Regular polytope

A polytope is regular in the strict sense when the automorphism group of the polytope is transitive on the set of flags. In this case, the automorphism group is transitive of the elements of each rank, as well as on sections of the same type. However, there are also looser definitions of regularity. For example, one definition requires transitivity at every level of element rather than flag-transitivity.

Regularity is defined separately for both abstract and concrete polytopes. Regularity of a concrete polytope is a stricter condition than that of an abstract polytope, as there are polytopes that are abstractly but not concretely regular such as the dodecadodecahedron.

0D, 1D
There is exactly one regular 0D polytope: the point. There is also 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: There also exists a regular polygon with an infinite number of sides: the apeirogon, which is also the only tiling of 1D space.
 * If construction depends on equally-spaced points on a circle corresponding to the vertices of the polygon, and these points are connected, a multiple 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.

Measures
A regular polygon with Schläfli symbol {n/d} has a circumradius given by $$\frac{1}{2\sin\frac{d\pi}{n}}$$, inradius given by $$\frac{1}{2\tan\frac{d\pi}{n}}$$, area given by $$\frac{n}{4\tan\frac{d\pi}{n}}$$, and vertex angle equal to $$\frac{\pi(n-2d)}{n}$$.

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

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 polytopes:


 * {3,3,3,3} - Hexateron
 * {4,3,3,3} - Penteract
 * {3,3,3,4} - Triacontaditeron

There are also 3 regular tilings of 4D space:


 * {4,3,3,4} - Tesseractic tetracomb
 * {3,3,4,3} - Hexadecachoric tetracomb
 * {3,4,3,3} - Icositetrachoric tetracomb

6D+
In all higher dimensions, there are only 3 infinite families of regular polytopes:


 * the simplex {3,3,...,3,3}
 * the hypercube {4,3,...,3,3}
 * the orthoplex {3,3,...,3,4}

and no nonconvex regular polytopes. There is also one regular honeycomb: the hypercubic honeycomb {4,3,...,3,4}.

Regular polytopes in spherical and hyperbolic space
Unique regular polytopes can also exist within non-Euclidean space, in particular spherical and hyperbolic space.

2D
In spherical space, there are also 2 additional degenerate regular polygons: the monogon {1} and the digon {2}.

In hyperbolic space, there are two different types of aperiogons: those that exist on horocycles (which have a center at an ideal point) and on hypercycles (which are the set of points on one side of a line that are a given distance from that line).

3D
Every regular convex polyhedron in Euclidean space has an embedding that becomes a tiling on the sphere. 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 every pair of convex regular polygons, which includes the apeirogon because it is convex and not flat in hyperbolic space (see the regular hyperbolic tilings). There are only two infinite sets of star tilings: those of the form {p/2,p} and their duals {p,p/2}, where p is odd (e.g. the stellated heptagonal tiling {7/2,7}).

Regular skew polytopes
Regular polytopes can also be skew, meaning that the whole polytope, some of its elements, or its vertex figure are rank $n$ but cannot lie within a flat $n$-dimensional space. In Euclidean space, it is known how many finite regular skew $n$-polytopes there are and the number of regular skew honeycombs there are in any dimension, but the number of regular apeirotopes in any dimension greater than 3 is unknown.

Regular skew polygons
There exist an infinite number of regular skew polygons, which are regular polygons that do not fit within a plane. The only exception to this is the skew apeirogon {∞}#{} (or zigzag) in 2D space, which is generally considered skew.

There are two sets of skew polygons that can exist in 3D space: the first set contains finite polygons formed by blending a polygon with a line segment (see below image), and the second is the set of helices which have an infinite number of vertices, and are formed by blending a regular polygon with an apeirogon.

Regular skew polyhedra
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.

