Regular convex polytope

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Points and edges[edit | edit source]

There is exactly one regular 0D polytope: the point. There is also exactly one regular 1D polytope: the line segment.

Polygons[edit | edit source]

The regular heptagon

For every integer n  > 2, there is exactly one regular convex polygon with n  sides and vertices. This polygon is called the n -gon, and is given the Schläfli symbol {n}.

Polyhedra[edit | edit source]

Regular polyhedra have Schläfli symbols of the form {p,q}, with p -gonal faces with a q -gonal vertex figure. There are five convex regular polyhedra, known as the Platonic solids:

Polychora[edit | edit source]

Regular polychora have Schläfli symbols of the form {p,q,r}, where the cells are regular convex polyhedra {p,q} and there is an r -gonal edge figure. Their vertex figure is then a regular polyhedron {q,r}. There are 6 convex regular polychora:

5-polytopes and higher[edit | edit source]

In all higher dimensions, there are only 3 infinite families of regular polytopes: