List of regular polytopes

Example regular polytopes

Regular (2D) polygons
Convex	Star
File:Regular pentagon.svg {5}	File:Star polygon 5-2.svg {5/2}
Regular (3D) polyhedra
Convex	Star
File:Dodecahedron.png {5,3}	File:Small stellated dodecahedron.png {5/2,5}
Regular 4D polytopes
Convex	Star
File:Schlegel wireframe 120-cell.png {5,3,3}	File:Ortho solid 010-uniform polychoron p53-t0.png {5/2,5,3}
Regular 2D tessellations
Euclidean	Hyperbolic
File:Uniform tiling 44-t0.svg {4,4}	File:H2-5-4-dual.svg {5,4}
Regular 3D tessellations
Euclidean	Hyperbolic
File:Cubic honeycomb.png {4,3,4}	File:Hyperbolic orthogonal dodecahedral honeycomb.png {5,3,4}

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

Overview

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

rowspan=3 Template:Verth	Finite			Euclidean		Hyperbolic			rowspan=3 Template:Verth
Compact						Paracompact
Convex	Star	SkewTemplate:Efn<ref name="RegPolyFull">Template:Citation</ref>	Convex	SkewTemplate:Efn<ref name="RegPolyFull"/>	Convex	Star	Convex
1	1	style="color:#999999; font-size:80%"Template:CNone	style="color:#999999; font-size:80%"Template:CNone	style="color:#999999; font-size:80%"Template:CNone	style="color:#999999; font-size:80%"Template:CNone	style="color:#999999; font-size:80%"Template:CNone	style="color:#999999; font-size:80%"Template:CNone	style="color:#999999; font-size:80%"Template:CNone	1
2	∞	∞	style="color:#999999; font-size:80%"Template:CNone	1	style="color:#999999; font-size:80%"Template:CNone	1	style="color:#999999; font-size:80%"Template:CNone	style="color:#999999; font-size:80%"Template:CNone	∞
3	5	4	9	3	3	∞	∞	∞	∞
4	6	10	18	1	7	4	style="color:#999999; font-size:80%"Template:CNone	11	∞
5	3	style="color:#999999; font-size:80%"Template:CNone	3	3	15	5	4	2	∞
6	3	style="color:#999999; font-size:80%"Template:CNone	3	1	7	style="color:#999999; font-size:80%"Template:CNone	style="color:#999999; font-size:80%"Template:CNone	5	∞
7+	3	style="color:#999999; font-size:80%"Template:CNone	3	1	7	style="color:#999999; font-size:80%"Template:CNone	style="color:#999999; font-size:80%"Template:CNone	style="color:#999999; font-size:80%"Template:CNone	∞

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

1-polytopes

File:Coxeter node markup1.png

A Coxeter diagram represent mirror "planes" as nodes, and puts a ring around a node if a point is not on the plane. A dion { }, Template:CDD, is a point Template:Mvar and its mirror image point Template:Mvar, and the line segment between them.

There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol { },Template:SfnpTemplate:Sfnp or a Coxeter diagram with a single ringed node, Template:CDD. Norman Johnson calls it a dion<ref>Template:Cite book</ref> and gives it the Schläfli symbol { }.

Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes.Template:Sfnp It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram Template:CDD as a Cartesian product of a line segment and a regular polygon.Template:Sfnp

2-polytopes (polygons)

The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A Template:Mvar-gonal regular polygon is represented by Schläfli symbol {p}.

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 {p} represents a [[Regular polygon|regular Template:Mvar-gon]].

Name	Triangle (2-simplex)	Square (2-orthoplex) (2-cube)	Pentagon (2-pentagonal polytope)	Hexagon	Heptagon	Octagon
Schläfli	{3}	{4}	{5}	{6}	{7}	{8}
Symmetry	D3, [3]	D4, [4]	D5, [5]	D6, [6]	D7, [7]	D8, [8]
Coxeter	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Image	File:Regular triangle.svg	File:Regular quadrilateral.svg	File:Regular pentagon.svg	File:Regular hexagon.svg	File:Regular heptagon.svg	File:Regular octagon.svg
Name	Nonagon (Enneagon)	Decagon	Hendecagon	Dodecagon	Tridecagon	Tetradecagon
Schläfli	{9}	{10}	{11}	{12}	{13}	{14}
Symmetry	D9, [9]	D10, [10]	D11, [11]	D12, [12]	D13, [13]	D14, [14]
Dynkin	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Image	File:Regular nonagon.svg	File:Regular decagon.svg	File:Regular hendecagon.svg	File:Regular dodecagon.svg	File:Regular tridecagon.svg	File:Regular tetradecagon.svg
Name	Pentadecagon	Hexadecagon	Heptadecagon	Octadecagon	Enneadecagon	Icosagon	p-gon
Schläfli	{15}	{16}	{17}	{18}	{19}	{20}	{p}
Symmetry	D15, [15]	D16, [16]	D17, [17]	D18, [18]	D19, [19]	D20, [20]	Dp, [p]
Dynkin	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Image	File:Regular pentadecagon.svg	File:Regular hexadecagon.svg	File:Regular heptadecagon.svg	File:Regular octadecagon.svg	File:Regular enneadecagon.svg	File:Regular icosagon.svg	File:Disk 1.svg

Spherical

The regular digon {2} 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 {1} could also be realised on the sphere as a single point with a great circle through it.<ref>Coxeter, Regular Complex Polytopes, p. 9</ref> 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	{1}	{2}
Symmetry	D1, [ ]	D2, [2]
Coxeter diagram	Template:CDD or Template:CDD	Template:CDD
Image	File:Monogon.svg	File:Digon.svg

Stars

There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers Template:Math. They are called star polygons and share the same vertex arrangements of the convex regular polygons.

In general, for any natural number Template:Mvar, there are regular Template:Mvar-pointed stars with Schläfli symbols Template:Math for all Template:Mvar such that Template:Math (strictly speaking Template:Math) and Template:Mvar and Template:Mvar are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Symbols where Template:Mvar and Template:Mvar are not coprime may be used to represent compound polygons.

Name	Pentagram	Heptagrams		Octagram	Enneagrams		Decagram	...n-grams
Schläfli	{5/2}	{7/2}	{7/3}	{8/3}	{9/2}	{9/4}	{10/3}	{p/q}
Symmetry	D5, [5]	D7, [7]		D8, [8]	D9, [9],		D10, [10]	Dp, [p]
Coxeter	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Image	File:Star polygon 5-2.svg	File:Star polygon 7-2.svg	File:Star polygon 7-3.svg	File:Star polygon 8-3.svg	File:Star polygon 9-2.svg	File:Star polygon 9-4.svg	File:Star polygon 10-3.svg

Regular star polygons up to 20 sides

File:Regular star polygon 11-2.svg {11/2}	File:Regular star polygon 11-3.svg {11/3}	File:Regular star polygon 11-4.svg {11/4}	File:Regular star polygon 11-5.svg {11/5}	File:Regular star polygon 12-5.svg {12/5}	File:Regular star polygon 13-2.svg {13/2}	File:Regular star polygon 13-3.svg {13/3}	File:Regular star polygon 13-4.svg {13/4}	File:Regular star polygon 13-5.svg {13/5}	File:Regular star polygon 13-6.svg {13/6}
File:Regular star polygon 14-3.svg {14/3}	File:Regular star polygon 14-5.svg {14/5}	File:Regular star polygon 15-2.svg {15/2}	File:Regular star polygon 15-4.svg {15/4}	File:Regular star polygon 15-7.svg {15/7}	File:Regular star polygon 16-3.svg {16/3}	File:Regular star polygon 16-5.svg {16/5}	File:Regular star polygon 16-7.svg {16/7}
File:Regular star polygon 17-2.svg {17/2}	File:Regular star polygon 17-3.svg {17/3}	File:Regular star polygon 17-4.svg {17/4}	File:Regular star polygon 17-5.svg {17/5}	File:Regular star polygon 17-6.svg {17/6}	File:Regular star polygon 17-7.svg {17/7}	File:Regular star polygon 17-8.svg {17/8}	File:Regular star polygon 18-5.svg {18/5}	File:Regular star polygon 18-7.svg {18/7}
File:Regular star polygon 19-2.svg {19/2}	File:Regular star polygon 19-3.svg {19/3}	File:Regular star polygon 19-4.svg {19/4}	File:Regular star polygon 19-5.svg {19/5}	File:Regular star polygon 19-6.svg {19/6}	File:Regular star polygon 19-7.svg {19/7}	File:Regular star polygon 19-8.svg {19/8}	File:Regular star polygon 19-9.svg {19/9}	File:Regular star polygon 20-3.svg {20/3}	File:Regular star polygon 20-7.svg {20/7}	File:Regular star polygon 20-9.svg {20/9}

Star polygons that can only exist as spherical tilings, similarly to the monogon and digon, may exist (for example: {3/2}, {5/3}, {5/4}, {7/4}, {9/5}), 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.<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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 Template:Mvar and Template:Mvar, written Template:Math, can be constructed as follows:

1. take the cartesian product of their vertices Template:Math.
2. add edges Template:Math where Template:Math is an edge of Template:Mvar and Template:Math is an edge of Template:Mvar.
3. select an arbitrary connected component of the result.

Alternatively, the blend is the polygon Template:Math where Template:Mvar and Template:Mvar are the generating mirrors of Template:Mvar and Template:Mvar placed in orthogonal subspaces.Template:Sfn The blending operation is commutative, associative and idempotent.

Every regular skew polygon can be expressed as the blend of a uniqueTemplate:Efn set of planar polygons.Template:Sfn If Template:Mvar and Template:Mvar share no factors then Template:Math.

In 3 space

The regular finite polygons in 3 dimensions are exactly the blends of the planar polygons (dimension 2) with the digon (dimension 1). They have vertices corresponding to a prism (Template:Math where Template:Mvar is odd) or an antiprism (Template:Math where Template:Mvar is even). 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 Template:Math, Coxeter diagrams Template:CDD, has a regular face type Template:Math, and regular vertex figure Template:Math.

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 Template:Math is constrained by an inequality, related to the vertex figure's angle defect: <math display=block> \begin{align} & \frac{1}{p} + \frac{1}{q} > \frac{1}{2} : \text{Polyhedron (existing in Euclidean 3-space)} \\[6pt] & \frac{1}{p} + \frac{1}{q} = \frac{1}{2} : \text{Euclidean plane tiling} \\[6pt] & \frac{1}{p} + \frac{1}{q} < \frac{1}{2} : \text{Hyperbolic plane tiling} \end{align} </math>

By enumerating the permutations, we find five convex forms, four star forms and three plane tilings, all with polygons Template:Math and Template:Math limited to: {3}, {4}, {5}, {5/2}, and {6}.

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 (χ) of 2.

Name	Schläfli Template:Math	Coxeter Template:CDD	Image (solid)	Image (sphere)	Faces Template:Math	Edges	Vertices Template:Math	Symmetry	Dual
Tetrahedron (3-simplex)	{3,3}	Template:CDD	File:Polyhedron 4b.png	File:Uniform tiling 332-t2.svg	4 {3}	6	4 {3}	Td [3,3] (*332)	(self)
Hexahedron Cube (3-cube)	{4,3}	Template:CDD	File:Polyhedron 6.png	File:Uniform tiling 432-t0.png	6 {4}	12	8 {3}	Oh [4,3] (*432)	Octahedron
Octahedron (3-orthoplex)	{3,4}	Template:CDD	File:Polyhedron 8.png	File:Uniform tiling 432-t2.png	8 {3}	12	6 {4}	Oh [4,3] (*432)	Cube
Dodecahedron	{5,3}	Template:CDD	File:Polyhedron 12.png	File:Uniform tiling 532-t0.png	12 {5}	30	20 {3}	Ih [5,3] (*532)	Icosahedron
Icosahedron	{3,5}	Template:CDD	File:Polyhedron 20.png	File:Uniform tiling 532-t2.png	20 {3}	30	12 {5}	Ih [5,3] (*532)	Dodecahedron

Spherical

In spherical geometry, regular spherical polyhedra (tilings of the sphere) exist that would otherwise be degenerate as polytopes. These are the hosohedra {2,n} and their dual dihedra {n,2}. Coxeter calls these cases "improper" tessellations.Template:Sfnp

The first few cases (n from 2 to 6) are listed below.

Hosohedra

Name	Schläfli {2,p}	Coxeter diagram	Image (sphere)	Faces {2}π/p	Edges	Vertices {p}	Symmetry	Dual
Digonal hosohedron	{2,2}	Template:CDD	File:Spherical digonal hosohedron.svg	2 {2}π/2	2	2 {2}π/2	D2h [2,2] (*222)	Self
Trigonal hosohedron	{2,3}	Template:CDD	File:Spherical trigonal hosohedron.svg	3 {2}π/3	3	2 {3}	D3h [2,3] (*322)	Trigonal dihedron
Square hosohedron	{2,4}	Template:CDD	File:Spherical square hosohedron.svg	4 {2}π/4	4	2 {4}	D4h [2,4] (*422)	Square dihedron
Pentagonal hosohedron	{2,5}	Template:CDD	File:Spherical pentagonal hosohedron.svg	5 {2}π/5	5	2 {5}	D5h [2,5] (*522)	Pentagonal dihedron
Hexagonal hosohedron	{2,6}	Template:CDD	File:Spherical hexagonal hosohedron.svg	6 {2}π/6	6	2 {6}	D6h [2,6] (*622)	Hexagonal dihedron

Dihedra

Name	Schläfli {p,2}	Coxeter diagram	Image (sphere)	Faces {p}	Edges	Vertices {2}	Symmetry	Dual
Digonal dihedron	{2,2}	Template:CDD	File:Digonal dihedron.png	2 {2}π/2	2	2 {2}π/2	D2h [2,2] (*222)	Self
Trigonal dihedron	{3,2}	Template:CDD	File:Trigonal dihedron.png	2 {3}	3	3 {2}π/3	D3h [3,2] (*322)	Trigonal hosohedron
Square dihedron	{4,2}	Template:CDD	File:Tetragonal dihedron.png	2 {4}	4	4 {2}π/4	D4h [4,2] (*422)	Square hosohedron
Pentagonal dihedron	{5,2}	Template:CDD	File:Pentagonal dihedron.png	2 {5}	5	5 {2}π/5	D5h [5,2] (*522)	Pentagonal hosohedron
Hexagonal dihedron	{6,2}	Template:CDD	File:Hexagonal dihedron.png	2 {6}	6	6 {2}π/6	D6h [6,2] (*622)	Hexagonal hosohedron

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

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 {5,3} and icosahedron {3,5}:

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 (skeletonic)	Image (solid)	Image (sphere)	Stellation diagram	Schläfli Template:Math and Coxeter	Faces Template:Math	Edges	Vertices Template:Math verf.	χ	Density	Symmetry	Dual
Small stellated dodecahedron	File:Skeleton St12, size m.png	File:Small stellated dodecahedron (gray with yellow face).svg	File:Small stellated dodecahedron tiling.png	File:First stellation of dodecahedron facets.svg	{5/2,5} Template:CDD	12 {5/2} File:Star polygon 5-2.svg	30	12 {5} File:Regular pentagon.svg	−6	3	Ih [5,3] (*532)	Great dodecahedron
Great dodecahedron	File:Skeleton Gr12, size m.png	File:Great dodecahedron (gray with yellow face).svg	File:Great dodecahedron tiling.svg	File:Second stellation of dodecahedron facets.svg	{5,5/2} Template:CDD	12 {5} File:Regular pentagon.svg	30	12 {5/2} File:Star polygon 5-2.svg	−6	3	Ih [5,3] (*532)	Small stellated dodecahedron
Great stellated dodecahedron	File:Skeleton GrSt12, size s.png	File:Great stellated dodecahedron (gray with yellow face).svg	File:Great stellated dodecahedron tiling.svg	File:Third stellation of dodecahedron facets.svg	{5/2,3} Template:CDD	12 {5/2} File:Star polygon 5-2.svg	30	20 {3} File:Regular triangle.svg	2	7	Ih [5,3] (*532)	Great icosahedron
Great icosahedron	File:Skeleton Gr20, size m.png	File:Great icosahedron (gray with yellow face).svg	File:Great icosahedron tiling.svg	File:Great icosahedron stellation facets.svg	{3,5/2} Template:CDD	20 {3} File:Regular triangle.svg	30	12 {5/2} File:Star polygon 5-2.svg	2	7	Ih [5,3] (*532)	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 {5/2,4}, {5/2,9}, {7/2,3}, {5/2,5/2}, {7/2,7/3}, {4,5/2}, and {3,7/3}.

Skew polyhedra

Template:Expand section 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 {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and Template:Mvar-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.

The regular skew polyhedra, represented by {l,m|n}, follow this equation:

<math display="block">2 \sin\left(\frac{\pi}{l}\right) \sin\left(\frac{\pi}{m}\right) = \cos\left(\frac{\pi}{n}\right)</math>

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:

File:4-simplex t03.svg	File:4-simplex t12.svg	File:24-cell t03 F4.svg	File:24-cell t12 F4.svg
{4, 6 Template:Pipe 3}	{6, 4 Template:Pipe 3}	{4, 8 Template:Pipe 3}	{8, 4 Template:Pipe 3}

4-polytopes

Regular 4-polytopes with Schläfli symbol <math>\{p,q,r\}</math> have cells of type <math>\{p,q\}</math>, faces of type <math>\{p\}</math>, edge figures <math>\{r\}</math>, and vertex figures <math>\{q,r\}</math>.

• A vertex figure (of a 4-polytope) 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 <math>\{p,q,r\}</math> is constrained by the existence of the regular polyhedra <math>\{p,q\}, \{q,r\}</math>. A suggested name for 4-polytopes is "polychoron".<ref>Template:Cite conference</ref>

Each will exist in a space dependent upon this expression:

<math>\sin \left ( \frac{\pi}{p} \right ) \sin \left(\frac{\pi}{r}\right) - \cos\left(\frac{\pi}{q}\right)</math>

<math>> 0</math> : Hyperspherical 3-space honeycomb or 4-polytope

<math>= 0</math> : Euclidean 3-space honeycomb

<math>< 0</math> : 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 <math>\chi</math> for convex 4-polytopes is zero: <math>\chi = V+F-E-C = 0</math>

Convex

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

Name	Schläfli {p,q,r}	Coxeter Template:CDD	Cells {p,q}	Faces {p}	Edges {r}	Vertices {q,r}	Dual {r,q,p}
5-cell (4-simplex)	{3,3,3}	Template:CDD	5 {3,3}	10 {3}	10 {3}	5 {3,3}	(self)
8-cell (4-cube) (Tesseract)	{4,3,3}	Template:CDD	8 {4,3}	24 {4}	32 {3}	16 {3,3}	16-cell
16-cell (4-orthoplex)	{3,3,4}	Template:CDD	16 {3,3}	32 {3}	24 {4}	8 {3,4}	Tesseract
24-cell	{3,4,3}	Template:CDD	24 {3,4}	96 {3}	96 {3}	24 {4,3}	(self)
120-cell	{5,3,3}	Template:CDD	120 {5,3}	720 {5}	1200 {3}	600 {3,3}	600-cell
600-cell	{3,3,5}	Template:CDD	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
File:Complete graph K5.svg	File:4-cube graph.svg	File:4-orthoplex.svg	File:24-cell graph F4.svg	File:Cell120Petrie.svg	File:Cell600Petrie.svg
Solid orthographic projections
File:Tetrahedron.png tetrahedral envelope (cell/ vertex-centered)	File:Hexahedron.png cubic envelope (cell-centered)	File:16-cell ortho cell-centered.png cubic envelope (cell-centered)	File:Ortho solid 24-cell.png cuboctahedral envelope (cell-centered)	File:Ortho solid 120-cell.png truncated rhombic triacontahedron envelope (cell-centered)	File:Ortho solid 600-cell.png Pentakis icosidodecahedral envelope (vertex-centered)
Wireframe Schlegel diagrams (Perspective projection)
File:Schlegel wireframe 5-cell.png (cell-centered)	File:Schlegel wireframe 8-cell.png (cell-centered)	File:Schlegel wireframe 16-cell.png (cell-centered)	File:Schlegel wireframe 24-cell.png (cell-centered)	File:Schlegel wireframe 120-cell.png (cell-centered)	File:Schlegel wireframe 600-cell vertex-centered.png (vertex-centered)
Wireframe stereographic projections (Hyperspherical)
File:Stereographic polytope 5cell.png	File:Stereographic polytope 8cell.png	File:Stereographic polytope 16cell.png	File:Stereographic polytope 24cell.png	File:Stereographic polytope 120cell.png	File:Stereographic polytope 600cell.png

Spherical

Di-4-topes and hoso-4-topes exist as regular tessellations of the 3-sphere.

Regular di-4-topes (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their hoso-4-tope duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,p}. 4-polytopes of the form {2,p,2} are the same as {2,2,p}. There are also the cases {p,2,q} which have dihedral cells and hosohedral vertex figures.

Regular hoso-4-topes as 3-sphere honeycombs

Schläfli {2,p,q}	Coxeter Template:CDD	Cells {2,p}π/q	Faces {2}π/p,π/q	Edges	Vertices	Vertex figure {p,q}	Symmetry	Dual
{2,3,3}	Template:CDD	4 {2,3}π/3 File:Spherical trigonal hosohedron.svg	6 {2}π/3,π/3	4	2	{3,3} File:Uniform tiling 332-t0-1-.svg	[2,3,3]	{3,3,2}
{2,4,3}	Template:CDD	6 {2,4}π/3 File:Spherical square hosohedron.svg	12 {2}π/4,π/3	8	2	{4,3} File:Uniform tiling 432-t0.png	[2,4,3]	{3,4,2}
{2,3,4}	Template:CDD	8 {2,3}π/4 File:Spherical trigonal hosohedron.svg	12 {2}π/3,π/4	6	2	{3,4} File:Uniform tiling 432-t2.png	[2,4,3]	{4,3,2}
{2,5,3}	Template:CDD	12 {2,5}π/3 File:Spherical trigonal hosohedron.svg	30 {2}π/5,π/3	20	2	{5,3} File:Uniform tiling 532-t0.png	[2,5,3]	{3,5,2}
{2,3,5}	Template:CDD	20 {2,3}π/5 File:Spherical pentagonal hosohedron.svg	30 {2}π/3,π/5	12	2	{3,5} File:Uniform tiling 532-t2.png	[2,5,3]	{5,3,2}

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 {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 regular star 4-polytopes, shown as orthogonal projections:

Name	Wireframe	Solid	Schläfli {p, q, r} Coxeter	Cells {p, q}	Faces {p}	Edges {r}	Vertices {q, r}	Density	χ	Symmetry group	Dual {r, q,p}
Icosahedral 120-cell (faceted 600-cell)	File:Schläfli-Hess polychoron-wireframe-3.png	File:Ortho solid 007-uniform polychoron 35p-t0.png	{3,5,5/2} Template:CDD	120 {3,5} File:Icosahedron.png	1200 {3} File:Regular triangle.svg	720 {5/2} File:Star polygon 5-2.svg	120 {5,5/2} File:Great dodecahedron.png	4	480	H4 [5,3,3]	Small stellated 120-cell
Small stellated 120-cell	File:Schläfli-Hess polychoron-wireframe-2.png	File:Ortho solid 010-uniform polychoron p53-t0.png	{5/2,5,3} Template:CDD	120 {5/2,5} File:Small stellated dodecahedron.png	720 {5/2} File:Star polygon 5-2.svg	1200 {3} File:Regular triangle.svg	120 {5,3} File:Dodecahedron.png	4	−480	H4 [5,3,3]	Icosahedral 120-cell
Great 120-cell	File:Schläfli-Hess polychoron-wireframe-3.png	File:Ortho solid 008-uniform polychoron 5p5-t0.png	{5,5/2,5} Template:CDD	120 {5,5/2} File:Great dodecahedron.png	720 {5} File:Regular pentagon.svg	720 {5} File:Regular pentagon.svg	120 {5/2,5} File:Small stellated dodecahedron.png	6	0	H4 [5,3,3]	Self-dual
Grand 120-cell	File:Schläfli-Hess polychoron-wireframe-3.png	File:Ortho solid 009-uniform polychoron 53p-t0.png	{5,3,5/2} Template:CDD	120 {5,3} File:Dodecahedron.png	720 {5} File:Regular pentagon.svg	720 {5/2} File:Star polygon 5-2.svg	120 {3,5/2} File:Great icosahedron.png	20	0	H4 [5,3,3]	Great stellated 120-cell
Great stellated 120-cell	File:Schläfli-Hess polychoron-wireframe-4.png	File:Ortho solid 012-uniform polychoron p35-t0.png	{5/2,3,5} Template:CDD	120 {5/2,3} File:Great stellated dodecahedron.png	720 {5/2} File:Star polygon 5-2.svg	720 {5} File:Regular pentagon.svg	120 {3,5} File:Icosahedron.png	20	0	H4 [5,3,3]	Grand 120-cell
Grand stellated 120-cell	File:Schläfli-Hess polychoron-wireframe-4.png	File:Ortho solid 013-uniform polychoron p5p-t0.png	{5/2,5,5/2} Template:CDD	120 {5/2,5} File:Small stellated dodecahedron.png	720 {5/2} File:Star polygon 5-2.svg	720 {5/2} File:Star polygon 5-2.svg	120 {5,5/2} File:Great dodecahedron.png	66	0	H4 [5,3,3]	Self-dual
Great grand 120-cell	File:Schläfli-Hess polychoron-wireframe-2.png	File:Ortho solid 011-uniform polychoron 53p-t0.png	{5,5/2,3} Template:CDD	120 {5,5/2} File:Great dodecahedron.png	720 {5} File:Regular pentagon.svg	1200 {3} File:Regular triangle.svg	120 {5/2,3} File:Great stellated dodecahedron.png	76	−480	H4 [5,3,3]	Great icosahedral 120-cell
Great icosahedral 120-cell (great faceted 600-cell)	File:Schläfli-Hess polychoron-wireframe-4.png	File:Ortho solid 014-uniform polychoron 3p5-t0.png	{3,5/2,5} Template:CDD	120 {3,5/2} File:Great icosahedron.png	1200 {3} File:Regular triangle.svg	720 {5} File:Regular pentagon.svg	120 {5/2,5} File:Small stellated dodecahedron.png	76	480	H4 [5,3,3]	Great grand 120-cell
Grand 600-cell	File:Schläfli-Hess polychoron-wireframe-4.png	File:Ortho solid 015-uniform polychoron 33p-t0.png	{3,3,5/2} Template:CDD	600 {3,3} File:Tetrahedron.png	1200 {3} File:Regular triangle.svg	720 {5/2} File:Star polygon 5-2.svg	120 {3,5/2} File:Great icosahedron.png	191	0	H4 [5,3,3]	Great grand stellated 120-cell
Great grand stellated 120-cell	File:Schläfli-Hess polychoron-wireframe-1.png	File:Ortho solid 016-uniform polychoron p33-t0.png	{5/2,3,3} Template:CDD	120 {5/2,3} File:Great stellated dodecahedron.png	720 {5/2} File:Star polygon 5-2.svg	1200 {3} File:Regular triangle.svg	600 {3,3} File:Tetrahedron.png	191	0	H4 [5,3,3]	Grand 600-cell

There are 4 failed potential regular star 4-polytopes 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.

Skew 4-polytopes

Template:Expand section In addition to the 16 planar 4-polytopes above there are 18 finite skew polytopes.Template:Sfnp 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 <math>\{p,q,r,s\}</math> where <math>\{p,q,r\}</math> is the 4-face type, <math>\{p,q\}</math> is the cell type, <math>\{p\}</math> is the face type, and <math>\{s\}</math> is the face figure, <math>\{r,s\}</math> is the edge figure, and <math>\{q,r,s\}</math> is the vertex figure.

A vertex figure (of a 5-polytope) is a 4-polytope, 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 <math>\{p,q,r,s\}</math> exists only if <math>\{p,q,r\}</math> and <math>\{q,r,s\}</math> are regular 4-polytopes.

The space it fits in is based on the expression:

<math>\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)}</math>

<math>< 1</math> : Spherical 4-space tessellation or 5-space polytope

<math>= 1</math> : Euclidean 4-space tessellation

<math>> 1</math> : hyperbolic 4-space tessellation

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.Template:Sfnp

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

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

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} Template:CDD	6 {3,3,3}	15 {3,3}	20 {3}	15	6	{3}	{3,3}	{3,3,3}
5-cube	{4,3,3,3} Template:CDD	10 {4,3,3}	40 {4,3}	80 {4}	80	32	{3}	{3,3}	{3,3,3}
5-orthoplex	{3,3,3,4} Template:CDD	32 {3,3,3}	80 {3,3}	80 {3}	40	10	{4}	{3,4}	{3,3,4}

File:5-simplex t0.svg 5-simplex

File:5-cube graph.svg 5-cube

File:5-orthoplex.svg 5-orthoplex

6 dimensions

Name	Schläfli	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

File:6-simplex t0.svg 6-simplex

File:6-cube graph.svg 6-cube

File:6-orthoplex.svg 6-orthoplex

7 dimensions

Name	Schläfli	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

File:7-simplex t0.svg 7-simplex

File:7-cube graph.svg 7-cube

File:7-orthoplex.svg 7-orthoplex

8 dimensions

Name	Schläfli	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

File:8-simplex t0.svg 8-simplex

File:8-cube.svg 8-cube

File:8-orthoplex.svg 8-orthoplex

9 dimensions

Name	Schläfli	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

File:9-simplex t0.svg 9-simplex

File:9-cube.svg 9-cube

File:9-orthoplex.svg 9-orthoplex

10 dimensions

Name	Schläfli	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

File:10-simplex t0.svg 10-simplex

File:10-cube.svg 10-cube

File:10-orthoplex.svg 10-orthoplex

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. Template:Abbr hosotopes and ditopes.

Regular projective polytopes

A projective regular Template:Math-polytope exists when an original regular Template:Mvar-spherical tessellation, {p,q,...}, is centrally symmetric. Such a polytope is named hemi-{p,q,...}, and contain half as many elements. Coxeter gives a symbol {p,q,...}/2, while McMullen writes {p,q,...}_h/2 with h as the coxeter number.Template:Sfnp

Even-sided regular polygons have hemi-2n-gon projective polygons, {2p}/2.

There are 4 regular projective polyhedra related to 4 of 5 Platonic solids.

The hemi-cube and hemi-octahedron generalize as hemi-[[hypercube|Template:Mvar-cubes]] and hemi-Template:Mvar-orthoplexes to any rank.

Regular projective polyhedra

rank 3 regular hemi-polytopes

Name	Coxeter McMullen	Image	Faces	Edges	Vertices	χ	skeleton graph
Hemi-cube	{4,3}/2 {4,3}3	File:Hemicube.svg	3	6	4	1	K4
Hemi-octahedron	{3,4}/2 {3,4}3	File:Hemi-octahedron2.png	4	6	3	1	Double-edged K3
Hemi-dodecahedron	{5,3}/2 {5,3}5	File:Hemi-dodecahedron.png	6	15	10	1	G(5,2)
Hemi-icosahedron	{3,5}/2 {3,5}5	File:Hemi-icosahedron2.png	10	15	6	1	K6

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.

Rank 4 regular hemi-polytopes

Name	Coxeter symbol	McMullen Symbol	Cells	Faces	Edges	Vertices	χ	Skeleton graph
Hemitesseract	{4,3,3}/2	{4,3,3}4	4	12	16	8	0	K4,4
Hemi-16-cell	{3,3,4}/2	{3,3,4}4	8	16	12	4	0	double-edged K4
Hemi-24-cell	{3,4,3}/2	{3,4,3}6	12	48	48	12	0
Hemi-120-cell	{5,3,3}/2	{5,3,3}15	60	360	600	300	0
Hemi-600-cell	{3,3,5}/2	{3,3,5}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	{4,3,3,3}/2	5	20	40	40	16	1	Tesseract skeleton + 8 central diagonals
hemi-pentacross	{3,3,3,4}/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 Template:Mvar-apeirotope is an infinite Template:Mvar-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:<ref>Template:Cite journal</ref>

• Regular honeycombs in Template:Mvar dimensions, which completely fill an Template:Mvar-dimensional space.
• Regular skew apeirotopes, comprising an Template:Mvar-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 Template:CDD.

...File:Regular apeirogon.svg...

It exists as the limit of the Template:Mvar-gon as Template:Mvar tends to infinity, as follows:

Name	Monogon	Digon	Triangle	Square	Pentagon	Hexagon	Heptagon	p-gon	Apeirogon
Schläfli	{1}	{2}	{3}	{4}	{5}	{6}	{7}	{p}	{∞}
Symmetry	D1, [ ]	D2, [2]	D3, [3]	D4, [4]	D5, [5]	D6, [6]	D7, [7]	[p]
Coxeter	Template:CDD or Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Image	File:Monogon.svg	File:Digon.svg	File:Regular triangle.svg	File:Regular quadrilateral.svg	File:Regular pentagon.svg	File:Regular hexagon.svg	File:Regular heptagon.svg		File:Regular apeirogon.svg

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.

{∞}	{iπ/λ}
File:Hyperbolic apeirogon example.png Apeirogon on horocycle	File:Pseudogon example.png 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
File:Regular zig-zag.svg Zig-zag apeirogon	File:Triangular helix.png Helix apeirogon

3-apeirotopes (apeirohedra)

Euclidean tilings

There are six regular tessellations of the plane: the three listed below, and their corresponding Petrials.

Name	Square tiling (quadrille)	Triangular tiling (deltille)	Hexagonal tiling (hextille)
Symmetry	p4m, [4,4], (*442)	p6m, [6,3], (*632)
Schläfli {p,q}	{4,4}	{3,6}	{6,3}
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD
Image	File:Uniform tiling 44-t0.svg	File:Uniform tiling 63-t2-red.svg	File:Uniform tiling 63-t0.svg

There are two improper regular tilings: {∞,2}, an apeirogonal dihedron, made from two apeirogons, each filling half the plane; and secondly, its dual, {2,∞}, an apeirogonal hosohedron, seen as an infinite set of parallel lines.

File:Apeirogonal tiling.png {∞,2}, Template:CDD

File:Apeirogonal hosohedron.png {2,∞}, Template:CDD

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

There are a number of different ways to display the hyperbolic plane, including the Poincaré 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 (apeirohedra) as regular tilings of the hyperbolic plane, of the form {p,q}, with p+q<pq/2.

• {3,7}, {3,8}, {3,9} ... {3,∞}
• {4,5}, {4,6}, {4,7} ... {4,∞}
• {5,4}, {5,5}, {5,6} ... {5,∞}
• {6,4}, {6,5}, {6,6} ... {6,∞}
• {7,3}, {7,4}, {7,5} ... {7,∞}
• {8,3}, {8,4}, {8,5} ... {8,∞}
• {9,3}, {9,4}, {9,5} ... {9,∞}
• ...
• {∞,3}, {∞,4}, {∞,5} ... {∞,∞}

A sampling: Template:Regular hyperbolic tiling table

The tilings {p, ∞} have ideal vertices, on the edge of the Poincaré disk model. Their duals {∞, p} have ideal apeirogonal faces, meaning that they are inscribed in horocycles. One could go further (as is done in the table above) and find tilings with ultra-ideal vertices, outside the Poincaré disk, which are dual to tiles inscribed in hypercycles; in what is symbolised {p, iπ/λ} above, infinitely many tiles still fit around each ultra-ideal vertex.<ref name=hyphon>Roice Nelson and Henry Segerman, Visualizing Hyperbolic Honeycombs Template:Webarchive</ref> (Parallel lines in extended hyperbolic space meet at an ideal point; ultraparallel lines meet at an ultra-ideal point.)<ref>Irving Adler, A New Look at Geometry (2012 Dover edition), p.233</ref>

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, ....Template:Sfnp 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 greateningsTemplate:Efn 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,Template:Sfnp and when m = 3, the case degenerates to a tetrahedron. The other two Kepler–Poinsot polyhedra (the great stellated dodecahedron and great icosahedron) do not have regular hyperbolic tiling analogues. 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}	Template:CDD	File:Hyperbolic tiling 7-2 7.png	{7/2} File:Star polygon 7-2.svg	{7} File:Regular heptagon.svg	3	*732 [7,3]	Heptagrammic-order heptagonal tiling
Heptagrammic-order heptagonal tiling	{7,7/2}	Template:CDD	File:Hyperbolic tiling 7 7-2.png	{7} File:Regular heptagon.svg	{7/2} File:Star polygon 7-2.svg	3	*732 [7,3]	Order-7 heptagrammic tiling
Order-9 enneagrammic tiling	{9/2,9}	Template:CDD	File:Hyperbolic tiling 9-2 9.png	{9/2} File:Star polygon 9-2.svg	{9} File:Regular nonagon.svg	3	*932 [9,3]	Enneagrammic-order enneagonal tiling
Enneagrammic-order enneagonal tiling	{9,9/2}	Template:CDD	File:Hyperbolic tiling 9 9-2.png	{9} File:Regular nonagon.svg	{9/2} File:Star polygon 9-2.svg	3	*932 [9,3]	Order-9 enneagrammic tiling
Order-11 hendecagrammic tiling	{11/2,11}	Template:CDD	File:Order-11 hendecagrammic tiling.png	{11/2} File:Star polygon 11-2.svg	{11} File:Regular hendecagon.svg	3	*11.3.2 [11,3]	Hendecagrammic-order hendecagonal tiling
Hendecagrammic-order hendecagonal tiling	{11,11/2}	Template:CDD	File:Hendecagrammic-order hendecagonal tiling.png	{11} File:Regular hendecagon.svg	{11/2} File:Star polygon 11-2.svg	3	*11.3.2 [11,3]	Order-11 hendecagrammic tiling
Order-p p-grammic tiling	{p/2,p}	Template:CDD		{p/2}	{p}	3	*p32 [p,3]	p-grammic-order p-gonal tiling
p-grammic-order p-gonal tiling	{p,p/2}	Template:CDD		{p}	{p/2}	3	*p32 [p,3]	Order-p p-grammic tiling

Skew apeirohedra in Euclidean 3-space

Template:Expand section There are three regular skew apeirohedra in Euclidean 3-space, with planar faces.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite book</ref> They share the same vertex arrangement and edge arrangement of 3 convex uniform honeycombs.

• 6 squares around each vertex: {4,6|4}
• 4 hexagons around each vertex: {6,4|4}
• 6 hexagons around each vertex: {6,6|3}

File:Pure 3-dimensional apeirohedra chart.png

12 3-dimensional "pure" apeirohedra based on the structure of the cubic honeycomb, {4,3,4}.Template:Sfnp 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.

Regular skew polyhedra with planar faces
File:Mucube external.png {4,6|4}	File:Muoctahedron external.png {6,4|4}	File:Mutetrahedron external.png {6,6|3}

Allowing for skew faces, there are 30 regular apeirohedra in Euclidean 3-space.Template:Sfnp 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:

• Template:Math, the mucube
• Template:Math, the Petrial of the mucube
• Template:Math, the mutetrahedron
• Template:Math, the Petrial of the mutetrahedron
• Template:Math, the muoctahedron
• Template:Math, the Petrial of the muoctahedron
• Template:Math, the halving of the mucube
• Template:Math, the Petrial of Template:Math
• Template:Math, the skewing of the muoctahedron
• Template:Math, the Petrial of Template:Math
• Template:Math
• Template:Math

Skew apeirohedra in hyperbolic 3-space

There are 31 regular skew apeirohedra with convex faces in hyperbolic 3-space with compact or paracompact symmetry:<ref>Template:Cite journal Note: His paper says there are 32, but one is self-dual, leaving 31.</ref>

• 14 are compact: {8,10|3}, {10,8|3}, {10,4|3}, {4,10|3}, {6,4|5}, {4,6|5}, {10,6|3}, {6,10|3}, {8,8|3}, {6,6|4}, {10,10|3},{6,6|5}, {8,6|3}, and {6,8|3}.
• 17 are paracompact: {12,10|3}, {10,12|3}, {12,4|3}, {4,12|3}, {6,4|6}, {4,6|6}, {8,4|4}, {4,8|4}, {12,6|3}, {6,12|3}, {12,12|3}, {6,6|6}, {8,6|4}, {6,8|4}, {12,8|3}, {8,12|3}, and {8,8|4}.

4-apeirotopes

Tessellations of Euclidean 3-space

File:Cubic honeycomb.png

Edge framework of cubic honeycomb, {4,3,4}

There is only one non-degenerate regular tessellation of 3-space (honeycombs), {4, 3, 4}:Template:Sfnp

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

Improper tessellations of Euclidean 3-space

File:Order-4 square hosohedral honeycomb-sphere.png

Regular {2,4,4} honeycomb, seen projected into a sphere.

There are six improper regular tessellations, pairs based on the three regular Euclidean tilings. Their cells and vertex figures are all regular hosohedra {2,n}, dihedra, {n,2}, 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 {p,q,r}	Coxeter diagram	Cell type {p,q}	Face type {p}	Edge figure {r}	Vertex figure {q,r}
{2,4,4}	Template:CDD	{2,4}	{2}	{4}	{4,4}
{2,3,6}	Template:CDD	{2,3}	{2}	{6}	{3,6}
{2,6,3}	Template:CDD	{2,6}	{2}	{3}	{6,3}
{4,4,2}	Template:CDD	{4,4}	{4}	{2}	{4,2}
{3,6,2}	Template:CDD	{3,6}	{3}	{2}	{6,2}
{6,3,2}	Template:CDD	{6,3}	{6}	{2}	{3,2}

Tessellations of hyperbolic 3-space

There are 15 flat regular honeycombs of hyperbolic 3-space:

• 4 are compact: {3,5,3}, {4,3,5}, {5,3,4}, and {5,3,5}
• while 11 are paracompact: {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}.

4 compact regular honeycombs File:H3 534 CC center.png {5,3,4} File:H3 535 CC center.png {5,3,5} File:H3 435 CC center.png {4,3,5} File:H3 353 CC center.png {3,5,3}

4 of 11 paracompact regular honeycombs File:H3 344 CC center.png {3,4,4} File:H3 363 FC boundary.png {3,6,3} File:H3 443 FC boundary.png {4,4,3} File:H3 444 FC boundary.png {4,4,4}

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

4 compact regular honeycombs

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

11 paracompact regular honeycombs

Name	Schläfli Symbol {p,q,r}	Coxeter Template:CDD	Cell type {p,q}	Face type {p}	Edge figure {r}	Vertex figure {q,r}	χ	Dual
Order-6 tetrahedral honeycomb	{3,3,6}	Template:CDD	{3,3}	{3}	{6}	{3,6}	0	{6,3,3}
Hexagonal tiling honeycomb	{6,3,3}	Template:CDD	{6,3}	{6}	{3}	{3,3}	0	{3,3,6}
Order-4 octahedral honeycomb	{3,4,4}	Template:CDD	{3,4}	{3}	{4}	{4,4}	0	{4,4,3}
Square tiling honeycomb	{4,4,3}	Template:CDD	{4,4}	{4}	{3}	{4,3}	0	{3,3,4}
Triangular tiling honeycomb	{3,6,3}	Template:CDD	{3,6}	{3}	{3}	{6,3}	0	Self-dual
Order-6 cubic honeycomb	{4,3,6}	Template:CDD	{4,3}	{4}	{4}	{3,6}	0	{6,3,4}
Order-4 hexagonal tiling honeycomb	{6,3,4}	Template:CDD	{6,3}	{6}	{4}	{3,4}	0	{4,3,6}
Order-4 square tiling honeycomb	{4,4,4}	Template:CDD	{4,4}	{4}	{4}	{4,4}	0	Self-dual
Order-6 dodecahedral honeycomb	{5,3,6}	Template:CDD	{5,3}	{5}	{5}	{3,6}	0	{6,3,5}
Order-5 hexagonal tiling honeycomb	{6,3,5}	Template:CDD	{6,3}	{6}	{5}	{3,5}	0	{5,3,6}
Order-6 hexagonal tiling honeycomb	{6,3,6}	Template:CDD	{6,3}	{6}	{6}	{3,6}	0	Self-dual

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental tetrahedron having ultra-ideal vertices). All honeycombs with hyperbolic cells or vertex figures and do not have 2 in their Schläfli symbol are noncompact. Template:Regular honeycomb table

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.

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 (Euclidean tilings rather than finite polyhedra). As the last number in the Schläfli symbol rises further, the vertex figure becomes hyperbolic, and vertices become ultra-ideal (so the edges do not meet within hyperbolic space). In honeycombs {p, q, ∞} 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 (though still infinitely many {p, q} would meet at such edges). In general, when the last number of the Schläfli symbol becomes ∞, faces of codimension two intersect the Poincaré hyperball only in one ideal point.<ref name=hyphon/>

5-apeirotopes

Tessellations of Euclidean 4-space

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

3 regular Euclidean honeycombs

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}

File:Tesseractic tetracomb.png Projected portion of {4,3,3,4} (Tesseractic honeycomb)

File:Demitesseractic tetra hc.png Projected portion of {3,3,4,3} (16-cell honeycomb)

File:Icositetrachoronic tetracomb.png Projected portion of {3,4,3,3} (24-cell honeycomb)

There are also the two improper cases {4,3,4,2} and {2,4,3,4}.

There are three flat regular honeycombs of Euclidean 4-space:Template:Sfnp

• {4,3,3,4}, {3,3,4,3}, and {3,4,3,3}.

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

• 5 are compact: {3,3,3,5}, {5,3,3,3}, {4,3,3,5}, {5,3,3,4}, {5,3,3,5}
• 2 are paracompact: {3,4,3,4}, and {4,3,4,3}.

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

• {5/2,5,3,3}, {3,3,5,5/2}, {3,5,5/2,5}, and {5,5/2,5,3}.

Tessellations of hyperbolic 4-space

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

Five compact regular honeycombs in H⁴:

5 compact regular honeycombs

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-5 120-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}.

2 paracompact regular honeycombs

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}

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental 5-cell having some parts inaccessible beyond infinity). All honeycombs which are not shown in the set of tables below and do not have 2 in their Schläfli symbol are noncompact. Template:Regular tetracomb table

Star tessellations of hyperbolic 4-space

There are four regular star-honeycombs in H⁴ space, all compact:

4 compact regular star-honeycombs

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	Density
Small stellated 120-cell honeycomb	{5/2,5,3,3}	{5/2,5,3}	{5/2,5}	{5/2}	{3}	{3,3}	{5,3,3}	{3,3,5,5/2}	5
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}	5
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}	10
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}	10

6-apeirotopes

There is only one flat regular honeycomb of Euclidean 5-space: (previously listed above as tessellations)Template:Sfnp

• {4,3,3,3,4}

There are five flat regular regular honeycombs of hyperbolic 5-space, all paracompact: (previously listed above as tessellations)Template:Sfnp

• {3,3,3,4,3}, {3,4,3,3,3}, {3,3,4,3,3}, {3,4,3,3,4}, and {4,3,3,4,3}

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 {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

In E⁵, there are also the improper cases {4,3,3,4,2}, {2,4,3,3,4}, {3,3,4,3,2}, {2,3,3,4,3}, {3,4,3,3,2}, and {2,3,4,3,3}. In Eⁿ, {4,3ⁿ⁻³,4,2} and {2,4,3ⁿ⁻³,4} are always improper Euclidean tessellations.

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.

5 paracompact regular honeycombs

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

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 {p,q,r,s,...} not covered above (p,q,r,s,... natural numbers above 2, or infinity) will form a noncompact tessellation of hyperbolic n-space.<ref name=hyphon/>

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 this atlas for a sample. Some notable examples of abstract regular polytopes that do not appear elsewhere in this list are the 11-cell, {3,5,3}, and the 57-cell, {5,3,5}, which have regular projective polyhedra as cells and vertex figures.

The elements of an abstract polyhedron are its body (the maximal element), 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 can not be realised faithfully and symmetrically, were identified by H. S. M. Coxeter in his book Regular Polytopes (1977) and again by J. M. Wills in his paper "The combinatorially regular polyhedra of index 2" (1987).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.

Polyhedron	File:DU36 medial rhombic triacontahedron.png Medial rhombic triacontahedron	File:Dodecadodecahedron.png Dodecadodecahedron	File:DU41 medial triambic icosahedron.png Medial triambic icosahedron	File:Ditrigonal dodecadodecahedron.png Ditrigonal dodecadodecahedron	File:Excavated dodecahedron.png Excavated dodecahedron
Vertex figure	{5}, {5/2} File:Regular polygon 5.svgFile:Pentagram green.svg	(5.5/2)2 File:Dodecadodecahedron vertfig.png	{5}, {5/2} File:Regular polygon 5.svgFile:Pentagram green.svg	(5.5/3)3 File:Ditrigonal dodecadodecahedron vertfig.png	File:Medial triambic icosahedron face.svg
Faces	30 rhombi File:Rhombus definition2.svg	12 pentagons 12 pentagrams File:Regular polygon 5.svgFile:Pentagram green.svg	20 hexagons File:Medial triambic icosahedron face.svg	12 pentagons 12 pentagrams File:Regular polygon 5.svgFile:Pentagram green.svg	20 hexagrams File:Star hexagon face.png
Tiling	File:Uniform tiling 45-t0.png {4, 5}	File:Uniform tiling 552-t1.png {5, 4}	File:Uniform tiling 65-t0.png {6, 5}	File:Uniform tiling 553-t1.png {5, 6}	File:Uniform tiling 66-t2.png {6, 6}
χ	−6	−6	−16	−16	−20

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.

See also

• List of regular polytope compounds
• Polygon
• Polyhedron
• Platonic solids
• Kepler–Poinsot solids
• 4-polytope
• Regular 4-polytope (16 regular 4-polytopes, 4 convex and 10 star (Schläfli–Hess))
• Tessellation
• Tilings of regular polygons
• Convex uniform honeycomb
• Regular map (graph theory)

References

Citations

• Template:Citation. See in particular Summary Tables II, III, IV, V, pp. 212–213.
  • Originally published in Template:Citation.
• Template:Cite book See in particular Tables I and II: Regular polytopes and honeycombs, pp. 294–296.
• Template:Citation
• Template:Citation
• Template:Citation.
• Template:Cite arXiv hyperbolichoneycombs.org/ Template:Webarchive
• Template:Citation. Reprint of 1930 ed., published by E. P. Dutton. See in particular Chapter X: The Regular Polytopes.

External links

• The Platonic Solids
• Kepler-Poinsot Polyhedra
• Regular 4d Polytope Foldouts
• Multidimensional Glossary (Look up Hexacosichoron and Hecatonicosachoron)
• Polytope Viewer
• Polytopes and optimal packing of p points in n dimensional spheres
• An atlas of small regular polytopes
• Regular polyhedra through time I. Hubard, Polytopes, Maps and their Symmetries
• Regular Star Polytopes, Nan Ma