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 can take any regular polyhedron and transform it into one sharing edges and vertices with the original, but with skew faces. Because of this, there is a Petrie dual to every previous regular polyhedron.

In a Schläfli symbol, the Petrials have more than one valid representation, but the Petrie dual operation can be represented with π. The Petrie dual of a Petrial polyhedron gives the original polytope again.


 * {3,3}π, {4,3}3 - Petrial tetrahedron
 * {4,3}π, {6,3}4 - Petrial cube
 * {3,4}π, {6,4}3 - Petrial octahedron
 * {5,3}π, {10,3}5 - Petrial dodecahedron
 * {3,5}π, {10,5}3 - Petrial icosahedron
 * {5,5/2}π, {6,5/2}5 - Petrial great dodecahedron
 * {5/2,5}π, {6,5}·,3 - Petrial small stellated dodecahedron
 * {3,5/2}π, {10/3,5/2}3 - Petrial great icosahedron
 * {5/2,3}π, {10/3,3}5/2 - Petrial great stellated dodecahedron
 * {4,4}π, {∞,4}4 - Petrial square tiling
 * {3,6}π, {∞,6}3 - Petrial triangular tiling
 * {6,3}π, {∞,3}6 - Petrial hexagonal tiling

We can also create new polyhedra by taking the tilings of the plane and 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#{}, {∞,4}4#{} - Petrial blended square tiling / Petrial square tiling blended with a line segment
 * {3,6}π#{}, {∞,6}3#{} - Petrial blended triangular tiling / Petrial triangular tiling blended with a line segment
 * {6,3}π#{}, {∞,3}6#{} - 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#{∞}, {∞,4}4#{∞} - Petrial helical square tiling / Petrial square tiling blended with an apeirogon
 * {3,6}π#{∞}, {∞,6}3#{∞} - Petrial helical triangular tiling / Petrial triangular tiling blended with an apeirogon
 * {6,3}π#{∞}, {∞,3}6#{∞} - Petrial helical hexagonal tiling / Petrial hexagonal tiling blended with an apeirogon

Finally, there are the pure aperiohedra, formed from polyhedra that extend infinitely in all three dimensions. The notation {p,q|r} means that there are q p-gons around a vertex, with r-gonal holes formed around the faces.


 * {4,6|4} - Mucube
 * {6,4|4} - Muoctahedron
 * {6,6|3} - Mutetrahedron
 * {4,6|4}π, {∞,6}4,4 - Petrial mucube
 * {6,4|4}π, {∞,4}6,4 - Petrial muoctahedron
 * {6,6|3}π, {∞,6}6,3 - Petrial mutetrahedron
 * {6,6}4 - Halved mucube
 * {4,6}6 - Petrial halved mucube
 * {6,4}6 - Dual of the petrial halved mucube / Skew petrial muoctahedron
 * {∞,3}(a) - Trihelical square tiling / Facetted halved mucube
 * {∞,3}(b) - Tetrahelical triangular tiling / Petrial facetted halved mucube
 * {∞, 4}·,∗3 - 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:

There is also a single regular honeycomb of 3D space:
 * {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


 * {4,3,4} - Cubic honeycomb

5D
In 5D, there are 3 convex regular polytera and 3 tetracombs:


 * {3,3,3,3} - Hexateron
 * {4,3,3,3} - Penteract
 * {3,3,3,4} - Triacontaditeron
 * {4,3,3,4} - Tesseractic tetracomb
 * {3,3,4,3} - Hexadecachoric tetracomb
 * {3,4,3,3} - Icositetrachoric tetracomb

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. There will also be one regular honeycomb: the hypercubic honeycomb {4,3,...,3,4}.

Pseudoregular polytope
Polytopes which are not regular, but all of whose orbits of flags are conjugates, are referred to as pseudoregular polytopes.