List of regular polytopes

This page lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. Clicking on any picture will magnify it.

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.

Overview

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

Dimension	Convex	Nonconvex	Convex Euclidean tessellations	Convex hyperbolic tessellations	Nonconvex hyperbolic tessellations	Hyperbolic Tessellations with infinite cells and/or vertex figures	Abstract Polytopes
1	1 line segment	0	1	0	0	0	1
2	∞ polygons	∞ star polygons	3 tilings	1	0	0	∞
3	5 Platonic solids	4 Kepler–Poinsot solids	1 honeycomb	∞	∞	0	∞
4	6 convex polychora	10 Schläfli–Hess polychora	3 tessellations	4	0	11	∞
5	3 convex 5-polytopes	0	1 tessellation	5	4	2	∞
6	3 convex 6-polytopes	0	1 tessellation	0	0	5	∞
7+	3	0	1	0	0	0	∞

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

Tessellations

The classical convex polytopes may be considered tessellations, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.

One-dimensional regular polytope

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

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.

Convex

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

Name	Triangle (2-simplex)	Square (2-orthoplex) (2-cube)	Pentagon	Hexagon	Heptagon	Octagon
Schläfli	{3}	{4}	{5}	{6}	{7}	{8}
Coxeter
Image
Name	Enneagon	Decagon	Hendecagon	Dodecagon	Tridecagon	Tetradecagon
Schläfli	{9}	{10}	{11}	{12}	{13}	{14}
Dynkin
Image
Name	Pentadecagon	Hexadecagon	Heptadecagon	Octadecagon	Enneadecagon	Icosagon	...p-gon
Schläfli	{15}	{16}	{17}	{18}	{19}	{20}	{p}
Dynkin
Image

Degenerate (spherical)

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

Name	Henagon	Digon
Schläfli symbol	{1}	{2}
Coxeter diagram
Image

Non-convex

There exist infinitely many non-convex regular 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 n-pointed non-convex regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are coprime.

Name	Pentagram	Heptagrams		Octagram	Enneagrams		Decagram	...n-agrams
Schläfli	{5/2}	{7/2}	{7/3}	{8/3}	{9/2}	{9/4}	{10/3}	{p/q}
Coxeter
Image

Tessellation

There is one tessellation of the line, giving one polytope, the (two-dimensional) apeirogon. This has infinitely many vertices and edges. Its Schläfli symbol is {∞}, and Coxeter diagram .

......

Three-dimensional regular polytopes

In three dimensions, 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: $\{3\}, \{4\}, \{5\}, \{\frac{5}{2}\}$, and $\{6\}$.

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

Convex

The convex regular polyhedra are called the 5 Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have an Euler characteristic (χ) of 2.

Name	Schläfli {p,q}	Coxeter	Image (transparent)	Image (solid)	Image (sphere)	Faces {p}	Edges	Vertices {q}	Symmetry	Dual
Tetrahedron (3-simplex) (Triangular pyramid)	{3,3}					4 {3}	6	4 {3}	Td	(self)
Cube (3-cube) (Hexahedron)	{4,3}					6 {4}	12	8 {3}	Oh	Octahedron
Octahedron (3-orthoplex)	{3,4}					8 {3}	12	6 {4}	Oh	Cube
Dodecahedron	{5,3}					12 {5}	30	20 {3}	Ih	Icosahedron
Icosahedron	{3,5}					20 {3}	30	12 {5}	Ih	Dodecahedron

Degenerate (spherical)

In spherical geometry, the hosohedra {2,n}, dihedra {n,2} and henagonal henahedron {1,1} can be considered regular polyhedra (tilings of the sphere).

