List of regular polytopes

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

Overview

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

1-polytopes

There is only one polytope of rank 1, the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the 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.

2-polytopes (polygons)

The polytopes of rank 2 are called polygons. Regular polygons are equilateral and cyclic. A -gonal regular polygon is represented by Schläfli symbol.
Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed.

Convex

The Schläfli symbol represents a regular -gon.

Name	Triangle	Square	Pentagon	Hexagon	Heptagon	Octagon
Schläfli
Symmetry	D₃,	D₄,	D₅,	D₆,	D₇,	D₈,
Coxeter
Image	60px	60px	60px	60px	60px	60px
Name	Nonagon	Decagon	Hendecagon	Dodecagon	Tridecagon	Tetradecagon
Schläfli
Symmetry	D₉,	D₁₀,	D₁₁,	D₁₂,	D₁₃,	D₁₄,
Dynkin
Image	60px	60px	60px	60px	60px	60px
Name	Pentadecagon	Hexadecagon	Heptadecagon	Octadecagon	Enneadecagon	Icosagon	p-gon
Schläfli
Symmetry	D₁₅,	D₁₆,	D₁₇,	D₁₈,	D₁₉,	D₂₀,	D_p,
Dynkin
Image	60px	60px	60px	60px	60px	60px	60px

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. For example, digon can be realised non-degenerately as a spherical lune. A monogon could also be realised on the sphere as a single point with a great circle through it. However, a monogon is not a valid abstract polytope because its single edge is incident to only one vertex rather than two.

Name	Monogon	Digon
Schläfli symbol
Symmetry	D₁,	D₂,
Coxeter diagram	or
Image		60px

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, there are regular -pointed stars with Schläfli symbols for all such that and and are coprime. Symbols where and are not coprime may be used to represent compound polygons.

		-
-	-	-
	-	-

Star polygons that can only exist as spherical tilings, similarly to the monogon and digon, may exist, however these have not 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 addition to the planar regular polygons there are infinitely many regular skew polygons. Skew polygons can be created via the blending operation.
The blend of two polygons and, written, can be constructed as follows:

take the cartesian product of their vertices.
add edges where is an edge of and is an edge of.
select an arbitrary connected component of the result.

Alternatively, the blend is the polygon where and are the generating mirrors of and placed in orthogonal subspaces.
The blending operation is commutative, associative and idempotent.
Every regular skew polygon can be expressed as the blend of a unique set of planar polygons. If and share no factors then.

In 3 space

The regular finite polygons in 3 dimensions are exactly the blends of the planar polygons with the digon. They have vertices corresponding to a prism or an antiprism. All polygons in 3 space have an even number of vertices and edges.
Several of these appear as the Petrie polygons of regular polyhedra.

In 4 space

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

3-polytopes (polyhedra)

Polytopes of rank 3 are called polyhedra:
A regular polyhedron with Schläfli symbol, 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 five convex regular polyhedra are called the 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

Stellation
diagram

Schläfli
and
Coxeter

Faces

Edges

Vertices

verf.

Density

Symmetry

Dual

Small stellated dodecahedron

80px

Image:Small stellated [dodecahedron (gray with yellow face).svg|80px]

80px

12

30px

−6

I_h

Great dodecahedron

80px

12

30px

−6

I_h

Small stellated dodecahedron

Great stellated dodecahedron

80px

12

30px

20

30px

I_h

Great icosahedron

80px

20

30px

12

30px

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

Regular 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 -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:

4-polytopes (polychora)

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.