There are multiple extensions to Schläfli symbols which allow the Petrials to be represented. Where the Petrie dual operation can be represented with π, and {p,q}r is defined as a regular map, or equivalently a polyhedron with q p-gons around a vertex, and an r-gonal Petrie polygon. 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}5/2 - 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 apeirogon (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, which are infinite polyhedra that cannot be described as a blend in a non-trivial way. The notation {p,q|r} means that there are q p-gons around a vertex, with r-gonal holes formed around the faces.

In total, there are 48 regular polyhedra in 3D Euclidean space, 36 of which are skew.
 * {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
 * {∞, 4}·,∗3 - Skewed muoctahedron
 * {6,4}6 - Petrial skewed muoctahedron / Dual of the petrial halved mucube
 * {∞,3}(a) - Tetrahelical triangular tiling / Facetted halved mucube
 * {∞,3}(b) - Trihelical square tiling / Petrial facetted halved mucube

Higher dimensional skews
Just as we can make skew polygons whose vertices lie in 3D space, polyhedra can be made with points lying in 4D space.

Within four dimensions we can take the comb product of any regular polygon with itself to produce a regular 4D skew polyhedron: Some non-prismatic regular skew 4D polyhedra include:
 * {4,4|3} - Triangular duocomb
 * {4,4|4} - Square duocomb
 * {4,4|5} - Pentagonal duocomb
 * {4,4|5/2} - Pentagrammic duocomb
 * etc.
 * {4,6|3}
 * {6,4|3}
 * {4,8|3}
 * {8,4|3}
 * {4,8/3|3}
 * {8/3,4|3}
 * {8,8/3|3}
 * {8/3,8|3}

Regular skew apeirohedra in 3D hyperbolic space
There are 31 regular skew apeirohedra without self-intersections in 3D hyperbolic space:


 * 14 of these skew polyhedra are compact: {8,10|3}, {10,8|3}, {10,4|3}, {4,10|3}, {6,4|5}, {4,6|5}, {10,6|3}, {6,10|3}, {8,8|3}, {6,6|4}, {10,10|3},{6,6|5}, {8,6|3}, {6,8|3}
 * The other 17 are paracompact: {12,10|3}, {10,12|3}, {12,4|3}, {4,12|3}, {6,4|6}, {4,6|6}, {8,4|4}, {4,8|4}, {12,6|3}, {6,12|3}, {12,12|3}, {6,6|6}, {8,6|4}, {6,8|4}, {12,8|3}, {8,12|3}, and {8,8|4}

There may be many noncompact regular skew apeirohedra, as a list has not yet been enumerated.The total number of regular skew polyhedra in 3D hyperbolic space is currently unknown.

Regular skew polychora
Although the total number of regular skew polychora is not known, it is known how many regular honeycombs there are (8, 7 skew) and how many finite regular polychora there are (34, 18 skew).

The 8 regular honeycombs are:

The mucubic honeycomb is the Petrie dual of the cubic honeycomb.
 * $$\{ 4,3,4 \}$$ - Cubic honeycomb
 * $$\{ \{4,6 \mid 4 \}, \{ 6,4 \}_3 \}$$ - Mucubic honeycomb
 * $$\{ \{ \infty, 3 \}_6 \# \{ \} , \{ 3,3 \} \}$$ - Apeirotetrahedron
 * $$\{ \{ \infty, 4 \}_4 \# \{ \infty \} , \{ 4, 3 \}_3 \}$$ - Petrial apeirotetrahedron
 * $$\{ \{ \infty, 3 \}_6 \# \{ \} , \{ 3, 4 \} \}$$ - Apeiroctahedron
 * $$\{ \{ \infty, 6 \}_3 \# \{ \infty \} , \{ 6, 4 \}_3 \}$$ - Petrial apeiroctahedron
 * $$\{ \{ \infty, 4 \}_4 \# \{ \} , \{ 4, 3 \} \}$$ - Apeirocube
 * $$\{ \{ \infty, 6 \}_3 \# \{ \infty \} , \{ 6, 3 \}_4 \}$$ - Petrial apeirocube