List of regular polytopes and compounds

This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an n-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an n-polytope equivalently describes a tessellation of an -sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol, and with its octahedral symmetry, or, it is represented by Coxeter diagram.
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 seven 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.
A more general definition of regular polytopes which do not have simple Schläfli symbols includes regular skew polytopes and regular skew apeirotopes with nonplanar facets or vertex figures.

Overview

This table shows a summary of regular polytope counts by dimension.
There are no Euclidean regular star tessellations in any number of dimensions.

One dimension

A one-dimensional polytope or 1-polytope is a closed line segment, bounded by its two endpoints. A 1-polytope is regular by definition and is represented by Schläfli symbol, or a Coxeter diagram with a single ringed node,. Norman Johnson calls it a dion and gives it the Schläfli symbol.
Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon.

Two dimensions (polygons)

The two-dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol.
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 be completed.
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 represents a regular p-gon.

Name	Triangle	Square	Pentagon	Hexagon	Heptagon	Octagon	-
Schläfli							-
Symmetry	D₃,	D₄,	D₅,	D₆,	D₇,	D₈,	-
Coxeter							-
Image							-
Name	Nonagon	Decagon	Hendecagon	Dodecagon	Tridecagon	Tetradecagon	-
Schläfli							-
Symmetry	D₉,	D₁₀,	D₁₁,	D₁₂,	D₁₃,	D₁₄,	-
Dynkin							-
Image							-
Name	Pentadecagon	Hexadecagon	Heptadecagon	Octadecagon	Enneadecagon	Icosagon	...p-gon
Schläfli
Symmetry	D₁₅,	D₁₆,	D₁₇,	D₁₈,	D₁₉,	D₂₀,	D_p,
Dynkin
Image							-

Spherical

The regular digon 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.

Name	Monogon	Digon
Schläfli symbol
Symmetry	D₁,	D₂,
Coxeter diagram	or
Image

Stars

There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers. 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 star regular polygonal stars with Schläfli symbols for all m such that m < n/2 and m and n are coprime. Cases where m and n are not coprime are called [|compound polygons].

'		'	'				-
'	'		'		-	-	-
'				'		-	-
'					'

Star polygons that can only exist as spherical tilings, similarly to the monogon and digon, may exist, 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.

Skew polygons

In 3-dimensional space, a regular skew polygon is called an antiprismatic polygon, with the vertex arrangement of an antiprism, and a subset of edges, zig-zagging between top and bottom polygons.
In 4-dimensions a regular skew polygon can have vertices on a Clifford torus and related by a Clifford displacement. Unlike antiprismatic skew polygons, skew polygons on double rotations can include an odd-number of sides.
They can be seen in the Petrie polygons of the convex regular 4-polytopes, seen as regular plane polygons in the perimeter of Coxeter plane projection:

Pentagon	Octagon	Dodecagon	Triacontagon
5-cell	16-cell	24-cell	600-cell

Three dimensions (polyhedra)

In three dimensions, polytopes are called polyhedra:
A regular polyhedron with Schläfli symbol, Coxeter diagrams, has a regular face type, and regular vertex figure.
A vertex figure 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 polygon.
Existence of a regular polyhedron is constrained by an inequality, related to the vertex figure's angle defect:
By enumerating the permutations, we find five convex forms, four star forms and three plane tilings, all with polygons and limited to:,,,, and.
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 of 2.

Name	Schläfli	Coxeter	Image	Image	Faces	Edges	Vertices	Symmetry	Dual
Tetrahedron					4	6	4	T_d
Hexahedron Cube					6	12	8	O_h	Octahedron
Octahedron					8	12	6	O_h	Cube
Dodecahedron					12	30	20	I_h	Icosahedron
Icosahedron					20	30	12	I_h	Dodecahedron

Spherical

In spherical geometry, regular spherical polyhedra exist that would otherwise be degenerate as polytopes. These are the hosohedra and their dual dihedra. Coxeter calls these cases "improper" tessellations.
The first few cases are listed below.