Some include:

Name	Schläfli {p,q}	Coxeter diagram	Image (sphere)	Faces {p}	Edges	Vertices {q}	Symmetry	Dual
Henagonal henahedron	{1,1}			1 {1}	0	1 {1}	C1 (*1)	Self
Henagonal dihedron	{1,2}			2 {1}	1	1 {2}	C1v (*22)	Henagonal hosohedron
Henagonal hosohedron	{2,1}			1 {2}	1	2 {1}	C1v (*22)	Henagonal dihedron
Digonal dihedron Digonal hosohedron	{2,2}			2 {2}	2	2 {2}	D2h (*222)	Self
Trigonal hosohedron	{2,3}			3 {2}	3	2 {3}	D3h (*322)	Trigonal dihedron
Trigonal dihedron	{3,2}			2 {3}	3	3 {2}	D3h (*322)	Trigonal hosohedron
Hexagonal hosohedron	{2,6}			6 {2}	6	2 {6}	D6h (*622)	Hexagonal dihedron
Hexagonal dihedron	{6,2}			2 {6}	6	6 {2}	D6h (*622)	Hexagonal hosohedron

Non-convex

The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}:

As spherical tilings, these nonconvex forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms. The tiling images show a single spherical polygon face in yellow.

Name	Image (transparent)	Image (solid)	Image (sphere)	Stellation diagram	Schläfli {p,q} and Coxeter-Dynkin	Faces {p}	Edges	Vertices {q} verf.	χ	Density	Symmetry	Dual
Small stellated dodecahedron					{5/2,5}	12 {5/2}	30	12 {5}	−6	3	Ih	Great dodecahedron
Great dodecahedron					{5,5/2}	12 {5}	30	12 {5/2}	−6	3	Ih	Small stellated dodecahedron
Great stellated dodecahedron					{5/2,3}	12 {5/2}	30	20 {3}	2	7	Ih	Great icosahedron
Great icosahedron					{3,5/2}	20 {3}	30	12 {5/2}	2	7	Ih	Great stellated dodecahedron

Tessellations

Euclidean tilings

There are three regular tessellations of the plane. All three have an Euler characteristic (χ) of 0.

Name	Square tiling (Quadrille)	Triangular tiling (Deltille)	Hexagonal tiling (Hextille)
Schläfli {p,q}	{4,4}	{3,6}	{6,3}
Coxeter diagram
Image
Symmetry	*442 (p4m)	*632 (p6m)

There is one degenerate regular tiling, {∞,2}, made from two apeirogons, each filling half the plane. This tiling is related to a 2-faced dihedron, {p,2}, on the sphere.

Euclidean star-tilings

There are no regular plane tilings of star polygons. There are many enumerations that fit in the plane (1/p + 1/q = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc., but none repeat periodically.

Hyperbolic tilings

Tessellations of hyperbolic 2-space can be called hyperbolic tilings. There are infinitely many regular tilings in H². As stated above, every positive integer pair {p,q} such that 1/p + 1/q < 1/2 gives a hyperbolic tiling. In fact, for the general Schwarz triangle (p, q, r) the same holds true for 1/p + 1/q + 1/r < 1.

There are a number of different ways to display the hyperbolic plane, including the Poincaré disc model which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens.

A sampling:

Spherical (Platonic)/Euclidean/hyperbolic (Poincaré disc) tessellations with their Schläfli symbol

p \ q	3	4	5	6	7	8	...	∞
3	(tetrahedron) {3,3}	(octahedron) {3,4}	(icosahedron) {3,5}	(deltille) {3,6}	{3,7}	{3,8}		{3,∞}
4	(cube) {4,3}	(quadrille) {4,4}	{4,5}	{4,6}	{4,7}	{4,8}		{4,∞}
5	(dodecahedron) {5,3}	{5,4}	{5,5}	{5,6}	{5,7}	{5,8}		{5,∞}
6	(hextille) {6,3}	{6,4}	{6,5}	{6,6}	{6,7}	{6,8}		{6,∞}
7	{7,3}	{7,4}	{7,5}	{7,6}	{7,7}	{7,8}		{7,∞}
8	{8,3}	{8,4}	{8,5}	{8,6}	{8,7}	{8,8}		{8,∞}
...
∞	{∞,3}	{∞,4}	{∞,5}	{∞,6}	{∞,7}	{∞,8}	{∞,∞}

Hyperbolic star-tilings

There are 2 infinite forms of hyperbolic tilings whose faces or vertex figures are star polygons: {m/2, m} and their duals {m, m/2} with m = 7, 9, 11, .... The {m/2, m} tilings are stellations of the {m, 3} tilings while the {m, m/2} dual tilings are facetings of the {3, m} tilings and greatenings of the {m, 3} tilings.