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

Di-4-topes 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	75px	75px		120 [Icosahedron\|] 25px	1200 [Triangle\|] 25px	720 [Pentagram\|] 25px	120 [Great dodecahedron\|] 25px	4	480	H₄	Small stellated 120-cell
Small stellated 120-cell	75px	75px		120 [Small stellated dodecahedron\|] 25px	720 [Pentagram\|] 25px	1200 [Triangle\|] 25px	120 [Dodecahedron\|] 25px	4	−480	H₄	Icosahedral 120-cell
Great 120-cell	75px	75px		120 [Great dodecahedron\|] 25px	720 [Pentagon\|] 25px	720 [Pentagon\|] 25px	120 [Small stellated dodecahedron\|] 25px	6	0	H₄	Self-dual
Grand 120-cell	75px	75px		120 [Dodecahedron\|] 25px	720 [Pentagon\|] 25px	720 [Pentagram\|] 25px	120 [Great icosahedron\|] 25px	20	0	H₄	Great stellated 120-cell
Great stellated 120-cell	75px	75px		120 [Great stellated dodecahedron\|] 25px	720 [Pentagram\|] 25px	720 [Pentagon\|] 25px	120 [Icosahedron\|] 25px	20	0	H₄	Grand 120-cell
Grand stellated 120-cell	75px	75px		120 [Small stellated dodecahedron\|] 25px	720 [Pentagram\|] 25px	720 [Pentagram\|] 25px	120 [Great dodecahedron\|] 25px	66	0	H₄	Self-dual
Great grand 120-cell	75px	75px		120 [Great dodecahedron\|] 25px	720 [Pentagon\|] 25px	1200 [Triangle\|] 25px	120 [Great stellated dodecahedron\|] 25px	76	−480	H₄	Great icosahedral 120-cell
Great icosahedral 120-cell	75px	75px		120 [Great icosahedron\|] 25px	1200 [Triangle\|] 25px	720 [Pentagon\|] 25px	120 [Small stellated dodecahedron\|] 25px	76	480	H₄	Great grand 120-cell
Grand 600-cell	75px	75px		600 [Tetrahedron\|] 25px	1200 [Triangle\|] 25px	720 [Pentagram\|] 25px	120 [Great icosahedron\|] 25px	191	0	H₄	Great grand stellated 120-cell
Great grand stellated 120-cell	75px	75px		120 [Great stellated dodecahedron\|] 25px	720 [Pentagram\|] 25px	1200 [Triangle\|] 25px	600 [Tetrahedron\|] 25px	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.

Skew 4-polytopes

In addition to the 16 planar 4-polytopes above there are 18 finite skew polytopes. One of these is obtained as the Petrial of the tesseract, and the other 17 can be formed by applying the kappa operation to the planar polytopes and the Petrial of the tesseract.

Ranks 5 and higher

5-polytopes can be given the symbol 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, no star polytopes, 3 tessellations of Euclidean 4-space, and 5 tessellations of paracompact hyperbolic 4-space. The only non-convex regular polytopes for ranks 5 and higher are skews.

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.

Star polytopes

There are no regular star polytopes of rank 5 or higher, with the exception of degenerate polytopes created by the star product of lower rank star polytopes. hosotopes and ditopes.

Regular projective polytopes

A projective regular -polytope exists when an original regular -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--cubes and hemi--orthoplexes to any rank.

Regular projective 4-polytopes

5 of 6 convex regular 4-polytopes are centrally symmetric generating 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	Skeleton graph
Hemitesseract	/2	₄	4	12	16	8	0	K_4,4
Hemi-16-cell	/2	₄	8	16	12	4	0	double-edged K₄
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

Only 2 of 3 regular spherical polytopes are centrally symmetric for ranks 5 or higher. The corresponding regular projective polytopes are the hemi versions of the regular hypercube and orthoplex. They are tabulated below for rank 5, for example:

Name	Schläfli	4-faces	Cells	Faces	Edges	Vertices	χ	Skeleton graph
hemi-penteract	/2	5	20	40	40	16	1	Tesseract skeleton + 8 central diagonals
hemi-pentacross	/2	16	40	40	20	5	1	double-edged K₅

Apeirotopes

An apeirotope or infinite polytope is a polytope which has infinitely many facets. An -apeirotope is an infinite -polytope: a 2-apeirotope or apeirogon 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 dimensions, which completely fill an -dimensional space.
Regular skew apeirotopes, comprising an -dimensional manifold in a higher space.

2-apeirotopes (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.
......
It exists as the limit of the -gon as tends to infinity, as follows:

Name	Monogon	Digon	Triangle	Square	Pentagon	Hexagon	Heptagon	p-gon	Apeirogon
Schläfli
Symmetry	D₁,	D₂,	D₃,	D₄,	D₅,	D₆,	D₇,		-
Coxeter	or
Image

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

3-apeirotopes (apeirohedra)

Euclidean tilings

There are six regular tessellations of the plane: the three listed below, and their corresponding Petrials.
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 disk 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:
The tilings have ideal vertices, on the edge of the Poincaré disk model. Their duals have ideal apeirogonal faces, meaning that they are inscribed in horocycles. One could go further and find tilings with ultra-ideal vertices, outside the Poincaré disk, which are dual to tiles inscribed in hypercycles; in what is symbolised above, infinitely many tiles still fit around each ultra-ideal vertex.

