Regular polytope

A regular or flag transitive polytope is a polytope where all flags are congruent.

2D
In 2d there are an infinite number of both convex and starry regular polygons. These have Schläfli symbols of the form {n/d} where n is the number of sides (or equivalently, vertices) and d is the number of times the polygon winds around the center (d = 1 in all convex cases). In order to form a polygon, n and d must be coprime; otherwise it will form a multiple-covered polygon. An alternate interpretation of the symbol {n/d} where n and d are not coprime gives regular polygon compounds.

3D
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:


 * {3,3} - Tetrahedron
 * {4,3} - Cube
 * {3,4} - Octahedron
 * {5,3} - Dodecahedron
 * {3,5} - Icosahedron

In addition there are 4 non-convex regular polyhedra, known as the Kepler-Poinsot solids:


 * {5,5/2} - Great dodecahedron
 * {5/2,5} - Small stellated dodecahedron
 * {3,5/2} - Great icosahedron
 * {5/2,3} - Great stellated dodecahedron

There are also an infinite amount of degenerate cases that can only exist in spherical space. They are the {n,2} cases (dihedra) and {2,n} cases (hosohedra).

4D
Regular polychora have Schläfli symbols of the form {p,q,r}, where the cells are {p,q} and there is an r-gonal edge figure. Their vertex figure then is {q,r}. There are 6 convex regular polychora:


 * {3,3,3} - Pentachoron
 * {4,3,3} - Tesseract
 * {3,3,4} - Hexadecachoron
 * {3,4,3} - Icositetrachoron
 * {5,3,3} - Hecatonicosachoron
 * {3,3,5} - Hexacosichoron

There are also 10 non-convex regular polychora, known as the Schläfli-Hess polychora:


 * {3,5,5/2} - Faceted hexacosichoron
 * {5,5/2,5} - Great hecatonicosachoron
 * {5,3,5/2} - Grand hecatonicosachoron
 * {5/2,5,3} - Small stellated hecatonicosachoron
 * {5,5/2,3} - Great grand hecatonicosachoron
 * {5/2,3,5} - Great stellated hecatonicosachoron
 * {5/2,5,5/2} - Grand stellated hecatonicosachoron
 * {3,5/2,5} - Great faceted hexacosichoron
 * {3,3,5/2} - Grand hexacosichoron
 * {5/2,3,3} - Great grand stellated hecatonicosachoron

Higher dimensions
In all higher dimensions, there are only the 3 infinite families of regular polytopes - the simplex {3,3,...,3,3}, the hypercube {4,3,...,3,3}, and the orthoplex {3,3,...,3,4} - and no nonconvex regular polytopes.