The patterns {m/2, m} and {m, m/2} continue for odd m < 7 as polyhedra: when m = 5, we obtain the small stellated dodecahedron and great dodecahedron, and when m = 3, we obtain the tetrahedron. If m is even, depending on how we choose to define {m/2}, we can either obtain degenerate double covers of other tilings or compound tilings.

Name	Schläfli	Coxeter diagram	Image	Face type {p}	Vertex figure {q}	Density	Symmetry	Dual
Order-7 heptagrammic tiling	{7/2,7}			{7/2}	{7}	3	*732	Heptagrammic-order heptagonal tiling
Heptagrammic-order heptagonal tiling	{7,7/2}			{7}	{7/2}	3	*732	Order-7 heptagrammic tiling
Order-9 enneagrammic tiling	{9/2,9}			{9/2}	{9}	3	*932	Enneagrammic-order enneagonal tiling
Enneagrammic-order enneagonal tiling	{9,9/2}			{9}	{9/2}	3	*932	Order-9 enneagrammic tiling
Order-p p-grammic tiling	{p/2,p}			{p/2}	{p}	3	*p32	p-grammic-order p-gonal tiling
p-grammic-order p-gonal tiling	{p,p/2}			{p}	{p/2}	3	*p32	Order-p p-grammic tiling

Four-dimensional regular polytopes

Regular 4-polytopes (called 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 $\chi$ for polychora is $\chi = V+F-E-C$ and is zero for all forms.

Convex

The 6 convex regular polychora are shown in the table below. All these polychora have an Euler characteristic (χ) of 0.

Name	Schläfli {p,q,r}	Coxeter	Cells {p,q}	Faces {p}	Edges {r}	Vertices {q,r}	Dual {r,q,p}
5-cell (4-simplex) (Pentachoron)	{3,3,3}		5 {3,3}	10 {3}	10 {3}	5 {3,3}	(self)
8-cell (4-cube) (Tesseract)	{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)
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 (Petrie polygon) skew orthographic projections
Solid orthographic projections
tetrahedral envelope (cell/vertex-centered)	cubic envelope (cell-centered)	Cubic envelope (cell-centered)	cuboctahedral envelope (cell-centered)	truncated rhombic triacontahedron envelope (cell-centered)	Pentakis icosidodecahedral envelope (vertex-centered)
Wireframe Schlegel diagrams (Perspective projection)
(Cell-centered)	(Cell-centered)	(Cell-centered)	(Cell-centered)	(Cell-centered)	(Vertex-centered)
Wireframe stereographic projections (Hyperspherical)

Degenerate (spherical)

Dichora and hosochora exist as regular tessellations of the 3-sphere.

Regular dichora (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their hosochora duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,p}. Polychora of the form {2,p,2} are both dichora and hosochora.

Non-convex

There are ten regular star polychora, which can be called Schläfli–Hess polychoron. Their vertices are based on the convex 120-cell {5,3,3} and 600-cell {3,3,5}.

Ludwig Schläfli found four of them and skipped the last six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F+V−E=2). Edmund Hess (1843–1903) completed the full list of ten in his German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder (1883)[1].