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			60px	30px	30px	3	*732	Heptagrammic-order heptagonal tiling
Heptagrammic-order heptagonal tiling			60px	30px	30px	3	*732	Order-7 heptagrammic tiling
Order-9 enneagrammic tiling			60px	30px	30px	3	*932	Enneagrammic-order enneagonal tiling
Enneagrammic-order enneagonal tiling			60px	30px	30px	3	*932	Order-9 enneagrammic tiling
Order-11 hendecagrammic tiling			60px	30px	30px	3	*11.3.2	Hendecagrammic-order hendecagonal tiling
Hendecagrammic-order hendecagonal tiling			60px	30px	30px	3	*11.3.2	Order-11 hendecagrammic tiling
Order-p ''p-grammic tiling						3	*p''32	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 planar faces. 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:

Allowing for skew faces, there are 30 regular apeirohedra in Euclidean 3-space. These include the 12 blended apeirohedra created by blends with the Euclidean planar apeirohedra, and 18 pure apeirohedra, which cannot be expressed as a non-trivial blend including the planar apeirohedra and the three 3-dimensional apeirohedra above.
The 3-dimensional pure apeirohedra are:

, the mucube
, the Petrial of the mucube
, the mutetrahedron
, the Petrial of the mutetrahedron
, the muoctahedron
, the Petrial of the muoctahedron
, the halving of the mucube
, the Petrial of
, the skewing of the muoctahedron
, the skewing of

Skew apeirohedra in hyperbolic 3-space

There are 31 regular skew apeirohedra with convex faces in hyperbolic 3-space with compact or paracompact symmetry:

14 are compact:,,,,,,,,,,,,, and.
17 are paracompact:,,,,,,,,,,,,,,,, and.

4-apeirotopes

Tessellations of Euclidean 3-space

[Image:Cubic honeycomb.png|upright=0.8|thumb|Edge framework of cubic honeycomb, ]
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 15 flat regular honeycombs of hyperbolic 3-space:

4 are compact:,,, and
while 11 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\|]			[triangular tiling\|]	0
Order-4 hexagonal tiling honeycomb			[hexagonal tiling\|]			[Octahedron\|]	0
Order-4 square tiling honeycomb			[square tiling\|]			[square tiling\|]	0	Self-dual
Order-6 dodecahedral honeycomb			[dodecahedron\|]			[triangular tiling\|]	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 compact or paracompact hyperbolic star-honeycombs in H³: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical.
Ideal vertices now appear when the vertex figure is a Euclidean tiling, becoming inscribable in a horosphere rather than a sphere. They are dual to ideal cells. As the last number in the Schläfli symbol rises further, the vertex figure becomes hyperbolic, and vertices become ultra-ideal. In honeycombs the edges intersect the Poincaré ball only in one ideal point; the rest of the edge has become ultra-ideal. Continuing further would lead to edges that are completely ultra-ideal, both for the honeycomb and for the fundamental simplex. In general, when the last number of the Schläfli symbol becomes ∞, faces of codimension two intersect the Poincaré hyperball only in one ideal point.

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

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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, all compact:

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

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.

Apeirotopes of rank 7 or more

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.

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, and of other manifolds. There are infinitely many of every rank greater than 1. 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 rank - 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 cannot be realised faithfully and symmetrically, 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.

List of regular polytopes

Overview

1-polytopes

2-polytopes (polygons)

Convex

Spherical

Stars

Skew polygons

In 3 space

In 4 space

3-polytopes (polyhedra)

Convex

Spherical

Stars

Skew polyhedra

4-polytopes (polychora)

Convex

Spherical

Stars

Skew 4-polytopes

Ranks 5 and higher

Convex

5 dimensions

6 dimensions

7 dimensions

8 dimensions

9 dimensions

10 dimensions

Star polytopes

Regular projective polytopes

Regular projective 4-polytopes

Regular projective 5-polytopes

Apeirotopes

2-apeirotopes (apeirogons)

Skew apeirogons

3-apeirotopes (apeirohedra)

Euclidean tilings

Euclidean star-tilings

Hyperbolic tilings

Hyperbolic star-tilings

Skew apeirohedra in Euclidean 3-space

Skew apeirohedra in hyperbolic 3-space

4-apeirotopes

Tessellations of Euclidean 3-space

Improper tessellations of Euclidean 3-space

Tessellations of hyperbolic 3-space

5-apeirotopes

Tessellations of Euclidean 4-space

Tessellations of hyperbolic 4-space

Star tessellations of hyperbolic 4-space

6-apeirotopes

Tessellations of Euclidean 5-space

Tessellations of hyperbolic 5-space

Apeirotopes of rank 7 or more

Tessellations of hyperbolic 6-space and higher

Abstract polytopes