List of regular polytopes and compounds

Short description: None

Example regular polytopes
Regular (2D) polygons
Convex	Star
{5}	{5/2}
Regular (3D) polyhedra
Convex	Star
{5,3}	{5/2,5}
Regular 4D polytopes
Convex	Star
{5,3,3}	{5/2,5,3}
Regular 2D tessellations
Euclidean	Hyperbolic
{4,4}	{5,4}
Regular 3D tessellations
Euclidean	Hyperbolic
{4,3,4}	{5,3,4}

This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.

Overview

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

Rank	Finite			Euclidean		Hyperbolic			Abstract	Compounds
	Finite			Euclidean		Compact		Paracompact		Compounds
	Convex	Star	Skew^{[lower-alpha 1]}^[1]	Convex	Skew^{[lower-alpha 1]}^[1]	Convex	Star	Convex		Convex	Star
1	1	none	none	none	none	none	none	none	1	none	none
2	[math]\displaystyle{ \infty }[/math]	[math]\displaystyle{ \infty }[/math]	none	1	none	1	none	none	[math]\displaystyle{ \infty }[/math]	[math]\displaystyle{ \infty }[/math]	[math]\displaystyle{ \infty }[/math]
3	5	4	9	3	3	[math]\displaystyle{ \infty }[/math]	[math]\displaystyle{ \infty }[/math]	[math]\displaystyle{ \infty }[/math]	[math]\displaystyle{ \infty }[/math]	5	none
4	6	10	18	1	7	4	none	11	[math]\displaystyle{ \infty }[/math]	26	20
5	3	none	3	3	15	5	4	2	[math]\displaystyle{ \infty }[/math]	none	none
6	3	none	3	1	7	none	none	5	[math]\displaystyle{ \infty }[/math]	none	none
7	3	none	3	1	7	none	none	none	[math]\displaystyle{ \infty }[/math]	3	none
8	3	none	3	1	7	none	none	none	[math]\displaystyle{ \infty }[/math]	6	none
9+	3	none	3	1	7	none	none	none	[math]\displaystyle{ \infty }[/math]	^{[lower-alpha 2]}	none

↑ ^1.0 ^1.1 Only counting polytopes of full rank. There are more regular polytopes of each rank > 1 in higher dimensions.
↑ [math]\displaystyle{ \begin{cases} 2, & \text{if the number of dimensions is of the form } 2^k \\ 1, & \text{if the number of dimensions is of the form } 2^k-1 \\ 0, & \text{otherwise} \\ \end{cases} }[/math]

There are no Euclidean regular star tessellations in any number of dimensions.

1-polytopes

	A Coxeter diagram represent mirror "planes" as nodes, and puts a ring around a node if a point is not on the plane. A dion { }, , is a point p and its mirror image point p', and the line segment between them.

There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol { },^[2]^[3] or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a dion^[4] and gives it the Schläfli symbol { }.

Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes.^[5] It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon.^[6]

2-polytopes (polygons)

The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A $p$ -gonal regular polygon is represented by Schläfli symbol {p}.

Many sources only cosider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed.

Convex

The Schläfli symbol {p} represents a regular $p$ -gon.

Name	Nonagon (Enneagon)	Decagon	Hendecagon	Dodecagon	Tridecagon	Tetradecagon
Name	Triangle (2-simplex)	Square (2-orthoplex) (2-cube)	Pentagon (2-pentagonal polytope)	Hexagon	Heptagon	Octagon
Schläfli	{3}	{4}	{5}	{6}	{7}	{8}
Symmetry	D₃, [3]	D₄, [4]	D₅, [5]	D₆, [6]	D₇, [7]	D₈, [8]
Coxeter
Image
Schläfli	{9}	{10}	{11}	{12}	{13}	{14}
Symmetry	D₉, [9]	D₁₀, [10]	D₁₁, [11]	D₁₂, [12]	D₁₃, [13]	D₁₄, [14]
Dynkin
Image
Name	Pentadecagon	Hexadecagon	Heptadecagon	Octadecagon	Enneadecagon	Icosagon	...p-gon
Schläfli	{15}	{16}	{17}	{18}	{19}	{20}	{p}
Symmetry	D₁₅, [15]	D₁₆, [16]	D₁₇, [17]	D₁₈, [18]	D₁₉, [19]	D₂₀, [20]	D_p, [p]
Dynkin
Image

Spherical

The regular digon {2} can be considered to be a degenerate regular polygon. It can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of a sphere or torus. For example, digon can be realised non-degenerately as a spherical lune. A monogon {1} could also be realised on the sphere as a single point with a great circle through it.^[7] However, a monogon is not a valid abstract polytope because its single edge is incident to only one vertex rather than two.