Name	Schläfli	Coxeter diagram	Image	Faces _π/p	Edges	Vertices	Symmetry	Dual
Digonal hosohedron				2 _π/2	2	2 _π/2	D_2h	Self
Trigonal hosohedron				3 _π/3	3	2	D_3h	Trigonal dihedron
Square hosohedron				4 _π/4	4	2	D_4h	Square dihedron
Pentagonal hosohedron				5 _π/5	5	2	D_5h	Pentagonal dihedron
Hexagonal hosohedron				6 _π/6	6	2	D_6h	Hexagonal dihedron

Name	Schläfli	Coxeter diagram	Image	Faces	Edges	Vertices	Symmetry	Dual
Digonal dihedron				2 _π/2	2	2 _π/2	D_2h	Self
Trigonal dihedron				2	3	3 _π/3	D_3h	Trigonal hosohedron
Square dihedron				2	4	4 _π/4	D_4h	Square hosohedron
Pentagonal dihedron				2	5	5 _π/5	D_5h	Pentagonal hosohedron
Hexagonal dihedron				2	6	6 _π/6	D_6h	Hexagonal hosohedron

Star-dihedra and hosohedra and also exist for any star polygon.

Stars

The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron and icosahedron :
As spherical tilings, these star 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	Image	Image	Stellation diagram	Schläfli and Coxeter	Faces	Edges	Vertices verf.	χ	Density	Symmetry	Dual
Small stellated dodecahedron						12	30	12	−6	3	I_h	Great dodecahedron
Great dodecahedron						12	30	12	−6	3	I_h	Small stellated dodecahedron
Great stellated dodecahedron						12	30	20	2	7	I_h	Great icosahedron
Great icosahedron						20	30	12	2	7	I_h	Great stellated dodecahedron

There are infinitely many failed star polyhedra. These are also spherical tilings with star polygons in their Schläfli symbols, but they do not cover a sphere finitely many times. Some examples are,,,,,, and.

Skew polyhedra

are generalizations to the set of regular polyhedron which include the possibility of nonplanar vertex figures.
For 4-dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol for these figures, with implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.
The regular skew polyhedra, represented by, follow this equation:
Four of them can be seen in 4-dimensions as a subset of faces of four regular 4-polytopes, sharing the same vertex arrangement and edge arrangement:

Four dimensions

Regular 4-polytopes with Schläfli symbol have cells of type, faces of type, edge figures
, and vertex figures.

A vertex figure is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular 4-polytopes, this vertex figure is a regular polyhedron.
An edge figure is a polygon, seen by the arrangement of faces around an edge. For regular 4-polytopes, this edge figure will always be a regular polygon.

The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra. A suggested name for 4-polytopes is "polychoron".
Each will exist in a space dependent upon this expression:
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 for convex 4-polytopes is zero:

Convex

The 6 convex regular 4-polytopes are shown in the table below. All these 4-polytopes have an Euler characteristic of 0.

Name	Schläfli	Coxeter	Cells	Faces	Edges	Vertices	Dual
5-cell			5	10	10	5
8-cell			8	24	32	16	16-cell
16-cell			16	32	24	8	Tesseract
24-cell			24	96	96	24
120-cell			120	720	1200	600	600-cell
600-cell			600	1200	720	120	120-cell

Spherical

and hoso-4-topes exist as regular tessellations of the 3-sphere.
Regular di-4-topes include:,,,,,, and their hoso-4-tope duals :,,,,,. 4-polytopes of the form are the same as. There are also the cases which have dihedral cells and hosohedral vertex figures.

Schläfli	Coxeter	Cells _π/q	Faces _π/p,π/q	Edges	Vertices	Vertex figure	Symmetry	Dual
		4 _π/3	6 _π/3,π/3	4	2
		6 _π/3	12 _π/4,π/3	8	2
		8 _π/4	12 _π/3,π/4	6	2
		12 _π/3	30 _π/5,π/3	20	2
		20 _π/5	30 _π/3,π/5	12	2

Stars

There are ten regular star 4-polytopes, which are called the Schläfli–Hess 4-polytopes. Their vertices are based on the convex 120-cell ' and 600-cell '.
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. Edmund Hess 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 .
There are 4 unique edge arrangements and 7 unique face arrangements from these 10 regular star 4-polytopes, shown as orthogonal projections:

Name	Wireframe	Solid	Schläfli Coxeter	Cells	Faces	Edges	Vertices	Density	χ	Symmetry group	Dual
Icosahedral 120-cell				120 Icosahedron\|	1200 Triangle\|	720 Pentagram\|	120 Great dodecahedron\|	4	480	H₄	Small stellated 120-cell
Small stellated 120-cell				120 Small stellated dodecahedron\|	720 Pentagram\|	1200 Triangle\|	120 Dodecahedron\|	4	−480	H₄	Icosahedral 120-cell
Great 120-cell				120 Great dodecahedron\|	720 Pentagon\|	720 Pentagon\|	120 Small stellated dodecahedron\|	6	0	H₄	Self-dual
Grand 120-cell				120 Dodecahedron\|	720 Pentagon\|	720 Pentagram\|	120 Great icosahedron\|	20	0	H₄	Great stellated 120-cell
Great stellated 120-cell				120 Great stellated dodecahedron\|	720 Pentagram\|	720 Pentagon\|	120 Icosahedron\|	20	0	H₄	Grand 120-cell
Grand stellated 120-cell				120 Small stellated dodecahedron\|	720 Pentagram\|	720 Pentagram\|	120 Great dodecahedron\|	66	0	H₄	Self-dual
Great grand 120-cell				120 Great dodecahedron\|	720 Pentagon\|	1200 Triangle\|	120 Great stellated dodecahedron\|	76	−480	H₄	Great icosahedral 120-cell
Great icosahedral 120-cell				120 Great icosahedron\|	1200 Triangle\|	720 Pentagon\|	120 Small stellated dodecahedron\|	76	480	H₄	Great grand 120-cell
Grand 600-cell				600 Tetrahedron\|	1200 Triangle\|	720 Pentagram\|	120 Great icosahedron\|	191	0	H₄	Great grand stellated 120-cell
Great grand stellated 120-cell				120 Great stellated dodecahedron\|	720 Pentagram\|	1200 Triangle\|	600 Tetrahedron\|	191	0	H₄	Grand 600-cell

There are 4 failed potential regular star 4-polytopes permutations:,,,. Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.

Five and more dimensions

In five dimensions, a regular polytope can be named as
where is the 4-face type, is the cell type, is the face type, and is the face figure, is the edge figure, and is the vertex figure.
A regular 5-polytope exists only if and are regular 4-polytopes.
The space it fits in is based on the expression:
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.

Convex

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

Name	Schläfli Symbol	Coxeter	`k`-faces	Facet type	Vertex figure	Dual
n-simplex		...	binomial coefficient\|			Self-dual
`n`-cube		...				`n`-orthoplex
`n`-orthoplex		...				`n`-cube

There are also improper cases where some numbers in the Schläfli symbol are 2. For example, is an improper regular spherical polytope whenever is a regular spherical polytope, and is an improper regular spherical polytope whenever is a regular spherical polytope. Such polytopes may also be used as facets, yielding forms such as.

5 dimensions

5-simplex

5-cube

5-orthoplex

6 dimensions

6-simplex

6-cube

6-orthoplex

7 dimensions

7-simplex

7-cube

7-orthoplex

8 dimensions

8-simplex

8-cube

8-orthoplex

9 dimensions

9-simplex

9-cube

9-orthoplex

10 dimensions

10-simplex

10-cube

10-orthoplex

...

Non-convex

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

Regular projective polytopes

A projective regular -polytope exists when an original regular n-spherical tessellation,, is centrally symmetric. Such a polytope is named hemi-, and contain half as many elements. Coxeter gives a symbol /2, while McMullen writes _h/2 with h as the coxeter number.
Even-sided regular polygons have hemi-2n-gon projective polygons, /2.
There are 4 regular projective polyhedra related to 4 of 5 Platonic solids.
The hemi-cube and hemi-octahedron generalize as hemi-n-cubes and hemi-n-orthoplexes in any dimensions.

Regular projective polyhedra

Regular projective 4-polytopes

In 4-dimensions 5 of 6 convex regular 4-polytopes generate projective 4-polytopes. The 3 special cases are hemi-24-cell, hemi-600-cell, and hemi-120-cell.

Name	Coxeter symbol	McMullen Symbol	Cells	Faces	Edges	Vertices	χ
Hemi-tesseract	/2	₄	4	12	16	8	0
Hemi-16-cell	/2	₄	8	16	12	4	0
Hemi-24-cell	/2	₆	12	48	48	12	0
Hemi-120-cell	/2	₁₅	60	360	600	300	0
Hemi-600-cell	/2	₁₅	300	600	360	60	0

Regular projective 5-polytopes

There are only 2 convex regular projective hemi-polytopes in dimensions 5 or higher.

Name	Schläfli	4-faces	Cells	Faces	Edges	Vertices	χ
hemi-penteract	/2	5	20	40	40	16	1
hemi-pentacross	/2	16	40	40	20	5	1

Apeirotopes

An apeirotope or infinite polytope is a polytope which has infinitely many facets. An n-apeirotope is an infinite n-polytope: a 2-apeirotope or apeirotope, is an infinite polygon, a 3-apeirotope, or apeirohedron, is an infinite polyhedron, etc.
There are two main geometric classes of apeirotope:

Regular honeycombs in n dimensions, which completely fill an n-dimensional space.
Regular skew apeirotopes, comprising an n-dimensional manifold in a higher space.
One dimension (apeirogons)

The straight apeirogon is a regular tessellation of the line, subdividing it into infinitely many equal segments. It has infinitely many vertices and edges. Its Schläfli symbol is, and Coxeter diagram.
......
Apeirogons in the hyperbolic plane, most notably the regular apeirogon,, can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed by horocycles or hypercycles rather than circles.
Regular apeirogons that are scaled to converge at infinity have the symbol and exist on horocycles, while more generally they can exist on hypercycles.


Apeirogon on horocycle	Apeirogon on hypercycle

Above are two regular hyperbolic apeirogons in the Poincaré disk model, the right one shows perpendicular reflection lines of divergent fundamental domains, separated by length λ.

Skew apeirogons

A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular.
Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left- or right-handed.

2-dimensions	3-dimensions
Zig-zag apeirogon	Helix apeirogon

Two dimensions (apeirohedra)

Euclidean tilings

There are three regular tessellations of the plane. All three have an Euler characteristic of 0.
There are two improper regular tilings:, an apeirogonal dihedron, made from two apeirogons, each filling half the plane; and secondly, its dual,, an apeirogonal hosohedron, seen as an infinite set of parallel lines.

Order-2 apeirogonal tiling|,

Apeirogonal hosohedron|,

Euclidean star-tilings

There are no regular plane tilings of star polygons. There are many enumerations that fit in the plane, like,,,, etc., but none repeat periodically.

Hyperbolic tilings

Tessellations of hyperbolic 2-space are hyperbolic tilings. There are infinitely many regular tilings in H². As stated above, every positive integer pair such that 1/p + 1/q < 1/2 gives a hyperbolic tiling. In fact, for the general Schwarz triangle 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.
There are infinitely many flat regular 3-apeirotopes as regular tilings of the hyperbolic plane, of the form, with p+q

,, ...
,, ...
,, ...
,, ...
,, ...
,, ...
,, ...
...
,, ...

A sampling:

Hyperbolic star-tilings

There are 2 infinite forms of hyperbolic tilings whose faces or vertex figures are star polygons: and their duals with m = 7, 9, 11,.... The tilings are stellations of the tilings while the dual tilings are facetings of the tilings and greatenings of the tilings.
The patterns and continue for odd m < 7 as polyhedra: when m = 5, we obtain the small stellated dodecahedron and great dodecahedron, and when m = 3, the case degenerates to a tetrahedron. The other two Kepler–Poinsot polyhedra do not have regular hyperbolic tiling analogues. If m is even, depending on how we choose to define, we can either obtain degenerate double covers of other tilings or compound tilings.

Name	Schläfli	Coxeter diagram	Image	Face type	Vertex figure	Density	Symmetry	Dual
Order-7 heptagrammic tiling						3	*732	Heptagrammic-order heptagonal tiling
Heptagrammic-order heptagonal tiling						3	*732	Order-7 heptagrammic tiling
Order-9 enneagrammic tiling						3	*932	Enneagrammic-order enneagonal tiling
Enneagrammic-order enneagonal tiling						3	*932	Order-9 enneagrammic tiling
Order-11 hendecagrammic tiling						3	*11.3.2	Hendecagrammic-order hendecagonal tiling
Hendecagrammic-order hendecagonal tiling						3	*11.3.2	Order-11 hendecagrammic tiling
Order-p p-grammic tiling						3	*p32	p-grammic-order p-gonal tiling
p-grammic-order p-gonal tiling						3	*p32	Order-p p-grammic tiling

Skew apeirohedra in Euclidean 3-space

There are three regular skew apeirohedra in Euclidean 3-space, with regular skew polygon vertex figures. They share the same vertex arrangement and edge arrangement of 3 convex uniform honeycombs.

6 squares around each vertex:
4 hexagons around each vertex:
6 hexagons around each vertex:

,. A π petrie dual operator replaces faces with petrie polygons; δ is a dual operator reverses vertices and faces; φ_k is a kth facetting operator; η is a halving operator, and σ skewing halving operator.
There are thirty regular apeirohedra in Euclidean 3-space. These include those [|listed above], as well as 8 other "pure" apeirohedra, all related to the cubic honeycomb,, with others having skew polygon faces: ₄, ₄, ₆, ^a, ^b, ^.*3, _6,4, _4,4, and _6,3.

