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{{Short description|None}}{{use dmy dates|cs1-dates=ly|date=December 2020}}{| class="wikitable floatright"|+ Example regular polytopes!colspan=2|Regular (2D) polygons!Convex!Star align=center

150px){5}	150px){5/2}

!colspan=2|Regular (3D) polyhedra!Convex!Star align=center

150px){5,3}	150px){5/2,5}

! colspan="2" |Regular 4D polytopes!Convex!Star

150px){5,3,3}	150px){5/2,5,3}

!colspan=2|Regular 2D tessellations!Euclidean!Hyperbolic align=center

150px){4,4}	150px){5,4}

!colspan=2|Regular 3D tessellations!Euclidean!Hyperbolic align=center

150px){4,3,4}	150px){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.{| class="wikitable"!rowspan=3|Rank!rowspan=2 colspan=3|Finite!rowspan=2 colspan=2|Euclidean!colspan=3|Hyperbolic!rowspan=3|Abstract!colspan=2|Compact!Paracompact!Convex!Star!Skew{{efn|name=full rank|Only counting polytopes of full rank. There are more regular polytopes of each rank > 1 in higher dimensions.}}{{citation|last=McMullen|first=Peter|year=2004|title=Regular polytopes of full rank|journal=Discrete & Computational Geometry|volume=32 |pages=1–35 |doi=10.1007/s00454-004-0848-5 |s2cid=46707382 |url=https://link.springer.com/article/10.1007/s00454-004-0848-5}}!Convex!Skew{{efn|name=full rank}}!Convex!Star!Convex align=center!1

1	style="color:#999999; font-size:80%"	none>	none	style="color:#999999; font-size:80%"	none>	none	style="color:#999999; font-size:80%"	none>	none	style="color:#999999; font-size:80%"	none>|1

align=center!2

infty	infty	style="color:#999999; font-size:80%"	none>		none	1	style="color:#999999; font-size:80%"	none>	none	infty

align=center!3

5	4	9	3	3	infty	infty	infty	infty

align=center!4

6	10	18	1	7	4	style="color:#999999; font-size:80%"	none>	|infty

align=center!5

3	style="color:#999999; font-size:80%"	none>						|infty

align=center!6

3	style="color:#999999; font-size:80%"	none>				none	style="color:#999999; font-size:80%"	none>	|infty

align=center!7+

3	style="color:#999999; font-size:80%"	none>				none	style="color:#999999; font-size:80%"	none>	none	infty

{{notelist}}There are no Euclidean regular star tessellations in any number of dimensions.">

1-polytopes{| classwikitable alignright width330 valign=top

frameless|upright)	Coxeter diagram represent mirror "planes" as nodes, and puts a ring around a node if a point is not on the plane. A dion { }, {{CDD>node_1}}, is a point {{mvar	p'}}, 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 { },{{sfnp|Coxeter|1973|p=129}}{{sfnp|McMullen|Schulte|2002|p=30}} or a Coxeter diagram with a single ringed node, {{CDD|node_1}}. Norman Johnson calls it a dionBOOK, Norman Johnson (mathematician), N.W., Johnson, Geometries and Transformations, 2018, 978-1-107-10340-5, Chapter 11: Finite symmetry groups, Cambridge University Press, 11.1 Polytopes and Honeycombs, p. 224, and gives it the Schläfli symbol { }.Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes.{{sfnp|Coxeter|1973|p=120}} It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram {{CDD|node_1|2|node_1|p|node}} as a Cartesian product of a line segment and a regular polygon.{{sfnp|Coxeter|1973|p=124}}

2-polytopes (polygons)

The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A {{mvar|p}}-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 {{mvar|p}}-gon.{| class="wikitable" style="text-align:center; border:0;" bgcolor="#e0e0e0" valign="top"!Name!Triangle(2-simplex)!Square(2-orthoplex)(2-cube)!Pentagon(2-pentagonalpolytope)!Hexagon!Heptagon!Octagon bgcolor="#ffd0d0"!Schläfli|{3}|{4}|{5}|{6}|{7}|{8}!Symmetry

				|D8, [8]

!Coxeter

node_1	node}}	node_1	node}}	node_1	node}}	node_1	node}}	node_1	node}}	node_1	node}}

!Image

Image:Regular triangle.svg>60px	Image:Regular quadrilateral.svg>60px	Image:Regular pentagon.svg>60px	Image:Regular hexagon.svg>60px	Image:Regular heptagon.svg>60px	Image:Regular octagon.svg>60px

!Name!Nonagon(Enneagon)!Decagon!Hendecagon!Dodecagon!Tridecagon!Tetradecagon bgcolor="#ffd0d0"!Schläfli|{9}|{10}|{11}|{12}|{13}|{14}!Symmetry

				|D14, [14]

!Dynkin

node_1	node}}	node_1	node}}	node_1	node}}	node_1	node}}	node_1	node}}	node_1	node}}

!Image

Image:Regular nonagon.svg>60px	Image:Regular decagon.svg>60px	Image:Regular hendecagon.svg>60px	Image:Regular dodecagon.svg>60px	Image:Regular tridecagon.svg>60px	Image:Regular tetradecagon.svg>60px

!Name!Pentadecagon!Hexadecagon!Heptadecagon!Octadecagon!Enneadecagon!Icosagon! ...p-gon  bgcolor="#ffd0d0"!Schläfli|{15}|{16}|{17}|{18}|{19}|{20}|{p}!Symmetry

					|Dp, [p]

!Dynkin

node_1

node}}

node_1

node}}

node_1

node}}

node_1

node}}

node_1

node}}

node_1

node}}

node_1

node}}

!Image

Image:Regular pentadecagon.svg>60px	Image:Regular hexadecagon.svg>60px	Image:Regular heptadecagon.svg>60px	Image:Regular octadecagon.svg>60px	Image:Regular enneadecagon.svg>60px	Image:Regular icosagon.svg>60px	Image:Disk 1.svg>60px

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.Coxeter, Regular Complex Polytopes, p. 9 However, a monogon is not a valid abstract polytope because its single edge is incident to only one vertex rather than two.{| class="wikitable" style="text-align:center;" bgcolor="#e0e0e0" valign="top"!Name|Monogon|Digon bgcolor="#eeeedd"!Schläfli symbol|{1}|{2}!Symmetry|D1, [ ]|D2, [2]!Coxeter diagram

node}} or {{CDD	2x|node}}	node_1	node}}

!Image

60px)	Image:Digon.svg>60px

Stars

There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {{math|{{mset|n/m}}}}. They are called star polygons and share the same vertex arrangements of the convex regular polygons.In general, for any natural number {{mvar|n}}, there are regular {{mvar|n}}-pointed stars with Schläfli symbols {{math|{{mset|n/m}}}} for all {{mvar|m}} such that {{math|m < n/2}} (strictly speaking {{math|{{mset|n/m}} {{=}} {{mset|n/(n − m)}}}}) and {{mvar|m}} and {{mvar|n}} are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Symbols where {{mvar|m}} and {{mvar|n}} are not coprime may be used to represent compound polygons.{| class="wikitable" style="text-align:center;" bgcolor="#e0e0e0"!Name|Pentagram

Heptagrams|Octagram		Enneagrams		Decagram (geometry)>Decagram	star polygon>n-grams

bgcolor="#ffd0d0"!Schläfli|{5/2}|{7/2}|{7/3}|{8/3}|{9/2}|{9/4}|{10/3}|{p/q}!Symmetry

	D7, [7]		D8, [8]	colspan=2		|Dp, [p]

!Coxeter

node_1

rat

node}}

node_1

rat

node}}

node_1

rat

node}}

node_1

rat

node}}

node_1

rat

node}}

node_1

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node_1

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node}}

node_1

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node}}

!Image

75px)	75px)	75px)	75px)	75px)	75px)	75px)|

{| class=wikitable|+ Regular star polygons up to 20 sides align=center

60px){11/2}	60px){11/3}	60px){11/4}	60px){11/5}	60px){12/5}	60px){13/2}	60px){13/3}	60px){13/4}	60px){13/5}	60px){13/6}

align=center

60px){14/3}	60px){14/5}	60px){15/2}	60px){15/4}	60px){15/7}	60px){16/3}	60px){16/5}	60px){16/7}

align=center

60px){17/2}	60px){17/3}	60px){17/4}	60px){17/5}	60px){17/6}	60px){17/7}	60px){17/8}	60px){18/5}	60px){18/7}

align=center

60px){19/2}	60px){19/3}	60px){19/4}	60px){19/5}	60px){19/6}	60px){19/7}	60px){19/8}	60px){19/9}	60px){20/3}	60px){20/7}	60px){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 do not appear to have been studied in detail.There also exist failed star polygons, such as the piangle, which do not cover the surface of a circle finitely many times.WEB,weblink Between a square rock and a hard pentagon: Fractional polygons, Duncan, Hugh, 28 September 2017, chalkdust,

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 {{mvar|P}} and {{mvar|Q}}, written {{math|P#Q}}, can be constructed as follows:

1. take the cartesian product of their vertices {{math|V{{Sub|P}} × V{{sub|Q}}}}.
2. add edges {{math|(p{{sub|0}} × q{{sub|0}}, p{{sub|1}} × q{{sub|1}})}} where {{math|(p{{sub|0}}, p{{sub|1}})}} is an edge of {{mvar|P}} and {{math|(q{{sub|0}}, q{{sub|1}})}} is an edge of {{mvar|Q}}.
3. select an arbitrary connected component of the result.

Alternatively, the blend is the polygon {{math|{{angbr|ρ{{sub|0}}σ{{sub|0}}, ρ{{sub|1}}σ{{sub|1}}}}}} where {{mvar|ρ}} and {{mvar|σ}} are the generating mirrors of {{mvar|P}} and {{mvar|Q}} placed in orthogonal subspaces.{{sfn|McMullen|Schulte|2002}}The blending operation is commutative, associative and idempotent.Every regular skew polygon can be expressed as the blend of a unique{{efn|up to identity and idempotency}} set of planar polygons.{{sfn|McMullen|Schulte|2002}} If {{mvar|P}} and {{mvar|Q}} share no factors then {{math|Dim(P#Q) {{=}} Dim(P) + Dim(Q)}}.

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 ({{math|{{mset|n/m}}#{{(}}{{)}}}} where {{mvar|n}} is odd) or an antiprism ({{math|{{mset|n/m}}#{{(}}{{)}}}} where {{mvar|n}} 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 {{math|{{mset|p, q}}}}, Coxeter diagrams {{CDD|node_1|p|node|q|node}}, has a regular face type {{math|{{mset|p}}}}, and regular vertex figure {{math|{{mset|1}}}}.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 {{math|{{mset|p, q}}}} is constrained by an inequality, related to the vertex figure's angle defect: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}By enumerating the permutations, we find five convex forms, four star forms and three plane tilings, all with polygons {{math|{{mset|p}}}} and {{math|{{mset|q}}}} 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.{| class="wikitable" style="text-align:center;"!Name!Schläfli{{math|{{mset|p, q}}}}!Coxeter{{CDD|node_1|p|node|q|node}}!Image(solid)!Image(sphere)!Faces{{math|{{mset|p}}}}!Edges!Vertices{{math|{{mset|q}}}}!Symmetry!Dual bgcolor="#e0e0e0"

Tetrahedron(Simplex>3-simplex)|{3,3}	node_1	node	node}}	76px)	76px)|4{3}|6|4{3}|Td[3,3](*332)|(self)

bgcolor="#ffe0e0"

Cube (hypercube>3-cube)|{4,3}	node_1	node	node}}	76px)	76px)|6{4}|12|8{3}|Oh[4,3](*432)|Octahedron

bgcolor="#e0e0ff"

Octahedron(Cross-polytope>3-orthoplex)|{3,4}	node_1	node	node}}	76px)	76px)|8{3}|12|6{4}|Oh[4,3](*432)|Cube

bgcolor="#ffe0e0"|Dodecahedron|{5,3}

node_1	node	node}}	76px)	76px)|12{5}|30|20{3}|Ih[5,3](*532)|Icosahedron

bgcolor="#e0e0ff"|Icosahedron|{3,5}

node_1	node	node}}	76px)	76px)|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.{{sfnp|Coxeter|1973|pp=66-67}}The first few cases (n from 2 to 6) are listed below.{| class="wikitable" style="text-align:center;"|+ Hosohedra valign="top"!Name!Schläfli{2,p}!Coxeterdiagram!Image(sphere)!Faces{2}π/p!Edges!Vertices{p}!Symmetry!Dual bgcolor="#e0e0e0"|Digonal hosohedron|{2,2}

node_1	node	node}}	75px)|2{2}π/2|2|2{2}π/2|D2h[2,2](*222)|Self

bgcolor="#ffe0e0"|Trigonal hosohedron|{2,3}

node_1	node	node}}	75px)|3{2}π/3|3|2{3}|D3h[2,3](*322)|Trigonal dihedron

bgcolor="#ffe0e0"|Square hosohedron|{2,4}

node_1	node	node}}	75px)|4{2}π/4|4|2{4}|D4h[2,4](*422)|Square dihedron

bgcolor="#ffe0e0"|Pentagonal hosohedron|{2,5}

node_1	node	node}}	75px)|5{2}π/5|5|2{5}|D5h[2,5](*522)|Pentagonal dihedron

bgcolor="#ffe0e0"|Hexagonal hosohedron|{2,6}

node_1	node	node}}	75px)|6{2}π/6|6|2{6}|D6h[2,6](*622)|Hexagonal dihedron

{| class="wikitable" style="text-align:center;"|+ Dihedra valign="top"!Name!Schläfli{p,2}!Coxeterdiagram!Image(sphere)!Faces{p}!Edges!Vertices{2}!Symmetry!Dual bgcolor="#e0e0e0"|Digonal dihedron|{2,2}

node_1	node	node}}	75px)|2{2}π/2|2|2{2}π/2|D2h[2,2](*222)|Self

bgcolor="#e0e0ff"|Trigonal dihedron|{3,2}

node_1	node	node}}	75px)|2{3}|3|3{2}π/3|D3h[3,2](*322)|Trigonal hosohedron

bgcolor="#e0e0ff"|Square dihedron|{4,2}

node_1	node	node}}	75px)|2{4}|4|4{2}π/4|D4h[4,2](*422)|Square hosohedron

bgcolor="#e0e0ff"|Pentagonal dihedron|{5,2}

node_1	node	node}}	75px)|2{5}|5|5{2}π/5|D5h[5,2](*522)|Pentagonal hosohedron

bgcolor="#e0e0ff"|Hexagonal dihedron|{6,2}

node_1	node	node}}	75px)|2{6}|6|6{2}π/6|D6h[6,2](*622)|Hexagonal hosohedron

Star-dihedra and hosohedra {{math|{{mset|p/q, 2}}}} and {{math|{{mset|2, p/q}}}} also exist for any star polygon {{math|{{mset|p/q}}}}.

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.{| class="wikitable"!Name!Image(skeletonic)!Image(solid)!Image(sphere)!Stellationdiagram!Schläfli{{math|{{mset|p, q}}}} andCoxeter!Faces{{math|{{mset|p}}}}!Edges!Vertices{{math|{{mset|q}}}}verf.!χ!Density!Symmetry!Dual BGCOLOR="#ffe0e0" align=center|Small stellated dodecahedron

Image:Skeleton St12, size m.png>80px

Image:Small stellated dodecahedron (gray with yellow face).svg>80px

Image:Small stellated dodecahedron tiling.png>80px

80px)

node

node

rat

node_1}}

Image:Star polygon 5-2.svg>30px

Image:Regular pentagon.svg>30px

|Great dodecahedron

BGCOLOR="#e0e0ff" align=center|Great dodecahedron

Image:Skeleton Gr12, size m.png>80px

Image:Great dodecahedron (gray with yellow face).svg>80px

Image:Great dodecahedron tiling.svg>80px

80px)

node_1

node

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node}}

Image:Regular pentagon.svg>30px

Image:Star polygon 5-2.svg>30px

|Small stellated dodecahedron

BGCOLOR="#ffe0e0" align=center|Great stellated dodecahedron

Image:Skeleton GrSt12, size s.png>80px

Image:Great stellated dodecahedron (gray with yellow face).svg>80px

Image:Great stellated dodecahedron tiling.svg>80px

80px)

node

node

rat

node_1}}

Image:Star polygon 5-2.svg>30px

Image:Regular triangle.svg>30px

|Great icosahedron

BGCOLOR="#e0e0ff" align=center|Great icosahedron

Image:Skeleton Gr20, size m.png>80px

Image:Great icosahedron (gray with yellow face).svg>80px

Image:Great icosahedron tiling.svg>80px

80px)

node_1

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Image:Regular triangle.svg>30px

Image:Star polygon 5-2.svg>30px

|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

{{Expand section|reason=This section is 75 years behind on mathematical liturature on this topic. It should include Grünbaumian skews.|date=January 2024}}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 {{mvar|n}}-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:2 sinleft(frac{pi}{l}right) sinleft(frac{pi}{m}right) = cosleft(frac{pi}{n}right)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:{| class=wikitable 100px)100px)100px)100px)!{4, 6 | 3}!{6, 4 | 3}!{4, 8 | 3}!{8, 4 | 3}

4-polytopes

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

• A vertex figure (of a 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 {p,q,r} is constrained by the existence of the regular polyhedra {p,q}, {q,r}. A suggested name for 4-polytopes is "polychoron".CONFERENCE, Convex and Abstract Polytopes (May 19–21, 2005) and Polytopes Day in Calgary (May 22, 2005), Abstracts,weblink Each will exist in a space dependent upon this expression: These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.The Euler characteristic chi for convex 4-polytopes is zero:chi = V+F-E-C = 0

Convex

The 6 convex regular 4-polytopes are shown in the table below. All these 4-polytopes have an Euler characteristic (χ) of 0.{| class="wikitable"! Name! Schläfli{p,q,r}! Coxeter{{CDD|node|p|node|q|node|r|node}}! Cells {p,q}! Faces {p}! Edges {r}! Vertices {q,r}! Dual {r,q,p} BGCOLOR="#e0e0e0" align=center

5-cell(Simplex>4-simplex)| {3,3,3}	node_1	node	node	node}}| 5{3,3}| 10{3}| 10 {3}| 5 {3,3}| (self)

BGCOLOR="#ffe0e0" align=center

8-cell(Hypercube>4-cube)(Tesseract)| {4,3,3}	node_1	node	node	node}}| 8 {4,3}| 24 {4}| 32 {3}| 16 {3,3}| 16-cell

BGCOLOR="#e0e0ff" align=center

16-cell(Cross-polytope>4-orthoplex)| {3,3,4}	node_1	node	node	node}}| 16 {3,3}| 32 {3}| 24 {4}| 8 {3,4}| Tesseract

BGCOLOR="#e0e0e0" align=center| 24-cell| {3,4,3}

node_1	node	node	node}}| 24 {3,4}| 96 {3}| 96 {3}| 24 {4,3}| (self)

BGCOLOR="#ffe0e0" align=center| 120-cell| {5,3,3}

node_1	node	node	node}}| 120 {5,3}| 720 {5}| 1200 {3}| 600 {3,3}| 600-cell

BGCOLOR="#e0e0ff" align=center| 600-cell| {3,3,5}

node_1	node	node	node}}| 600 {3,3}| 1200 {3}| 720 {5}| 120 {3,5}| 120-cell

{| class="wikitable"! 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}!colspan=6|Wireframe (Petrie polygon) skew orthographic projections

105px)	105px)	105px)	105px)	105px)	105px)

!colspan=6|Solid orthographic projections

Image:Tetrahedron.png>105pxtetrahedralenvelope (cell/vertex-centered)	Image:Hexahedron.png>105pxcubic envelope(cell-centered)	105px)cubic envelope(cell-centered)	Image:Ortho solid 24-cell.png>105pxcuboctahedralenvelope(cell-centered)	Image:Ortho solid 120-cell.png>105pxtruncated rhombictriacontahedronenvelope(cell-centered)	Image:Ortho solid 600-cell.png>105pxPentakisicosidodecahedralenvelope(vertex-centered)

!colspan=6|Wireframe Schlegel diagrams (Perspective projection)

Image:Schlegel wireframe 5-cell.png>105px(cell-centered)	Image:Schlegel wireframe 8-cell.png>105px(cell-centered)	Image:Schlegel wireframe 16-cell.png>105px(cell-centered)	Image:Schlegel wireframe 24-cell.png>105px(cell-centered)	Image:Schlegel wireframe 120-cell.png>105px(cell-centered)	Image:Schlegel wireframe 600-cell vertex-centered.png>105px(vertex-centered)

!colspan=6|Wireframe stereographic projections (Hyperspherical)

Image:Stereographic polytope 5cell.png>105px	Image:Stereographic polytope 8cell.png>105px	Image:Stereographic polytope 16cell.png>105px	Image:Stereographic polytope 24cell.png>105px	Image:Stereographic polytope 120cell.png>105px	Image:Stereographic polytope 600cell.png>105px

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.{| class="wikitable" style="text-align:center;" width=640|+ Regular hoso-4-topes as 3-sphere honeycombs!Schläfli{2,p,q}!Coxeter{{CDD|node_1|2x|node|p|node|q|node}}!Cells{2,p}π/q!Faces{2}π/p,π/q!Edges!Vertices!Vertex figure{p,q}!Symmetry!Dual bgcolor="#e0e0e0"|{2,3,3}

node_1	node	node	node}}	50px)|6{2}π/3,π/3|4|2	75px)|[2,3,3]|{3,3,2}

bgcolor="#ffe0e0"|{2,4,3}

node_1	node	node	node}}	50px)|12{2}π/4,π/3|8|2	75px)|[2,4,3]|{3,4,2}

bgcolor="#e0e0ff"|{2,3,4}

node_1	node	node	node}}	50px)|12{2}π/3,π/4|6|2	75px)|[2,4,3]|{4,3,2}

bgcolor="#ffe0e0"|{2,5,3}

node_1	node	node	node}}	50px)|30{2}π/5,π/3|20|2	75px)|[2,5,3]|{3,5,2}

bgcolor="#e0e0ff"|{2,3,5}

node_1	node	node	node}}	50px)|30{2}π/3,π/5|12|2	75px)|[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)weblink.There are 4 unique edge arrangements and 7 unique face arrangements from these 10 regular star 4-polytopes, shown as orthogonal projections:{| class="wikitable"! 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} align=center BGCOLOR="#e0e0ff"| Icosahedral 120-cell(faceted 600-cell)

Image:Schläfli-Hess polychoron-wireframe-3.png>75px	Image:ortho solid 007-uniform polychoron 35p-t0.png>75px	node_1	node	node	rat	node}}	Icosahedron>{3,5} missing image! - Icosahedron.png - 25px	Triangle>{3}25px	Pentagram>{5/2}25px	Great dodecahedron>{5,5/2} missing image! - Great dodecahedron.png - 25px | 4| 480| H4[5,3,3]| Small stellated 120-cell

align=center BGCOLOR="#ffe0e0"| Small stellated 120-cell

Image:Schläfli-Hess polychoron-wireframe-2.png>75px	Image:ortho solid 010-uniform polychoron p53-t0.png>75px	node	node	node	rat	node_1}}	Small stellated dodecahedron>{5/2,5} missing image! - Small stellated dodecahedron.png - 25px	Pentagram>{5/2}25px	Triangle>{3}25px	Dodecahedron>{5,3} missing image! - Dodecahedron.png - 25px | 4| −480| H4[5,3,3]| Icosahedral 120-cell

align=center BGCOLOR="#e0ffe0"| Great 120-cell

Image:Schläfli-Hess polychoron-wireframe-3.png>75px	Image:ortho solid 008-uniform polychoron 5p5-t0.png>75px	node_1	node	rat	node	node}}	Great dodecahedron>{5,5/2} missing image! - Great dodecahedron.png - 25px	Pentagon>{5}25px	Pentagon>{5}25px	Small stellated dodecahedron>{5/2,5} missing image! - Small stellated dodecahedron.png - 25px | 6| 0| H4[5,3,3]| Self-dual

align=center BGCOLOR="#e0e0ff"| Grand 120-cell

Image:Schläfli-Hess polychoron-wireframe-3.png>75px	Image:ortho solid 009-uniform polychoron 53p-t0.png>75px	node_1	node	node	rat	node}}	Dodecahedron>{5,3} missing image! - Dodecahedron.png - 25px	Pentagon>{5}25px	Pentagram>{5/2}25px	Great icosahedron>{3,5/2} missing image! - Great icosahedron.png - 25px | 20| 0| H4[5,3,3]| Great stellated 120-cell

align=center BGCOLOR="#ffe0e0"| Great stellated 120-cell

Image:Schläfli-Hess polychoron-wireframe-4.png>75px	Image:ortho solid 012-uniform polychoron p35-t0.png>75px	node	node	node	rat	node_1}}	Great stellated dodecahedron>{5/2,3} missing image! - Great stellated dodecahedron.png - 25px	Pentagram>{5/2}25px	Pentagon>{5}25px	Icosahedron>{3,5} missing image! - Icosahedron.png - 25px | 20| 0| H4[5,3,3]| Grand 120-cell

align=center BGCOLOR="#e0ffe0"| Grand stellated 120-cell

Image:Schläfli-Hess polychoron-wireframe-4.png>75px	Image:ortho solid 013-uniform polychoron p5p-t0.png>75px	node_1	rat	node	node	rat	node}}	Small stellated dodecahedron>{5/2,5} missing image! - Small stellated dodecahedron.png - 25px	Pentagram>{5/2}25px	Pentagram>{5/2}25px	Great dodecahedron>{5,5/2} missing image! - Great dodecahedron.png - 25px | 66| 0| H4[5,3,3]| Self-dual

align=center BGCOLOR="#e0e0ff"| Great grand 120-cell

Image:Schläfli-Hess polychoron-wireframe-2.png>75px	Image:ortho solid 011-uniform polychoron 53p-t0.png>75px	node_1	node	rat	node	node}}	Great dodecahedron>{5,5/2} missing image! - Great dodecahedron.png - 25px	Pentagon>{5}25px	Triangle>{3}25px	Great stellated dodecahedron>{5/2,3} missing image! - Great stellated dodecahedron.png - 25px | 76| −480| H4[5,3,3]| Great icosahedral 120-cell

align=center BGCOLOR="#ffe0e0"| Great icosahedral 120-cell(great faceted 600-cell)

Image:Schläfli-Hess polychoron-wireframe-4.png>75px	Image:ortho solid 014-uniform polychoron 3p5-t0.png>75px	node	node	rat	node	node_1}}	Great icosahedron>{3,5/2} missing image! - Great icosahedron.png - 25px	Triangle>{3}25px	Pentagon>{5}25px	Small stellated dodecahedron>{5/2,5} missing image! - Small stellated dodecahedron.png - 25px | 76| 480| H4[5,3,3]| Great grand 120-cell

align=center BGCOLOR="#e0e0ff"| Grand 600-cell

Image:Schläfli-Hess polychoron-wireframe-4.png>75px	Image:ortho solid 015-uniform polychoron 33p-t0.png>75px	node_1	node	node	rat	node}}	Tetrahedron>{3,3} missing image! - Tetrahedron.png - 25px	Triangle>{3}25px	Pentagram>{5/2}25px	Great icosahedron>{3,5/2} missing image! - Great icosahedron.png - 25px | 191| 0| H4[5,3,3]| Great grand stellated 120-cell

align=center BGCOLOR="#ffe0e0"| Great grand stellated 120-cell

Image:Schläfli-Hess polychoron-wireframe-1.png>75px	Image:ortho solid 016-uniform polychoron p33-t0.png>75px	node	node	node	rat	node_1}}	Great stellated dodecahedron>{5/2,3} missing image! - Great stellated dodecahedron.png - 25px	Pentagram>{5/2}25px	Triangle>{3}25px	Tetrahedron>{3,3} missing image! - Tetrahedron.png - 25px | 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

{{Expand section|date=January 2024}}In addition to the 16 planar 4-polytopes above there are 18 finite skew polytopes.{{sfnp|McMullen|2004}} 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 {p,q,r,s} where {p,q,r} is the 4-face type, {p,q} is the cell type, {p} is the face type, and {s} is the face figure, {r,s} is the edge figure, and {q,r,s} is the vertex figure. A regular 5-polytope {p,q,r,s} exists only if {p,q,r} and {q,r,s} 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 no 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.{{sfnp|Coxeter|1973|loc=Table I: Regular polytopes, (iii) The three regular polytopes in {{mvar|n}} dimensions (n>=5), pp. 294–295}}{| class="wikitable"!Name!SchläfliSymbol{p1,...,pn−1}!Coxeter!k-faces!Facettype!Vertexfigure!Dual BGCOLOR="#e0e0e0" align=center

Simplex>n-simplex	{3n−1}	{{CDD	3	3}}...{{CDD	node	node}}	binomial coefficient	>		|Self-dual

BGCOLOR="#ffe0e0" align=center

Hypercube>n-cube	{4,3n−2}	{{CDD	4	3}}...{{CDD	node	node}}	2^{n-k}{n choose k}	{4,3n−3}	{3n−2}	n-orthoplex

BGCOLOR="#e0e0ff" align=center

Cross-polytope>n-orthoplex	{3n−2,4}	{{CDD	3	3}}...{{CDD	node	node}}	2^{k+1}{n choose {k+1}}	{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{| class"wikitable"

!Name!SchläfliSymbol{p,q,r,s}Coxeter!Facets{p,q,r}!Cells{p,q}!Faces{p}!Edges!Vertices!Facefigure{s}!Edgefigure{r,s}!Vertexfigure{q,r,s} BGCOLOR="#e0e0e0" align=center|5-simplex

node_1

node

node

node

node}}

6{3,3,3}

15{3,3}

20{3}

15

6

{3}

{3,3}

{3,3,3}

BGCOLOR="#ffe0e0" align=center|5-cube

node_1

node

node

node

node}}

10{4,3,3}

40{4,3}

80{4}

80

32

{3}

{3,3}

{3,3,3}

BGCOLOR="#e0e0ff" align=center|5-orthoplex

node_1

node

node

node

node}}

32{3,3,3}

80{3,3}

80{3}

40

10

{4}

{3,4}

{3,3,4}

{| class=wikitable align=center valign=top

Image:5-simplex t0.svg>150px5-simplex	Image:5-cube graph.svg>150px5-cube	150px)5-orthoplex

6 dimensions{| classwikitable

!Name!!Schläfli!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!χ BGCOLOR="#e0e0e0" align=center

6-simplex>							|0

BGCOLOR="#ffe0e0" align=center

6-cube>							|0

BGCOLOR="#e0e0ff" align=center

6-orthoplex>							|0

{| class=wikitable align=center valign=top

Image:6-simplex t0.svg>150px6-simplex	Image:6-cube graph.svg>150px6-cube	150px)6-orthoplex

7 dimensions{| classwikitable

!Name!!Schläfli!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!χ BGCOLOR="#e0e0e0" align=center

7-simplex>								|2

BGCOLOR="#ffe0e0" align=center

7-cube>								|2

BGCOLOR="#e0e0ff" align=center

7-orthoplex>								|2

{| class=wikitable align=center valign=top

Image:7-simplex t0.svg>150px7-simplex	Image:7-cube graph.svg>150px7-cube	150px)7-orthoplex

8 dimensions{| classwikitable

!Name!!Schläfli!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!7-faces!!χ BGCOLOR="#e0e0e0" align=center

8-simplex>									|0

BGCOLOR="#ffe0e0" align=center

8-cube>									|0

BGCOLOR="#e0e0ff" align=center

8-orthoplex>									|0

{| class=wikitable align=center valign=top

Image:8-simplex t0.svg>150px8-simplex	Image:8-cube.svg>150px8-cube	150px)8-orthoplex

9 dimensions{| classwikitable

!Name!!Schläfli!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!7-faces!!8-faces!!χ BGCOLOR="#e0e0e0" align=center

9-simplex>										|2

BGCOLOR="#ffe0e0" align=center

9-cube>										|2

BGCOLOR="#e0e0ff" align=center

9-orthoplex>										|2

{| class=wikitable align=center valign=top

Image:9-simplex t0.svg>150px9-simplex	Image:9-cube.svg>150px9-cube	150px)9-orthoplex

10 dimensions{| classwikitable

!Name!!Schläfli!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!7-faces!!8-faces!!9-faces!!χ BGCOLOR="#e0e0e0" align=center

10-simplex>											|0

BGCOLOR="#ffe0e0" align=center

10-cube>											|0

BGCOLOR="#e0e0ff" align=center

10-orthoplex>											|0

{| class=wikitable align=center valign=top

Image:10-simplex t0.svg>150px10-simplex	150px)10-cube	150px)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. {{abbr|e.g.}} hosotopes and ditopes.

Regular projective polytopes

A projective regular {{math|({{mvar|n}}+1)}}-polytope exists when an original regular {{mvar|n}}-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.{{sfnp|McMullen|Schulte|2002|loc="6C Projective Regular Polytopes" pp. 162-165}}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-{{mvar|n}}-cubes and hemi-{{mvar|n}}-orthoplexes to any rank.

Regular projective polyhedra{| classwikitable|+ rank 3 regular hemi-polytopes

!Name||CoxeterMcMullen||Image||Faces||Edges||Vertices||χ align=center!Hemi-cube||{4,3}/2{4,3}3

60px)	3	6	4	1

align=center!Hemi-octahedron||{3,4}/2{3,4}3

60px)	4	6	3	1

align=center!Hemi-dodecahedron||{5,3}/2{5,3}5

60px)	6	15	10	1

align=center!Hemi-icosahedron||{3,5}/2{3,5}5

60px)	10	15	6	1

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.{| class=wikitable|+ Rank 4 regular hemi-polytopes!Name||Coxetersymbol||McMullenSymbol||Cells||Faces||Edges||Vertices||χ align=center!Hemi-tesseract

{4,3,3}/2	{4,3,3}4				| 0

align=center!Hemi-16-cell

{3,3,4}/2	{3,3,4}4				| 0

align=center!Hemi-24-cell

{3,4,3}/2	{3,4,3}6				| 0

align=center!Hemi-120-cell

{5,3,3}/2	{5,3,3}15				| 0

align=center!Hemi-600-cell

{3,3,5}/2	{3,3,5}15				| 0

Regular projective 5-polytopes

Only 2 of 3 regular spereical polytopes are centrally symmetric for ranks 5 or higher: they are the hemi versions of the regular hypercube and orthoplex. They are tabulated below for rank 5, for example:{| class="wikitable"!Name

Schläfli symbol	>						Euler characteristic>χ

align=center!hemi-penteract

					|1

align=center!hemi-pentacross

					|1

Apeirotopes

An apeirotope or infinite polytope is a polytope which has infinitely many facets. An {{mvar|n}}-apeirotope is an infinite {{mvar|n}}-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:JOURNAL, Grünbaum, B., Regular Polyhedra—Old and New, Aequationes Mathematicae, 16, 1977, 1–2, 1–20, 10.1007/BF01836414, 125049930,

• Regular honeycombs in {{mvar|n}} dimensions, which completely fill an {{mvar|n}}-dimensional space.
• Regular skew apeirotopes, comprising an {{mvar|n}}-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 {{CDD|node_1|infin|node}}....(File:Regular apeirogon.svg|320px)...It exists as the limit of the {{mvar|p}}-gon as {{mvar|p}} tends to infinity, as follows:{| class="wikitable" style="text-align:center;" bgcolor="#e0e0e0" valign="top"!Name|Monogon|Digon

Equilateral triangle>Triangle|Square|Pentagon|Hexagon|Heptagon	Regular polygon>p-gon|Apeirogon

bgcolor="#ffd0d0"!Schläfli

{1}	{2}|{3}|{4}|{5}|{6}|{7}|{p}	{∞}

!Symmetry|D1, [ ]|D2, [2]

				|[p]

!Coxeter

node}} or {{CDD

2x|node}}

node_1

node}}

node_1

node}}

node_1

node}}

node_1

node}}

node_1

node}}

node_1

node}}

node_1

node}}

node_1

node}}

!Image

60px)	60px)	60px)	60px)	60px)	60px)	60px)|	60px)

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.{| class=wikitable!{∞}!{πi/λ}

160px)Apeirogon on horocycle	160px)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.{| class=wikitable!2 dimensions!3 dimensions align=center

400px)Zig-zag apeirogon	160px)Helix apeirogon

2-apeirotopes (apeirohedra)

Euclidean tilings

There are three regular tessellations of the plane.{| class=wikitable!Name!Square tiling(quadrille)!Triangular tiling(deltille)!Hexagonal tiling(hextille) align=center!Symmetry|p4m, [4,4], (*442)

p6m, [6,3], (*632)

align=center!Schläfli {p,q}|{4,4}|{3,6}|{6,3} align=center!Coxeter diagram

node_1	node	node}}	node	node	node_1}}	node_1	node	node}}

align=center!Image

100px)	100px)	100px)

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.{| class=wikitable align=center

210px)Order-2 apeirogonal tiling	, {{CDD>node_1	node	node}}	240px)Apeirogonal hosohedron	, {{CDD>node_1	node	node}}

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 disc model which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens.There are infinitely many flat regular 3-apeirotopes (apeirohedra) as regular tilings of the hyperbolic plane, of the form {p,q}, with p+q