# Regular polytope

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A regular polytope is a polytope whose symmetry group acts transitively on its flags. This makes regular polytopes have the highest possible degree of symmetry. All regular polytopes are vertex-transitive, edge-transitive, face-transitive, etc.

A polytope whose abstract structure is regular does not itself need to be regular, as such polytopes will not contain all of the symmetries that their abstract structure does. As an example, the dodecadodecahedron is not regular but is abstractly regular.

## Background and motivation

Symmetry is one of the most important properties of a polytope that mathematicians are concerned with. Naturally polytopes with unusually high symmetry are of considerable interest. Larger polytopes tend to have more symmetries than smaller ones: for example, the dodecagonal prism has 48 symmetries, while the tetrahedron has only 24 - half as many. Rather than being interested with polytopes whose symmetries exceed an arbitrary threshold, mathematicians are interested in polytopes who have a lot of symmetries relative to their size. The measure of size mathematicians end up using for this purpose is the number of flags. Flags are essentially the atoms that make up polytopes. Unlike vertices, edges, faces etc. flags are indivisible by the symmetry of a polytope. A mirror plane of a polyhedron can pass through a face, dividing it into two parts, however a mirror symmetry in some sense never passes through a flag. In the mathematical language, we can say that any symmetry of the polytope that maps a flag to itself must be the identity symmetry. This makes flags a very good way to measure the size of a polytope relative to the symmetry group.

Using this measure we can see that the dodecagonal prism has 144 flags and 48 symmetries while the tetrahedron has 24 flags and 24 symmetries. The tetrahedron is, relative to its number of flags, much more symmetric. It turns out that because flags are indivisible, the number of symmetries of a polytope can never exceed the number of flags, so the most symmetric polytopes are those who have the same number of flags as symmetries. This gives a definition for regular:

A polytope is regular iff it has the same number of flags as symmetries.

However this definition breaks down if we want to consider polytopes with infinitely many flags; for example tilings. To apply it to apeirotopes, we must tweak the definition slightly, so that there is a specific relationship between the flags and the symmetries. We can tweak it to the following:

A polytope 𝓟 is regular iff there is some flag f  such that for every flag g  there is a symmetry of 𝓟 which maps f  to g .

Since a symmetry can only map one flag to one place, it is clear that this implies that the number of symmetries and flags must be the same, but it offers a strong condition for apeirotopes. It essentially says that every flag is the same as f , which essentially tells us that all flags are the same. This is a very high degree of symmetry.

The whole idea can be simplified using transitivity into the modern definition of regular:

A polytope is regular iff its symmetry acts transitively on its flags.

## Overview

This table shows a summary of regular polytope counts by rank.

Rank Finite Euclidean Hyperbolic Abstract
Compact Paracompact
Convex Star Skew Convex Skew Convex Star Convex
d  = r [1] d  = r +1 d  = r +2 d  = r -1[1] d  = r  d  = r +1
1 1 0 0 0 0 0 0 0 0 0 0 0 1
2 0 1 0 1 1 0 0
3 5 4 9 3 3 24[2]
4 6 10 18 ? 1 7 ? 4 0 11
5 3 0 3 ? 3 15 ? 5 4 2
6 3 0 3 ? 1 7 ? 0 0 5
7+ 3 0 3 ? 1 7 ? 0 0 0

## Ranks 0 and 1

There is exactly one regular 0D polytope: the point. There is also exactly one regular 1D polytope: the line segment.

There are no non-convex polytopes of rank 0 or 1, neither star nor skew.

## Polygons

### Convex

For every integer n  > 2, there is exactly one regular convex polygon with n  sides and vertices. This polygon is called the n -gon, and is given the Schläfli symbol {n}.

### Star polygons

There is also an infinite number of regular star 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 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.

### Tilings

There also exists a regular polygon with an infinite number of sides: the apeirogon, which is also the only tiling of 1D space.

### 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 zigzag in 2D space, which is generally considered skew.

The pure regular polygons are exactly the planar (non-skew) polygons, thus every regular polygon is the blend of finitely many planar polygons.

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 dyad, 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. These helices include blends of dense polygons with the apeirogon, which produces a non-dense result.

### Measures

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

## Polyhedra

### Convex

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:

### Star

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

### Tilings

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

### Skews

There are 12 skew polyhedra of full rank, formed as the Petrials of the above.

## Polychora

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:

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

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

## 5-polytopes

There are 3 convex regular 5-polytopes:

There are also 3 regular tilings of 4D space:

## Ranks 6 and up

For each higher rank, there are only 3 regular finite planar polytopes:

All three of these families are convex, there are no regular star polytopes beyond rank 4.

There is also one regular planar 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 apeirogons: 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 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.

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.

Finally, there are the pure apeirohedra, 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.

### 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:

• ${\displaystyle \{4,3,4\}}$ - Cubic honeycomb
• ${\displaystyle \{\{4,6\mid 4\},\{6,4\}_{3}\}}$ - Mucubic honeycomb
• ${\displaystyle \{\{\infty ,3\}_{6}\#\{\},\{3,3\}\}}$ - Apeir tetrahedron
• ${\displaystyle \{\{\infty ,4\}_{4}\#\{\infty \},\{4,3\}_{3}\}}$ - Petrial apeir tetrahedron
• ${\displaystyle \{\{\infty ,3\}_{6}\#\{\},\{3,4\}\}}$ - Apeir octahedron
• ${\displaystyle \{\{\infty ,6\}_{3}\#\{\infty \},\{6,4\}_{3}\}}$ - Petrial apeir octahedron
• ${\displaystyle \{\{\infty ,4\}_{4}\#\{\},\{4,3\}\}}$ - Apeir cube
• ${\displaystyle \{\{\infty ,6\}_{3}\#\{\infty \},\{6,3\}_{4}\}}$ - Petrial apeir cube

The mucubic honeycomb is the Petrie dual of the cubic honeycomb.

## References

1. McMullen (2004)
2. McMullen

## Bibliography

• Coxeter, Donald (1973). Regular polytopes (3 ed.). Dover. ISBN 0-486-61480-8. OCLC 798003.
• McMullen, Peter (2004). "Regular Polytopes of Full Rank" (PDF). Discrete Computational Geometry.
• McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.