Skew apeirohedra in hyperbolic 3-space

There are 31 regular skew apeirohedra in hyperbolic 3-space:

14 are compact:,,,,,,,,,,,,, and.
17 are paracompact:,,,,,,,,,,,,,,,, and.
Three dimensions (4-apeirotopes)

Tessellations of Euclidean 3-space

There is only one non-degenerate regular tessellation of 3-space, :

Name	Schläfli	Coxeter	Cell type	Face type	Edge figure	Vertex figure	χ	Dual
Cubic honeycomb							0	Self-dual

Improper tessellations of Euclidean 3-space

There are six improper regular tessellations, pairs based on the three regular Euclidean tilings. Their cells and vertex figures are all regular hosohedra, dihedra,, and Euclidean tilings. These improper regular tilings are constructionally related to prismatic uniform honeycombs by truncation operations. They are higher-dimensional analogues of the order-2 apeirogonal tiling and apeirogonal hosohedron.

Schläfli	Coxeter diagram	Cell type	Face type	Edge figure	Vertex figure
order-4 square hosohedral honeycomb\|
Order-6 triangular hosohedral honeycomb\|
hexagonal hosohedral honeycomb\|
Order-2 square tiling honeycomb\|
Order-2 triangular tiling honeycomb\|
Order-2 hexagonal tiling honeycomb\|

Tessellations of hyperbolic 3-space

There are ten flat regular honeycombs of hyperbolic 3-space:

4 are compact:,,, and
while 6 are paracompact:,,,,,,,,,, and.

Tessellations of hyperbolic 3-space can be called hyperbolic honeycombs. There are 15 hyperbolic honeycombs in H³, 4 compact and 11 paracompact.

Name	Schläfli Symbol	Coxeter	Cell type	Face type	Edge figure	Vertex figure	χ	Dual
Icosahedral honeycomb			icosahedron\|			dodecahedron\|	0	Self-dual
Order-5 cubic honeycomb			cube\|			icosahedron\|	0
Order-4 dodecahedral honeycomb			Dodecahedron\|			octahedron\|	0
Order-5 dodecahedral honeycomb			Dodecahedron\|			icosahedron\|	0	Self-dual

There are also 11 paracompact H³ honeycombs :,,,,,,,,,, and.

Name	Schläfli Symbol	Coxeter	Cell type	Face type	Edge figure	Vertex figure	χ	Dual
Order-6 tetrahedral honeycomb			tetrahedron\|			triangular tiling\|	0
Hexagonal tiling honeycomb			hexagonal tiling\|			tetrahedron\|	0
Order-4 octahedral honeycomb			octahedron\|			square tiling\|	0
Square tiling honeycomb			square tiling\|			cube\|	0
Triangular tiling honeycomb			triangular tiling\|			hexagonal tiling\|	0	Self-dual
Order-6 cubic honeycomb			Cube\|			octahedron\|	0
Order-4 hexagonal tiling honeycomb			hexagonal tiling\|			Octahedron\|	0
Order-4 square tiling honeycomb			square tiling\|			square tiling\|	0
Order-6 dodecahedral honeycomb			dodecahedron\|			icosahedron\|	0
Order-5 hexagonal tiling honeycomb			hexagonal tiling\|			icosahedron\|	0
Order-6 hexagonal tiling honeycomb			hexagonal tiling\|			triangular tiling\|	0	Self-dual

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space. All honeycombs with hyperbolic cells or vertex figures and do not have 2 in their Schläfli symbol are noncompact.
There are no regular hyperbolic star-honeycombs in H³: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical.

Four dimensions (5-apeirotopes)

Tessellations of Euclidean 4-space

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

Name	Schläfli Symbol	Facet type	Cell type	Face type	Face figure	Edge figure	Vertex figure	Dual
Tesseractic honeycomb								Self-dual
16-cell honeycomb
24-cell honeycomb

Projected portion of

There are also the two improper cases and.
There are three flat regular honeycombs of Euclidean 4-space:

,, and.

There are seven flat regular convex honeycombs of hyperbolic 4-space:

5 are compact:,,,,
2 are paracompact:, and.

There are four flat regular star honeycombs of hyperbolic 4-space:

,,, and.
Tessellations of hyperbolic 4-space

There are seven convex regular honeycombs and four star-honeycombs in H⁴ space. Five convex ones are compact, and two are paracompact.
Five compact regular honeycombs in H⁴:

Name	Schläfli Symbol	Facet type	Cell type	Face type	Face figure	Edge figure	Vertex figure	Dual
Order-5 5-cell honeycomb		5-cell\|	tetrahedron\|			icosahedron\|	600-cell\|
120-cell honeycomb		120-cell\|	dodecahedron\|			tetrahedron\|	5-cell\|
Order-5 tesseractic honeycomb		tesseract\|	cube\|			icosahedron\|	600-cell\|
Order-4 120-cell honeycomb		120-cell\|	dodecahedron\|			octahedron\|	16-cell\|
Order-5 120-cell honeycomb		120-cell\|	dodecahedron\|			icosahedron\|	600-cell\|	Self-dual

The two paracompact regular H⁴ honeycombs are:,.

Name	Schläfli Symbol	Facet type	Cell type	Face type	Face figure	Edge figure	Vertex figure	Dual
Order-4 24-cell honeycomb		24-cell\|	octahedron\|			octahedron\|	cubic honeycomb\|
Cubic honeycomb honeycomb		cubic honeycomb\|	cube\|			cube\|	24-cell\|

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space. All honeycombs which are not shown in the set of tables below and do not have 2 in their Schläfli symbol are noncompact.

Star tessellations of hyperbolic 4-space

There are four regular star-honeycombs in H⁴ space:

Name	Schläfli Symbol	Facet type	Cell type	Face type	Face figure	Edge figure	Vertex figure	Dual	Density
Small stellated 120-cell honeycomb		Small stellated 120-cell\|					120-cell\|		5
Pentagrammic-order 600-cell honeycomb		600-cell\|	tetrahedron\|						5
Order-5 icosahedral 120-cell honeycomb			icosahedron\|						10
Great 120-cell honeycomb						dodecahedron\|			10

Five dimensions (6-apeirotopes)

There is only one flat regular honeycomb of Euclidean 5-space:

There are five flat regular regular honeycombs of hyperbolic 5-space, all paracompact:

,,,, and
Tessellations of Euclidean 5-space

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

Name	Schläfli	Facet type	Vertex figure	Dual
Square tiling				Self-dual
Cubic honeycomb				Self-dual
Tesseractic honeycomb				Self-dual
5-cube honeycomb				Self-dual
6-cube honeycomb				Self-dual
7-cube honeycomb				Self-dual
8-cube honeycomb				Self-dual
n-hypercubic honeycomb				Self-dual

In E⁵, there are also the improper cases,,,,, and. In Eⁿ, and are always improper Euclidean tessellations.

Tessellations of hyperbolic 5-space

There are 5 regular honeycombs in H⁵, all paracompact, which include infinite facets or vertex figures:,,,, and.
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	Facet type	4-face type	Cell type	Face type	Cell figure	Face figure	Edge figure	Vertex figure	Dual
5-orthoplex honeycomb		5-orthoplex\|	5-cell\|	tetrahedron\|			cube\|	24-cell\|	16-cell honeycomb\|
24-cell honeycomb honeycomb		24-cell honeycomb\|	24-cell\|	octahedron\|			tetrahedron\|	5-cell\|	5-cube\|
16-cell honeycomb honeycomb		16-cell honeycomb\|	16-cell\|	tetrahedron\|			tetrahedron\|	tesseract\|	24-cell honeycomb\|	self-dual
Order-4 24-cell honeycomb honeycomb		16-cell honeycomb\|	24-cell\|	octahedron\|			octahedron\|	16-cell\|	Tesseractic honeycomb\|
Tesseractic honeycomb honeycomb		Tesseractic honeycomb\|	Tesseract\|	Cube\|			cube\|	24-cell\|	16-cell honeycomb\|

Since there are no regular star n-polytopes for n ≥ 5, that could be potential cells or vertex figures, there are no more hyperbolic star honeycombs in Hⁿ for n ≥ 5.

6 dimensions and higher (7-apeirotopes+)

Tessellations of hyperbolic 6-space and higher

There are no regular compact or paracompact tessellations of hyperbolic space of dimension 6 or higher. However, any Schläfli symbol of the form not covered above will form a noncompact tessellation of hyperbolic n-space.

Compound polytopes

Two dimensional compounds

For any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols for all m such that m < n/2 and m and n are coprime. When m and n are not coprime, the star polygon obtained will be a regular polygon with n/m sides. A new figure is obtained by rotating these regular n/m-gons one vertex to the left on the original polygon until the number of vertices rotated equals n/m minus one, and combining these figures. An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon.
In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called star figures, improper star polygons or compound polygons. The same notation is often used for them, although authorities such as Grünbaum regard the form k as being more correct, where usually k = m.
A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon. This is correctly written in the form k, as 2, rather than the commonly used.
Coxeter's extended notation for compounds is of the form c'e, indicating that d distinct 's together cover the vertices of c times and the facets of e times. If no regular exists, the first part of the notation is removed, leaving e; the opposite holds if no regular exists. The dual of c'e is ec. If c or e are 1, they may be omitted. For compound polygons, this notation reduces to : for example, the hexagram may be written thus as .

2	3	4	5	6	7	8	9	10	11	12	13	14	15	-
2	3	4	5	6	7	8	9	10	2	3	4	5	6	7
2	3	4	5	6	2	3	4	5	6	2	3	4	5	-
2	3	4	2	3	4	2	3	4	2	3	2	3	-	-
2	3	2	3	2	3	2	3	2	3	-	-	-	-	-
2	2	2	2	2	2	2	2	2	2	2	2	2	-	-
2	2	2	2	2	2	2	-	-	-	-	-	-	-	-

Regular skew polygons also create compounds, seen in the edges of prismatic compound of antiprisms, for instance:

Three dimensional compounds

A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. With this definition there are 5 regular compounds.
Coxeter's notation for regular compounds is given in the table above, incorporating Schläfli symbols. The material inside the square brackets, , denotes the components of the compound: d separate 's. The material before the square brackets denotes the vertex arrangement of the compound: c is a compound of d 's sharing the vertices of an counted c times. The material after the square brackets denotes the facet arrangement of the compound: e is a compound of d 's sharing the faces of counted e times. These may be combined: thus ce is a compound of d 's sharing the vertices of counted c times and the faces of counted e times. This notation can be generalised to compounds in any number of dimensions.

Euclidean and hyperbolic plane compounds

There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not yet been proven.
The Euclidean and hyperbolic compound families 2 are analogous to the spherical stella octangula, 2.

Four dimensional compounds

lists 32 regular compounds of regular 4-polytopes in his book Regular Polytopes:. McMullen adds six in his paper New Regular Compounds of 4-Polytopes. In the following tables, the superscript indicates that the labeled compounds are distinct from the other compounds with the same symbols.

Compound	Constituent	Symmetry	Vertex arrangement	Cell arrangement
120	5-cell	, order 14400
120	5-cell	order 1200
720	5-cell	, order 14400	6	6
5	24-cell	, order 14400

Compound 1	Compound 2	Symmetry	Vertex arrangement	Cell arrangement	Vertex arrangement	Cell arrangement
3 16-cell\|	3 tesseract\|	, order 1152		2	2
15 16-cell\|	15 tesseract\|	, order 14400		2	2
75 16-cell\|	75 tesseract\|	, order 14400	5	10	10	5
75 16-cell\|	75 tesseract\|	, order 14400		2	2
75 16-cell\|	75 tesseract\|	order 600		2	2
300 16-cell\|	300 tesseract\|	⁺, order 7200	4	8	8	4
600 16-cell\|	600 tesseract\|	, order 14400	8	16	16	8
25 24-cell\|	25 24-cell\|	, order 14400		5	5

There are two different compounds of 75 tesseracts: one shares the vertices of a 120-cell, while the other shares the vertices of a 600-cell. It immediately follows therefore that the corresponding dual compounds of 75 16-cells are also different.

Compound	Symmetry	Vertex arrangement	Cell arrangement
5 Great 120-cell\|	⁺, order 7200
10 Great 120-cell\|	, order 14400	2	2
5 Grand stellated 120-cell\|	⁺, order 7200
10 Grand stellated 120-cell\|	, order 14400	2	2

Compound 1	Compound 2	Symmetry	Vertex arrangement	Cell arrangement	Vertex arrangement	Cell arrangement
5 Icosahedral 120-cell\|	5 Small stellated 120-cell\|	⁺, order 7200
10 Icosahedral 120-cell\|	10 Small stellated 120-cell\|	, order 14400	2	2	2	2
5 Great grand 120-cell\|	5 Great icosahedral 120-cell\|	⁺, order 7200
10 Great grand 120-cell\|	10 Great icosahedral 120-cell\|	, order 14400	2	2	2	2
5 Great stellated 120-cell\|	5 Grand 120-cell\|	⁺, order 7200
10 Great stellated 120-cell\|	10 Grand 120-cell\|	, order 14400	2	2	2	2

There are also fourteen partially regular compounds, that are either vertex-transitive or cell-transitive but not both. The seven vertex-transitive partially regular compounds are the duals of the seven cell-transitive partially regular compounds.

Compound 1 Vertex-transitive	Compound 2 Cell-transitive	Symmetry
2 16-cells	2 tesseracts	, order 384
25 24-cell	25 24-cell	order 600
100 24-cell	100 24-cell	⁺, order 7200
200 24-cell	200 24-cell	, order 14400
5 600-cell	5 120-cell	⁺, order 7200
10 600-cell	10 120-cell	, order 14400

Although the 5-cell and 24-cell are both self-dual, their dual compounds are not considered to be regular, unlike the compound of two tetrahedra and the various dual polygon compounds, because they are neither vertex-regular nor cell-regular: they are not facetings or stellations of any regular 4-polytope.

Euclidean 3-space compounds

The only regular Euclidean compound honeycombs are an infinite family of compounds of cubic honeycombs, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is .

Five dimensions and higher compounds

There are no regular compounds in five or six dimensions. There are three known seven-dimensional compounds, and six known eight-dimensional ones. There is also one compound of n-simplices in n-dimensional space provided that n is one less than a power of two, and also two compounds in n-dimensional space if n is a power of two.
The Coxeter notation for these compounds are :

Simplexes: γ_n−1β_n−1
Orthoplexes: γ_n
Hypercubes: β_n
Euclidean honeycomb compounds

A known family of regular Euclidean compound honeycombs in five or more dimensions is an infinite family of compounds of hypercubic honeycombs, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs. The Coxeter notation is δ_nδ_n where δ_n = when n = 2 and when n ≥ 3.

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 for a sample. Some notable examples of abstract regular polytopes that do not appear elsewhere in this list are the 11-cell,, and the 57-cell,, which have regular projective polyhedra as cells and vertex figures.
The elements of an abstract polyhedron are its body, its faces, edges, vertices and the null polytope or empty set. These abstract elements can be mapped into ordinary space or realised as geometrical figures. Some abstract polyhedra have well-formed or faithful realisations, others do not. A flag is a connected set of elements of each dimension - for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be regular if its combinatorial symmetries are transitive on its flags - that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.
Five such regular abstract polyhedra, which can not be realised faithfully, were identified by H. S. M. Coxeter in his book Regular Polytopes and again by J. M. Wills in his paper "The combinatorially regular polyhedra of index 2". They are all topologically equivalent to toroids. Their construction, by arranging n faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.
These occur as dual pairs as follows:

The medial rhombic triacontahedron and dodecadodecahedron are dual to each other.
The medial triambic icosahedron and ditrigonal dodecadodecahedron are dual to each other.
The excavated dodecahedron is self-dual.

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List of regular polytopes and compounds

Overview

One dimension

Two dimensions (polygons)

Convex

Spherical

Stars

Skew polygons

Three dimensions (polyhedra)

Convex

Spherical

Stars

Skew polyhedra

Four dimensions

Convex

Spherical

Stars

Five and more dimensions

Convex

5 dimensions

6 dimensions

7 dimensions

8 dimensions

9 dimensions

10 dimensions

Non-convex

Regular projective polytopes

Regular projective polyhedra

Regular projective 4-polytopes

Regular projective 5-polytopes

Apeirotopes

One dimension (apeirogons)

Skew apeirogons

Two dimensions (apeirohedra)

Euclidean tilings

Euclidean star-tilings

Hyperbolic tilings

Hyperbolic star-tilings

Skew apeirohedra in Euclidean 3-space

Skew apeirohedra in hyperbolic 3-space

Three dimensions (4-apeirotopes)

Tessellations of Euclidean 3-space

Improper tessellations of Euclidean 3-space

Tessellations of hyperbolic 3-space

Four dimensions (5-apeirotopes)

Tessellations of Euclidean 4-space

Tessellations of hyperbolic 4-space

Star tessellations of hyperbolic 4-space

Five dimensions (6-apeirotopes)

Tessellations of Euclidean 5-space

Tessellations of hyperbolic 5-space

6 dimensions and higher (7-apeirotopes+)

Tessellations of hyperbolic 6-space and higher

Compound polytopes

Two dimensional compounds

Three dimensional compounds

Euclidean and hyperbolic plane compounds

Four dimensional compounds

Euclidean 3-space compounds

Five dimensions and higher compounds

Euclidean honeycomb compounds

Abstract polytopes