There are 4 unique edge arrangements and 7 unique face arrangements from these 10 nonconvex polychora, shown as orthogonal projections:

Name	Wireframe	Solid	Schläfli {p, q, r} Coxeter–Dynkin	Cells {p, q}	Faces {p}	Edges {r}	Vertices {q, r}	Density	χ	Symmetry group	Dual {r, q,p}
Icosahedral 120-cell			{3,5,5/2}	120 {3,5}	1200 {3}	720 {5/2}	120 {5,5/2}	4	480	H4	Small stellated 120-cell
Small stellated 120-cell			{5/2,5,3}	120 {5/2,5}	720 {5/2}	1200 {3}	120 {5,3}	4	−480	H4	Icosahedral 120-cell
Great 120-cell			{5,5/2,5}	120 {5,5/2}	720 {5}	720 {5}	120 {5/2,5}	6	0	H4	Self-dual
Grand 120-cell			{5,3,5/2}	120 {5,3}	720 {5}	720 {5/2}	120 {3,5/2}	20	0	H4	Great stellated 120-cell
Great stellated 120-cell			{5/2,3,5}	120 {5/2,3}	720 {5/2}	720 {5}	120 {3,5}	20	0	H4	Grand 120-cell
Grand stellated 120-cell			{5/2,5,5/2}	120 {5/2,5}	720 {5/2}	720 {5/2}	120 {5,5/2}	66	0	H4	Self-dual
Great grand 120-cell			{5,5/2,3}	120 {5,5/2}	720 {5}	1200 {3}	120 {5/2,3}	76	−480	H4	Great icosahedral 120-cell
Great icosahedral 120-cell			{3,5/2,5}	120 {3,5/2}	1200 {3}	720 {5}	120 {5/2,5}	76	480	H4	Great grand 120-cell
Grand 600-cell			{3,3,5/2}	600 {3,3}	1200 {3}	720 {5/2}	120 {3,5/2}	191	0	H4	Great grand stellated 120-cell
Great grand stellated 120-cell			{5/2,3,3}	120 {5/2,3}	720 {5/2}	1200 {3}	600 {3,3}	191	0	H4	Grand 600-cell

There are 4 failed potential nonconvex regular polychora permutations: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}. Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.

Tessellations of Euclidean 3-space

Perspective view within wireframe cubic honeycomb{4,3,4}

There is only one regular tessellation of 3-space (honeycombs):

Name	Schläfli symbol {p,q,r}	Coxeter	Cell type {p,q}	Face type {p}	Edge figure {r}	Vertex figure {q,r}	χ	Dual
Cubic honeycomb	{4,3,4}		{4,3}	{4}	{4}	{3,4}	0	Self-dual

Tessellations of hyperbolic 3-space

Tessellations of hyperbolic 3-space can be called hyperbolic honeycombs. There are 4 regular honeycombs in H³:

Name	Schläfli Symbol {p,q,r}	Coxeter	Cell type {p,q}	Face type {p}	Edge figure {r}	Vertex figure {q,r}	χ	Dual
Icosahedral honeycomb	{3,5,3}		{3,5}	{3}	{3}	{5,3}	0	Self-dual
Order-5 cubic honeycomb	{4,3,5}		{4,3}	{4}	{5}	{3,5}	0	{5,3,4}
Order-4 dodecahedral honeycomb	{5,3,4}		{5,3}	{5}	{4}	{3,4}	0	{4,3,5}
Order-5 dodecahedral honeycomb	{5,3,5}		{5,3}	{5}	{5}	{3,5}	0	Self-dual

Here are some projected images: The first shows the perspective from the center of the disc in a Beltrami–Klein model, and the second and third from the outside with a Poincaré ball model.

{5,3,4} (8 dodecahedra at a vertex)

{4,3,5} (20 cubes at a vertex)

{3,5,3} (12 icosahedra at a vertex)

There are also 11 H3 honeycombs which have infinite (Euclidean) cells and/or vertex figures: {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, {6,3,6}. And infintely many honeycombs with hyperbolic cells and/or vertex figures:{7,3,3}, {5,4,3}, {5,4,4}, {7,3,7}, etc.

Five-dimensional regular polytopes and higher

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 tetracomb.

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. There are no non-convex regular polytopes in five dimensions or higher.

Higher-dimensional polytopes have sometimes received names. 6-polytopes have sometimes been called polypeta, 7-polytopes polyexa, 8-polytopes polyzetta, and 9-polytopes polyyotta.

