Selected regular polytopes

Regular (2D) polygons
Convex	Star
{5}	{5/2}
Regular (3D) polyhedra
Convex	Star
{5,3}	{5/2,5}
Regular 4D polytopes
Convex	Star
{5,3,3}	{5/2,5,3}
Regular 2D tessellations
Euclidean	Hyperbolic
{4,4}	{5,4}
Regular 3D tessellations
Euclidean	Hyperbolic
{4,3,4}	{5,3,4}

This article lists theregular polytopes inEuclidean,spherical andhyperbolic spaces.

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

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

ACoxeter diagram represent mirror "planes" as nodes, and puts a ring around a node if a point isnot on the plane. Adion { },, is a pointp and its mirror image pointp', and the line segment between them.

There is only one polytope of rank 1 (1-polytope), the closedline segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has theSchläfli symbol { },^[2][3] or aCoxeter diagram with a single ringed node,.Norman Johnson calls it adion^[4] and gives it the Schläfli symbol { }.

Although trivial as a polytope, it appears as theedges of polygons and other higher dimensional polytopes.^[5] It is used in the definition ofuniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as aCartesian product of a line segment and a regular polygon.^[6]

The polytopes of rank 2 (2-polytopes) are calledpolygons. Regular polygons areequilateral andcyclic. Ap-gonal regular polygon is represented bySchläfli symbol {p}.

Many sources only considerconvex polygons, butstar polygons, like thepentagram, 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.

The Schläfli symbol {p} represents aregularp-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
Image
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
Image
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
Image

The regulardigon {2} can be considered to be adegenerate regular polygon. It can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of asphere ortorus. For example, digon can be realised non-degenerately as aspherical lune. Amonogon {1} could also be realised on the sphere as a single point with a great circle through it.^[7] However, a monogon is not a validabstract 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	or
Image

There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers{n/m}. They are calledstar polygons and share the samevertex arrangements of the convex regular polygons.

In general, for any natural numbern, there are regularn-pointed stars with Schläfli symbols{n/m} for allm such thatm <n/2 (strictly speaking{n/m} = {n/(n −m)}) andm andn arecoprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Symbols wherem andn are not coprime may be used to representcompound 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
Image

Regular star polygons up to 20 sides

{11/2}	{11/3}	{11/4}	{11/5}	{12/5}	{13/2}	{13/3}	{13/4}	{13/5}	{13/6}
{14/3}	{14/5}	{15/2}	{15/4}	{15/7}	{16/3}	{16/5}	{16/7}
{17/2}	{17/3}	{17/4}	{17/5}	{17/6}	{17/7}	{17/8}	{18/5}	{18/7}
{19/2}	{19/3}	{19/4}	{19/5}	{19/6}	{19/7}	{19/8}	{19/9}	{20/3}	{20/7}	{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 thepiangle, which do not cover the surface of a circle finitely many times.^[8]

In addition to the planar regular polygons there are infinitely manyregular skew polygons. Skew polygons can be created via the blending operation.

The blend of two polygonsP andQ, writtenP#Q, can be constructed as follows:

1. take the cartesian product of their verticesV_P ×V_Q.
2. add edges(p₀ ×q₀,p₁ ×q₁) where(p₀,p₁) is an edge ofP and(q₀,q₁) is an edge ofQ.
3. select an arbitrary connected component of the result.

Alternatively, the blend is the polygon⟨ρ₀σ₀,ρ₁σ₁⟩ whereρ andσ are the generating mirrors ofP andQ placed in orthogonal subspaces.^[9]The blending operation is commutative, associative and idempotent.

Every regular skew polygon can be expressed as the blend of a unique^[i] set of planar polygons.^[9] IfP andQ share no factors thenDim(P#Q) = Dim(P) + Dim(Q).

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 ({n/m}#{} wheren is odd) or an antiprism ({n/m}#{} wheren 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.

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 aClifford torus and related by aClifford displacement. Unlike 3-dimensional polygons, skew polygons on double rotations can include an odd-number of sides.

Polytopes of rank 3 are calledpolyhedra:

A regular polyhedron withSchläfli symbol{p,q}, Coxeter diagrams, has a regular face type{p}, and regularvertex figure{q}.

