Regular polytope

A polytope is regular or flag transitive when all flags of the polytope are transitive.

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

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).

Generally, in order to form a nondegenerate polygon, n and d must be coprime. If n and d have a common divisor, the resulting figure depends on the interpretation of how polygons are derived from symbols:
 * If construction depends on equally-spaced points on a circle corresponding to the vertices of the polygon, and these points are connected, a double covering of the polygon corresponding to the cancelled fraction is obtained, and multiple vertices go unused. For example, {10/2} under this interpretation would result in a figure visually resembling {5} (trivially {5/1}), but has edges that overlap twice. It is not to be viewed as two coincident pentagons, as the figure is unicursal.
 * If construction depends on the stellation of a core regular polygon, a compound polygon is obtained.

There are also the regular skew polygons, which have vertices alternating between planes in 3D, as well as the apeirogon and the skew apeirogon (sometimes called a zigzag). Finally, there are also helical aperiogons, which also exist in three dimensions but have an infinite number of edges, each lying on a different plane.

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

The regular tilings of the plane can also be considered regular polyhedra; three exist in Euclidean space:


 * {4,4} - Square tiling
 * {3,6} - Triangular tiling
 * {6,3} - Hexagonal tiling

Regular polyhedra can also have skew faces. The Petrie dual or Petrial of a polytope (represented with π in Schläfli symbols) can take any regular polyhedron and transform it into one sharing edges and vertices. Because of this, there is a Petrie dual to every previous regular polyhedron. The Petrie dual of a Petrial gives the original polytope again.


 * {3,3}π - Petrial tetrahedron
 * {4,3}π - Petrial cube
 * {3,4}π - Petrial octahedron
 * {5,3}π - Petrial dodecahedron
 * {3,5}π - Petrial icosahedron
 * {5,5/2}π - Petrial great dodecahedron
 * {5/2,5}π - Petrial small stellated dodecahedron
 * {3,5/2}π - Petrial great icosahedron
 * {5/2,3}π - Petrial great stellated dodecahedron
 * {4,4}π - Petrial square tiling
 * {3,6}π - Petrial triangular tiling
 * {6,3}π - Petrial hexagonal tiling

We can also take the tilings of the plane by blending (unrelated to this blending) them with either a line segment (every other vertex in a new plane) or an aperiogon (take every face and turn it into a helix). These also have Petrials.


 * {4,4}#{} - Blended square tiling / Square tiling blended with a line segment
 * {3,6}#{} - Blended triangular tiling / Triangular tiling blended with a line segment
 * {6,3}#{} - Blended hexagonal tiling / Hexagonal tiling blended with a line segment
 * {4,4}π#{} - Petrial blended square tiling / Petrial square tiling blended with a line segment
 * {3,6}π#{} - Petrial blended triangular tiling / Petrial triangular tiling blended with a line segment
 * {6,3}π#{} - Petrial blended hexagonal tiling / Petrial hexagonal tiling blended with a line segment
 * {4,4}#{∞} - Helical square tiling / Square tiling blended with an apeirogon
 * {3,6}#{∞} - Helical triangular tiling / Triangular tiling blended with an apeirogon
 * {6,3}#{∞} - Helical hexagonal tiling / Hexagonal tiling blended with an apeirogon
 * {4,4}π#{∞} - Petrial helical square tiling / Petrial square tiling blended with an apeirogon
 * {3,6}π#{∞} - Petrial helical triangular tiling / Petrial triangular tiling blended with an apeirogon
 * {6,3}π#{∞} - Petrial helical hexagonal tiling / Petrial hexagonal tiling blended with an apeirogon

Finally, there are the pure aperiohedra, formed from polyhedra that may contain skew vertex figures and helical faces.


 * {4,6|4} - Mucube
 * {6,4|4} - Muoctahedron
 * {6,6|3} - Mutetrahedron
 * {4,6|4}π - Petrial mucube
 * {6,4|4}π - Petrial muoctahedron
 * {6,6|3}π - Petrial mutetrahedron
 * h{4,6|4} - Halved mucube
 * h{4,6|4}π - Petrial halved mucube
 * (no known symbol) - Dual of the petrial halved mucube / Skew petrial muoctahedron
 * (no known symbol) - Trihelical square tiling / Facetted halved mucube
 * (no known symbol) - Tetrahelical triangular tiling / Petrial facetted halved mucube
 * (no known symbol) - Skew muoctahedron

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).

In hyperbolic space, there are an infinite number of tilings for each regular polygon.

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.

Pseudoregular polytope
Polytopes which have multiple orbits of flags under Euclidean transformations but only one with the addition of conjugation are referred to as pseudoregular polytopes.