Name	Monogon	Digon
Schläfli symbol	{1}	{2}
Symmetry	D₁, [ ]	D₂, [2]
Coxeter diagram	or
Image

Stars

There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers ${n / m}$ . They are called star polygons and share the same vertex arrangements of the convex regular polygons.

In general, for any natural number $n$ , there are regular $n$ -pointed stars with Schläfli symbols ${n / m}$ for all $m$ such that $m < n /2$ (strictly speaking ${{{1}}}$ ) and $m$ and $n$ are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Symbols where $m$ and $n$ are not coprime may be used to represent compound polygons.

Name	Pentagram	Heptagrams		Octagram	Enneagrams		Decagram	...n-grams
Schläfli	{5/2}	{7/2}	{7/3}	{8/3}	{9/2}	{9/4}	{10/3}	{p/q}
Symmetry	D₅, [5]	D₇, [7]		D₈, [8]	D₉, [9],		D₁₀, [10]	D_p, [p]
Coxeter
Image

Regular star polygons up to 20 sides
{11/2}	{11/3}	{11/4}	{11/5}	{12/5}	{13/2}	{13/3}	{13/4}	{13/5}	{13/6}
{14/3}	{14/5}	{15/2}	{15/4}	{15/7}	{16/3}	{16/5}	{16/7}
{17/2}	{17/3}	{17/4}	{17/5}	{17/6}	{17/7}	{17/8}	{18/5}	{18/7}
{19/2}	{19/3}	{19/4}	{19/5}	{19/6}	{19/7}	{19/8}	{19/9}	{20/3}	{20/7}	{20/9}

Star polygons that can only exist as spherical tilings, similarly to the monogon and digon, may exist (for example: {3/2}, {5/3}, {5/4}, {7/4}, {9/5}), however these do not appear to have been studied in detail.

There also exist failed star polygons, such as the piangle, which do not cover the surface of a circle finitely many times.^[8]

Skew polygons

In addition to the planar regular polygons there are infinitely many regular skew polygons. Skew polygons can be created via the blending operation.

The blend of two polygons $P$ and $Q$ , written $P # Q$ , can be constructed as follows:

take the cartesian product of their vertices $V P \times V Q$ .
add edges $(p 0 \times q 0, p 1 \times q 1)$ where $(p 0, p 1)$ is an edge of $P$ and $(q 0, q 1)$ is an edge of $Q$ .
select an arbitrary connected component of the result.

Alternatively, the blend is the polygon $⟨ ρ 0 σ 0, ρ 1 σ 1 ⟩$ where $ρ$ and $σ$ are the generating mirrors of $P$ and $Q$ placed in orthogonal subspaces.^[9] The blending operation is commutative, associative and idempotent.

Every regular skew polygon can be expressed as the blend of a unique^{[lower-alpha 1]} set of planar polygons.^[9] If $P$ and $Q$ share no factors then $Dim(P # Q) = Dim(P) + Dim(Q)$ .

In 3 space

The regular finite polygons in 3 dimensions are exactly the blends of the planar polygons (dimension 2) with the digon (dimension 1). They have vertices corresponding to a prism ( ${n / m}#{}$ where $n$ is odd) or an antiprism ( ${n / m}#{}$ where $n$ is even). All polygons in 3 space have an even number of vertices and edges.

Several of these appear as the Petrie polygons of regular polyhedra.

In 4 space

The regular finite polygons in 4 dimensions are exactly the polygons formed as a blend of two distinct planar polygons. They have vertices lying on a Clifford torus and related by a Clifford displacement. Unlike 3-dimensional polygons, skew polygons on double rotations can include an odd-number of sides.

3-polytopes (polyhedra)

Polytopes of rank 3 are called polyhedra:

A regular polyhedron with Schläfli symbol ${p, q}$ , Coxeter diagrams , has a regular face type ${p}$ , and regular vertex figure ${1}$ .

A vertex figure (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.

Existence of a regular polyhedron ${p, q}$ is constrained by an inequality, related to the vertex figure's angle defect: [math]\displaystyle{ \begin{align} & \frac{1}{p} + \frac{1}{q} \gt \frac{1}{2} : \text{Polyhedron (existing in Euclidean 3-space)} \\[6pt] & \frac{1}{p} + \frac{1}{q} = \frac{1}{2} : \text{Euclidean plane tiling} \\[6pt] & \frac{1}{p} + \frac{1}{q} \lt \frac{1}{2} : \text{Hyperbolic plane tiling} \end{align} }[/math]

By enumerating the permutations, we find five convex forms, four star forms and three plane tilings, all with polygons ${p}$ and ${q}$ limited to: {3}, {4}, {5}, {5/2}, and {6}.

Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.