List of regular polytopes

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This page lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.

The Schläfli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.

The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one lower dimensional Euclidean space.

Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with 7 equilateral triangles and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.

1 Regular polytope summary count by dimension
2 One-dimensional regular polytopes
3 Two-dimensional regular polytopes
4 Three-dimensional regular polytopes
5 Four-dimensional regular polytopes
6 Five-dimensional regular polytopes
7 Classical convex polytopes
8 Finite non-convex polytopes - star-polytopes
9 Tessellations
10 Apeirotopes
11 Abstract polytopes
12 See also
13 References
14 External links

[edit] Regular polytope summary count by dimension

Dimension	Convex	Nonconvex	Convex Euclidean tessellations	Convex hyperbolic tessellations	Nonconvex hyperbolic tessellations	Abstract Polytopes
1	1 line segment	0	0	0	0	1
2	∞ polygons	∞ star polygons	1	1	0	∞
3	5 Platonic solids	4 Kepler-Poinsot solids	3 tilings	∞	∞	∞
4	6 convex polychora	10 Schläfli-Hess polychora	1 honeycomb	4	0	∞
5	3 convex 5-polytopes	0 nonconvex 5-polytopes	3 tessellations	5	4	∞
6+	3	0	1	0	0	∞

[edit] One-dimensional regular polytopes

There is only one polytope in 1 dimensions, whose boundaries are the two endpoints of a line segment, represented by the empty Schläfli symbol {}.

[edit] Two-dimensional regular polytopes

The two dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol {p}.

Usually only convex polygons are considered regular, but star polygons, like the pentagram, can also be considered 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 complete.

Star polygons should be called nonconvex rather than concave because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary.

[edit] Three-dimensional regular polytopes

In three dimensions, the regular polytopes are called polyhedra:

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

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:

1 / p + 1 / q > 1 / 2

: Polyhedron (existing in Euclidean 3-space)

1 / p + 1 / q = 1 / 2

: Euclidean plane tiling

1 / p + 1 / q < 1 / 2

: Hyperbolic plane tiling

By enumerating the permutations, we find 5 convex forms, 4 nonconvex forms and 3 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.

[edit] Four-dimensional regular polytopes

Regular polychora with Schläfli symbol ${p, q, r}$ have cells of type ${p, q}$ , faces of type ${p}$ , edge figures ${r}$ , and vertex figures ${q, r}$ .

A vertex figure (of a polychoron) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular polychora, this vertex figure is a regular polyhedron.
An edge figure is a polygon, seen by the arrangement of faces around an edge. For regular polychora, this edge figure will always be a regular polygon.

The existence of a regular polychoron ${p, q, r}$ is constrained by the existence of the regular polyhedra ${p, q},{q, r}$ .

Each will exist in a space dependent upon this expression:

$\sin \left ( \frac{\pi}{p} \right ) \sin \left(\frac{\pi}{r}\right) - \cos\left(\frac{\pi}{q}\right)$

> 0

: Hyperspherical 3-space honeycomb or 4-space polychoron

= 0

: Euclidean 3-space honeycomb

< 0

: Hyperbolic 3-space honeycomb

These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.

The Euler characteristic $χ$ for polychora is $χ = V + F - E - C$ and is zero for all forms.

[edit] Five-dimensional regular polytopes

In five dimensions, a regular polytope can be named as ${p, q, r, s}$ where ${p, q, r}$ is the hypercell (or teron) type, ${p, q}$ is the cell type, ${p}$ is the face type, and ${s}$ is the face figure, ${r, s}$ is the edge figure, and ${q, r, s}$ is the vertex figure.

A 5-polytope has been called a polyteron, and if infinite (i.e. a honeycomb).

A vertex figure (of a 5-polytope) is a polychoron, seen by the arrangement of neighboring vertices to each vertex.

An edge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.

A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.

A regular polytope ${p, q, r, s}$ exists only if ${p, q, r}$ and ${q, r, s}$ are regular polychora.

The space it fits in is based on the expression:

$\frac{\cos^2\left(\frac{\pi}{q}\right)}{\sin^2\left(\frac{\pi}{p}\right)} + \frac{\cos^2\left(\frac{\pi}{r}\right)}{\sin^2\left(\frac{\pi}{s}\right)}$

< 1

: Spherical 4-space tessellation or 5-space polytope

= 1

: Euclidean 4-space tessellation

> 1

: hyperbolic 4-space tessellation

Enumeration of these constraints produce 3 convex polytopes, zero nonconvex polytopes, 3 4-space tessellations, and 5 hyperbolic 4-space tessellations.

[edit] Classical convex polytopes

[edit] Two dimensions

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

A regular henagon {1}, and regular digon {2}, can be considered a degenerate regular polygon. They can exist nondegenerately in non-Euclidean spaces like on the surface of a sphere or torus.

The regular apeirogon exists in the limit as {∞}, and can be considered as a tessellation of 1-dimensional space.

