# 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
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 Schläfli symbol Henagon Digon {1} {2}

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

### 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
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 (pqr) 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}

{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

{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 15 hyperbolic honeycombs in H3, 4 compact and 11 paracompact.

 {5,3,4} (8 dodecahedra at a vertex) Beltrami–Klein model center view {5,3,5} (20 dodecahedra at a vertex) Poincaré disk model center view {5,3,6} Poincaré disk model {4,3,5} (20 cubes at a vertex) {3,5,3} (12 icosahedra at a vertex) Poincaré disk model exterior view {6,3,3} {6,3,4} Poincaré disk model center view
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

There are also 11 paracompact H3 honeycombs (those with 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}, and {6,3,6}.

Name Schläfli
Symbol
{p,q,r}
Coxeter
Cell
type
{p,q}
Face
type
{p}
Edge
figure
{r}
Vertex
figure

{q,r}
χ Dual
Order-6 tetrahedral honeycomb {3,3,6} {3,3} {3} {6} {3,6} 0 {6,3,3}
Hexagonal tiling honeycomb {6,3,3} {6,3} {6} {3} {3,3} 0 {3,3,6}
Order-4 octahedral honeycomb {3,4,4} {3,4} {3} {4} {4,4} 0 {4,4,3}
Square tiling honeycomb {4,4,3} {4,4} {4} {3} {4,3} 0 {3,3,4}
Triangular tiling honeycomb {3,6,3} {3,6} {3} {3} {6,3} 0 Self-dual
Order-6 cubic honeycomb {4,3,6} {4,3} {4} {4} {3,4} 0 {6,3,4}
Order-4 hexagonal tiling honeycomb {6,3,4} {6,3} {6} {4} {3,4} 0 {4,3,6}
Order-4 square tiling honeycomb {4,4,4} {4,4} {4} {4} {4,4} 0 {4,4,4}
Order-6 dodecahedral honeycomb {5,3,6} {5,3} {5} {5} {3,5} 0 {6,3,5}
Order-5 hexagonal tiling honeycomb {6,3,5} {6,3} {6} {5} {3,5} 0 {5,3,6}
Order-6 hexagonal tiling honeycomb {6,3,6} {6,3} {6} {6} {3,6} 0 Self-dual

## 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 5-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.[1]

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}

#### 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

#### 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

#### 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

#### 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

#### 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

...

### 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 seven convex regular honeycombs and four star-honeycombs in H4 space.[2] Five convex ones are compact, and two are paracompact.

Five compact 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}
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

The two paracompact regular H4 honeycombs are: {3,4,3,4}, {4,3,4,3}.

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-4 24-cell honeycomb {3,4,3,4} {3,4,3} {3,4} {3} {4} {3,4} {4,3,4} {4,3,4,3}
Cubic honeycomb honeycomb {4,3,4,3} {4,3,4} {4,3} {4} {3} {4,3} {3,4,3} {3,4,3,4}

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
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}
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}

#### Tessellations of hyperbolic 5-space

There are 5 regular honeycombs in H5, all paracompact, which include 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}, and {4,3,3,4,3}.

There are no compact regular tessellations of hyperbolic space of dimension 5 or higher and no paracompact regular tessellations in hyperbolic space of dimension 6 or higher.

Name Schläfli
Symbol
{p,q,r,s,t}
Facet
type
{p,q,r,s}
4-face
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Cell
figure
{t}
Face
figure
{s,t}
Edge
figure
{r,s,t}
Vertex
figure

{q,r,s,t}
Dual
5-orthoplex honeycomb {3,3,3,4,3} {3,3,3,4} {3,3,3} {3,3} {3} {3} {4,3} {3,4,3} {3,3,4,3} {3,4,3,3,3}
24-cell honeycomb honeycomb {3,4,3,3,3} {3,4,3,3} {3,4,3} {3,4} {3} {3} {3,3} {3,3,3} {3,3,3,3} {3,3,3,4,3}
16-cell honeycomb honeycomb {3,3,4,3,3} {3,3,4,3} {3,3,4} {3,3} {3} {3} {3,3} {4,3,3} {3,4,3,3} self-dual
Order-4 24-cell honeycomb honeycomb {3,4,3,3,4} {3,4,3,3} {3,4,3} {3,4} {3} {4} {3,4} {3,3,4} {4,3,3,4} {4,3,3,4,3}
Tesseractic honeycomb honeycomb {4,3,3,4,3} {4,3,3,4} {4,3,3} {4,3} {4} {3} {4,3} {3,4,3} {3,3,4,3} {3,4,3,3,4}

#### 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.[3] 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)

### 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.