Regular polytope

A regular polytope is one where all the elements of any dimension are the same, and it is flag transitive as well.

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, m and n must be coprime; otherwise it will form a multiple-covered polygon.

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 {2,n} cases (dihedra) and {n,2} 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.