Convex regular polygons by name and schläfli symbol
henagon {1}	digon {2}	triangle (2-simplex) {3}	square (2-cube) (2-orthoplex) {4}	pentagon {5}
hexagon {6}	heptagon {7}	octagon {8}	enneagon {9}	decagon {10}
hendecagon {11}	dodecagon {12}	tridecagon {13}	tetradecagon {14}	pentadecagon {15}
hexadecagon {16}	heptadecagon {17}	octadecagon {18}	enneadecagon {19}	icosagon {20}
n-gon {n}	...... apeirogon {∞}

[edit] Three dimensions

The convex regular polyhedra are called the 5 Platonic solids. (The vertex figure is given with each vertex count.)

Name	Schläfli {p,q}	Faces {p}	Edges	Vertices {q}	χ	Symmetry	Dual
Tetrahedron (3-simplex)	{3,3}	4 {3}	6	4 {3}	2	T_d	Self-dual
Cube (hexahedron) (3-cube)	{4,3}	6 {4}	12	8 {3}	2	O_h	Octahedron
Octahedron (3-orthoplex)	{3,4}	8 {3}	12	6 {4}	2	O_h	Cube
Dodecahedron	{5,3}	12 {5}	30	20 {3}	2	I_h	Icosahedron
Icosahedron	{3,5}	20 {3}	30	12 {5}	2	I_h	Dodecahedron

{3,3}	{4,3}	{3,4}	{5,3}	{3,5}

In spherical geometry, hosohedron, {2,n} and dihedron {n,2} can be considered regular polyhedra (tilings of the sphere).

[edit] Four dimensions

The 6 convex polychora are as follows:

Name	Schläfli {p,q,r}	Cells {p,q}	Faces {p}	Edges {r}	Vertices {q,r}	Dual {r,q,p}
5-cell (pentachoron) (4-simplex)	{3,3,3}	5 {3,3}	10 {3}	10 {3}	5 {3,3}	Self-dual
8-cell (Tesseract) (4-cube)	{4,3,3}	8 {4,3}	24 {4}	32 {3}	16 {3,3}	16-cell
16-cell (4-orthoplex)	{3,3,4}	16 {3,3}	32 {3}	24 {4}	8 {3,4}	Tesseract
24-cell	{3,4,3}	24 {3,4}	96 {3}	96 {3}	24 {4,3}	Self-dual
120-cell	{5,3,3}	120 {5,3}	720 {5}	1200 {3}	600 {3,3}	600-cell
600-cell	{3,3,5}	600 {3,3}	1200 {3}	720 {5}	120 {3,5}	120-cell

5-cell	8-cell	16-cell	24-cell	120-cell	600-cell
{3,3,3}	{4,3,3}	{3,3,4}	{3,4,3}	{5,3,3}	{3,3,5}
Wireframe orthographic projections

Solid orthographic projections (cell-centered)
tetrahedral envelope	cubic envelope	octahedral envelope	cuboctahedral envelope	truncated rhombic triacontahedron envelope	pentakis dodecahedral envelope
Wireframe Schlegel diagrams (Perspective projection)
(Cell-centered)	(Cell-centered)	(Cell-centered)	(Cell-centered)	(Cell-centered)	(Vertex-centered)
Wireframe stereographic projections (Hyperspherical)

[edit] Five dimensions

There are three kinds of convex regular polytopes in five dimensions:

Name	Schläfli Symbol {p,q,r,s}	Facets {p,q,r}	Cells {p,q}	Faces {p}	Edges	Vertices	Face figure {s}	Edge figure {r,s}	Vertex figure {q,r,s}	Dual
5-simplex (or hexateron)	{3,3,3,3}	6 {3,3,3}	15 {3,3}	20 {3}	15	6	{3}	{3,3}	{3,3,3}	Self-dual
5-hypercube (or decateron or penteract)	{4,3,3,3}	10 {4,3,3}	40 {4,3}	80 {4}	80	32	{3}	{3,3}	{3,3,3}	pentacross
5-orthoplex (or triacontakaiditeron or pentacross)	{3,3,3,4}	32 {3,3,3}	80 {3,3}	80 {3}	40	10	{4}	{3,4}	{3,3,4}	penteract

[edit] Higher dimensions

In dimensions 5 and higher, there are only three kinds of convex regular polytopes. [Coxeter, Regular Polytopes, Table I: Regular polytopes, (iii) The three regular polytopes in n dimensions (n>=5), pp. 294-295]

Name	Schläfli Symbol {p₁,p₂,...,p_n-1}	Facet type	Vertex figure	Dual
n-simplex	{3,3,3,...,3}	{3,3,...,3}	{3,3,...,3}	Self-dual
n-cube	{4,3,3,...,3}	{4,3,...,3}	{3,3,...,3}	n-orthoplex
n-orthoplex	{3,...,3,3,4}	{3,...,3,3}	{3,...,3,4}	n-cube