]]; hence, the [[dual polytope]] of a regular polytope is also regular.

There are three main classes of regular polytope which occur in any number of dimensions:
*[[Simplex|Simplices]], including the [[equilateral triangle]] and the [[regular tetrahedron]].
*[[Hypercube]]s or measure polytopes, including the [[square]] and the [[cube]].
*[[Orthoplex]]es or cross polytopes, including the [[square]] and [[regular octahedron]].

Dimensions two, three and four include regular figures which have fivefold symmetries and some of which are non-convex stars, and in two dimensions there are infinitely many [[regular polygon]]s of ''n''-fold symmetry, both convex and (for ''n'' ≥ 5) star. But in higher dimensions there are no other regular polytopes.<ref name="coxeter1973"/>

In three dimensions the convex [[Platonic solid]]s include the fivefold-symmetric [[dodecahedron]] and [[icosahedron]], and there are also four star [[Kepler-Poinsot polyhedra]] with fivefold symmetry, bringing the total to nine regular polyhedra.

In four dimensions the [[regular 4-polytope]]s include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten star [[Schläfli-Hess 4-polytope]]s, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.