Convex

In dimensions 5 and higher, there are only three kinds of convex regular polytopes.

Name	Schläfli Symbol {p1,...,pn−1}	Coxeter	k-faces	Facet type	Vertex figure	Dual
n-simplex	{3n−1}	...	${{n+1} \choose {k+1}}$	{3n−2}	{3n−2}	Self-dual
n-cube	{4,3n−2}	...	$2^{n-k}{n \choose k}$	{4,3n−3}	{3n−2}	n-orthoplex
n-orthoplex	{3n−2,4}	...	$2^{k+1}{n \choose {k+1}}$	{3n−2}	{3n−3,4}	n-cube

5 dimensions

Name	Schläfli Symbol {p,q,r,s} Coxeter	Facets {p,q,r}	Cells {p,q}	Faces {p}	Edges	Vertices	Face figure {s}	Edge figure {r,s}	Vertex figure {q,r,s}
5-simplex	{3,3,3,3}	6 {3,3,3}	15 {3,3}	20 {3}	15	6	{3}	{3,3}	{3,3,3}
5-cube	{4,3,3,3}	10 {4,3,3}	40 {4,3}	80 {4}	80	32	{3}	{3,3}	{3,3,3}
5-orthoplex	{3,3,3,4}	32 {3,3,3}	80 {3,3}	80 {3}	40	10	{4}	{3,4}	{3,3,4}

5-simplex

5-cube

5-orthoplex

6 dimensions

Name	Schläfli symbol	Vertices	Edges	Faces	Cells	4-faces	5-faces	χ
6-simplex	{3,3,3,3,3}	7	21	35	35	21	7	0
6-cube	{4,3,3,3,3}	64	192	240	160	60	12	0
6-orthoplex	{3,3,3,3,4}	12	60	160	240	192	64	0

6-simplex

6-cube

6-orthoplex

7 dimensions

Name	Schläfli symbol	Vertices	Edges	Faces	Cells	4-faces	5-faces	6-faces	χ
7-simplex	{3,3,3,3,3,3}	8	28	56	70	56	28	8	2
7-cube	{4,3,3,3,3,3}	128	448	672	560	280	84	14	2
7-orthoplex	{3,3,3,3,3,4}	14	84	280	560	672	448	128	2

7-simplex

7-cube

7-orthoplex

8 dimensions

Name	Schläfli symbol	Vertices	Edges	Faces	Cells	4-faces	5-faces	6-faces	7-faces	χ
8-simplex	{3,3,3,3,3,3,3}	9	36	84	126	126	84	36	9	0
8-cube	{4,3,3,3,3,3,3}	256	1024	1792	1792	1120	448	112	16	0
8-orthoplex	{3,3,3,3,3,3,4}	16	112	448	1120	1792	1792	1024	256	0

8-simplex

8-cube

8-orthoplex

9 dimensions

Name	Schläfli symbol	Vertices	Edges	Faces	Cells	4-faces	5-faces	6-faces	7-faces	8-faces	χ
9-simplex	{38}	10	45	120	210	252	210	120	45	10	2
9-cube	{4,37}	512	2304	4608	5376	4032	2016	672	144	18	2
9-orthoplex	{37,4}	18	144	672	2016	4032	5376	4608	2304	512	2

9-simplex

9-cube

9-orthoplex

10 dimensions

Name	Schläfli symbol	Vertices	Edges	Faces	Cells	4-faces	5-faces	6-faces	7-faces	8-faces	9-faces	χ
10-simplex	{39}	11	55	165	330	462	462	330	165	55	11	0
10-cube	{4,38}	1024	5120	11520	15360	13440	8064	3360	960	180	20	0
10-orthoplex	{38,4}	20	180	960	3360	8064	13440	15360	11520	5120	1024	0

10-simplex

10-cube

10-orthoplex

...

Non-convex

There are no non-convex regular polytopes in five dimensions or higher.

Tessellations of Euclidean space

Tessellations of Euclidean 4-space

There are three kinds of infinite regular tessellations (honeycombs) that can tessellate Euclidean four-dimensional space:

Name	Schläfli Symbol {p,q,r,s}	Facet type {p,q,r}	Cell type {p,q}	Face type {p}	Face figure {s}	Edge figure {r,s}	Vertex figure {q,r,s}	Dual
Tesseractic honeycomb	{4,3,3,4}	{4,3,3}	{4,3}	{4}	{4}	{3,4}	{3,3,4}	Self-dual
16-cell honeycomb	{3,3,4,3}	{3,3,4}	{3,3}	{3}	{3}	{4,3}	{3,4,3}	{3,4,3,3}
24-cell honeycomb	{3,4,3,3}	{3,4,3}	{3,4}	{3}	{3}	{3,3}	{4,3,3}	{3,3,4,3}

Projected portion of {4,3,3,4} (Tesseractic honeycomb)

Projected portion of {3,3,4,3} (16-cell honeycomb)

Projected portion of {3,4,3,3} (24-cell honeycomb)

Tessellations of Euclidean 5-space and higher

The hypercubic honeycomb is the only family of regular honeycomb that can tessellate each dimension, five or higher, formed by hypercube facets, four around every ridge.

Name	Schläfli {p1, p2, ..., pn−1}	Facet type	Vertex figure	Dual
Square tiling	{4,4}	{4}	{4}	Self-dual
Cubic honeycomb	{4,3,4}	{4,3}	{3,4}	Self-dual
Tesseractic honeycomb	{4,32,4}	{4,32}	{32,4}	Self-dual
5-cube honeycomb	{4,33,4}	{4,33}	{33,4}	Self-dual
6-cube honeycomb	{4,34,4}	{4,34}	{34,4}	Self-dual
7-cube honeycomb	{4,35,4}	{4,35}	{35,4}	Self-dual
8-cube honeycomb	{4,36,4}	{4,36}	{36,4}	Self-dual
n-hypercubic honeycomb	{4,3n−2,4}	{4,3n−2}	{3n−2,4}	Self-dual

Tessellations of hyperbolic space

Tessellations of hyperbolic 4-space

There are five kinds of convex regular honeycombs and four kinds of star-honeycombs in H⁴ space.

Five convex regular honeycombs in H⁴:

Name	Schläfli Symbol {p,q,r,s}	Facet type {p,q,r}	Cell type {p,q}	Face type {p}	Face figure {s}	Edge figure {r,s}	Vertex figure {q,r,s}	Dual
Order-5 5-cell honeycomb	{3,3,3,5}	{3,3,3}	{3,3}	{3}	{5}	{3,5}	{3,3,5}	{5,3,3,3}
Order-3 120-cell honeycomb	{5,3,3,3}	{5,3,3}	{5,3}	{5}	{3}	{3,3}	{3,3,3}	{3,3,3,5}
Order-5 tesseractic honeycomb	{4,3,3,5}	{4,3,3}	{4,3}	{4}	{5}	{3,5}	{3,3,5}	{5,3,3,4}
Order-4 120-cell honeycomb	{5,3,3,4}	{5,3,3}	{5,3}	{5}	{4}	{3,4}	{3,3,4}	{4,3,3,5}
Order-5 600-cell honeycomb	{5,3,3,5}	{5,3,3}	{5,3}	{5}	{5}	{3,5}	{3,3,5}	Self-dual

There are four regular star-honeycombs in H⁴ space:

Name	Schläfli Symbol {p,q,r,s}	Facet type {p,q,r}	Cell type {p,q}	Face type {p}	Face figure {s}	Edge figure {r,s}	Vertex figure {q,r,s}	Dual
Order-3 small stellated 120-cell honeycomb	{5/2,5,3,3}	{5/2,5,3}	{5/2,5}	{5}	{5}	{3,3}	{5,3,3}	{3,3,5,5/2}
Pentagrammic-order 600-cell honeycomb	{3,3,5,5/2}	{3,3,5}	{3,3}	{3}	{5/2}	{5,5/2}	{3,5,5/2}	{5/2,5,3,3}
Order-5 icosahedral 120-cell honeycomb	{3,5,5/2,5}	{3,5,5/2}	{3,5}	{3}	{5}	{5/2,5}	{5,5/2,5}	{5,5/2,5,3}
Order-3 great 120-cell honeycomb	{5,5/2,5,3}	{5,5/2,5}	{5,5/2}	{5}	{3}	{5,3}	{5/2,5,3}	{3,5,5/2,5}

There are also 2 H⁴ honeycombs with infinite (Euclidean) facets or vertex figures: {3,4,3,4}, {4,3,4,3}.

Tessellations of hyperbolic 5-space

There are no finite-faceted regular tessellations of hyperbolic space of dimension 5 or higher.

There are 5 regular honeycombs in H⁵ with infinite (Euclidean) facets or vertex figures: {3,4,3,3,3}, {3,3,4,3,3}, {3,3,3,4,3}, {3,4,3,3,4}, {4,3,3,4,3}.

Tessellations of hyperbolic 6-space and higher

Even allowing for infinite (Euclidean) facets and/or vertex figures, there are no regular tessellations of hyperbolic space of dimension 6 or higher.

Apeirotopes

An apeirotope is, like any other polytope, an unbounded hyper-surface. The difference is that whereas a polytope's hyper-surface curls back on itself to close round a finite volume of hyperspace, an apeirotope does not curl back.

Some people regard apeirotopes as just a special kind of polytope, while others regard them as rather different things.

Two dimensions

A regular apeirogon is a regular division of an infinitely long line into equal segments, joined by vertices. It has regular embeddings in the plane, and in higher-dimensional spaces. In two dimensions it can form a straight line or a zig-zag. In three dimensions, it traces out a helical spiral. The zig-zag and spiral forms are said to be skew.

Three dimensions

An apeirohedron is an infinite polyhedral surface. Like an apeirogon, it can be flat or skew. A flat apeirohedron is just a tiling of the plane. A skew apeirohedron is an intricate honeycomb-like structure which divides space into two regions.

There are thirty regular apeirohedra in Euclidean space. These include the tessellations of type {4,4}, {6,3}, {3,6} above, as well as (in the plane) polytopes of type: {∞,3}, {∞,4}, {∞,6} and in 3-dimensional space, blends of these with either an apeirogon or a line segment, and the "pure" 3-dimensional apeirohedra (12 in number)

See also regular skew polyhedron.

Four and higher dimensions

The apeirochora have not been completely classified as of 2006.

Abstract polytopes

The abstract polytopes arose out of an attempt to study polytopes apart from the geometrical space they are embedded in. They include the tessellations of spherical, euclidean and hyperbolic space, tessellations of other manifolds, and many other objects that do not have a well-defined topology, but instead may be characterised by their "local" topology. There are infinitely many in every dimension. See this atlas for a sample. Some notable examples of abstract polytopes that do not appear elsewhere in this list are the 11-cell and the 57-cell.

See also

• Polygon
  • Regular polygon
  • Star polygon
• Polyhedron
  • Regular polyhedron (5 regular Platonic solids and 4 Kepler–Poinsot solids)
    • Uniform polyhedron
• Polychoron
  • Convex regular 4-polytope (6 regular polychora)
    • Uniform polychoron
  • Schläfli–Hess polychoron (10 regular star polychora)
• Tessellation
  • Tilings of regular polygons
  • Convex uniform honeycomb
• Regular polytope
  • Uniform polytope

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, pp. 212–213)
• D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes

External links

• The Platonic Solids
• Kepler-Poinsot Polyhedra
• Regular 4d Polytope Foldouts
• Multidimensional Glossary (Look up Hexacosichoron and Hecatonicosachoron)
• Polytope Viewer
• Polytopes and optimal packing of p points in n dimensional spheres
• An atlas of small regular polytopes