Avertex figure (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. Forregular polyhedra, this vertex figure is always a regular (and planar) polygon.

Existence of a regular polyhedron{p,q} is constrained by an inequality, related to the vertex figure'sangle defect:

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

Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.

The five convex regularpolyhedra are called thePlatonic solids. Thevertex figure is given with each vertex count. All these polyhedra have anEuler characteristic (χ) of 2.

Name	Schläfli {p,q}	Coxeter	Image (solid)	Image (sphere)	Faces {p}	Edges	Vertices {q}	Symmetry	Dual
Tetrahedron (3-simplex)	{3,3}				4 {3}	6	4 {3}	Td [3,3] (*332)	(self)
Hexahedron Cube (3-cube)	{4,3}				6 {4}	12	8 {3}	Oh [4,3] (*432)	Octahedron
Octahedron (3-orthoplex)	{3,4}				8 {3}	12	6 {4}	Oh [4,3] (*432)	Cube
Dodecahedron	{5,3}				12 {5}	30	20 {3}	Ih [5,3] (*532)	Icosahedron
Icosahedron	{3,5}				20 {3}	30	12 {5}	Ih [5,3] (*532)	Dodecahedron

Inspherical geometry, regularspherical polyhedra (tilings of thesphere) exist that would otherwise be degenerate as polytopes. These are thehosohedra {2,n} and their dualdihedra {n,2}.Coxeter calls these cases "improper" tessellations.^[10]

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}			2 {2}π/2	2	2 {2}π/2	D2h [2,2] (*222)	Self
Trigonal hosohedron	{2,3}			3 {2}π/3	3	2 {3}	D3h [2,3] (*322)	Trigonal dihedron
Square hosohedron	{2,4}			4 {2}π/4	4	2 {4}	D4h [2,4] (*422)	Square dihedron
Pentagonal hosohedron	{2,5}			5 {2}π/5	5	2 {5}	D5h [2,5] (*522)	Pentagonal dihedron
Hexagonal hosohedron	{2,6}			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}			2 {2}π/2	2	2 {2}π/2	D2h [2,2] (*222)	Self
Trigonal dihedron	{3,2}			2 {3}	3	3 {2}π/3	D3h [3,2] (*322)	Trigonal hosohedron
Square dihedron	{4,2}			2 {4}	4	4 {2}π/4	D4h [4,2] (*422)	Square hosohedron
Pentagonal dihedron	{5,2}			2 {5}	5	5 {2}π/5	D5h [5,2] (*522)	Pentagonal hosohedron
Hexagonal dihedron	{6,2}			2 {6}	6	6 {2}π/6	D6h [6,2] (*622)	Hexagonal hosohedron

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

The regularstar polyhedra are called theKepler–Poinsot polyhedra and there are four of them, based on thevertex arrangements of thedodecahedron {5,3} andicosahedron {3,5}:

Asspherical tilings, these star forms overlap the sphere multiple times, called itsdensity, being 3 or 7 for these forms. The tiling images show a singlespherical polygon face in yellow.

Name	Image (skeletonic)	Image (solid)	Image (sphere)	Stellation diagram	Schläfli {p,q} and Coxeter	Faces {p}	Edges	Vertices {q} verf.	χ	Density	Symmetry	Dual
Small stellated dodecahedron					{5/2,5}	12 {5/2}	30	12 {5}	−6	3	Ih [5,3] (*532)	Great dodecahedron
Great dodecahedron					{5,5/2}	12 {5}	30	12 {5/2}	−6	3	Ih [5,3] (*532)	Small stellated dodecahedron
Great stellated dodecahedron					{5/2,3}	12 {5/2}	30	20 {3}	2	7	Ih [5,3] (*532)	Great icosahedron
Great icosahedron					{3,5/2}	20 {3}	30	12 {5/2}	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}.

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Regular skew polyhedra are generalizations to the set ofregular polyhedron which include the possibility of nonplanarvertex figures.

For 4-dimensional skew polyhedra, Coxeter offered a modifiedSchläfli symbol {l,m|n} for these figures, with {l,m} implying thevertex figure,m l-gons around a vertex, andn-gonal holes. Their vertex figures areskew polygons, zig-zagging between two planes.

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

$${\displaystyle 2\sin \left(\frac{\pi }{l}\right)\sin \left(\frac{\pi }{m}\right)=\cos \left(\frac{\pi }{n}\right)}$$

Four of them can be seen in 4-dimensions as a subset of faces of fourregular 4-polytopes, sharing the samevertex arrangement andedge arrangement:

{4, 6 | 3}	{6, 4 | 3}	{4, 8 | 3}	{8, 4 | 3}

Regular4-polytopes withSchläfli symbol${\displaystyle \{p,q,r\}}$ have cells of type${\displaystyle \{p,q\}}$, faces of type${\displaystyle \{p\}}$, edge figures${\displaystyle \{r\}}$, and vertex figures${\displaystyle \{q,r\}}$.

• Avertex 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.
• Anedge 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${\displaystyle \{p,q,r\}}$ is constrained by the existence of the regular polyhedra${\displaystyle \{p,q\},\{q,r\}}$. A suggested name for 4-polytopes is "polychoron".^[11]

Each will exist in a space dependent upon this expression:

${\displaystyle \sin \left(\frac{\pi }{p}\right)\sin \left(\frac{\pi }{r}\right)-\cos \left(\frac{\pi }{q}\right)}$

${\displaystyle >0}$ : Hyperspherical 3-space honeycomb or 4-polytope

${\displaystyle =0}$ : Euclidean 3-space honeycomb

 : 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.

TheEuler characteristic${\displaystyle \chi }$ for convex 4-polytopes is zero:${\displaystyle \chi =V+F-E-C=0}$

The 6 convexregular 4-polytopes are shown in the table below. All these 4-polytopes have anEuler characteristic (χ) of 0.

Name	Schläfli {p,q,r}	Coxeter	Cells {p,q}	Faces {p}	Edges {r}	Vertices {q,r}	Dual {r,q,p}
5-cell (4-simplex)	{3,3,3}		5 {3,3}	10 {3}	10 {3}	5 {3,3}	(self)
8-cell (4-cube) (Tesseract)	{4,3,3}		8 {4,3}	24 {4}	32 {3}	16 {3,3}	16-cell
16-cell (4-orthoplex)	{3,3,4}		16 {3,3}	32 {3}	24 {4}	8 {3,4}	Tesseract
24-cell	{3,4,3}		24 {3,4}	96 {3}	96 {3}	24 {4,3}	(self)
120-cell	{5,3,3}		120 {5,3}	720 {5}	1200 {3}	600 {3,3}	600-cell
600-cell	{3,3,5}		600 {3,3}	1200 {3}	720 {5}	120 {3,5}	120-cell

5-cell	8-cell	16-cell	24-cell	120-cell	600-cell
{3,3,3}	{4,3,3}	{3,3,4}	{3,4,3}	{5,3,3}	{3,3,5}
Wireframe (Petrie polygon) skeworthographic projections
Solidorthographic projections
tetrahedral envelope (cell/ vertex-centered)	cubic envelope (cell-centered)	cubic envelope (cell-centered)	cuboctahedral envelope (cell-centered)	truncated rhombic triacontahedron envelope (cell-centered)	Pentakis icosidodecahedral envelope (vertex-centered)
WireframeSchlegel diagrams (Perspective projection)
(cell-centered)	(cell-centered)	(cell-centered)	(cell-centered)	(cell-centered)	(vertex-centered)
Wireframestereographic projections (Hyperspherical)

Di-4-topes andhoso-4-topes exist as regular tessellations of the3-sphere.

Regulardi-4-topes (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and theirhoso-4-topeduals (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 as3-sphere honeycombs

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

There are tenregular star 4-polytopes, which are called theSchläfli–Hess 4-polytopes. Their vertices are based on the convex120-cell{5,3,3} and600-cell{3,3,5}.

Ludwig Schläfli found four of them and skipped the last six because he would not allow forms that failed theEuler 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 bookEinleitung 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 uniqueedge arrangements and 7 uniqueface arrangements from these 10 regular star 4-polytopes, shown asorthogonal 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)			{3,5,5/2}	120 {3,5}	1200 {3}	720 {5/2}	120 {5,5/2}	4	480	H4 [5,3,3]	Small stellated 120-cell
Small stellated 120-cell			{5/2,5,3}	120 {5/2,5}	720 {5/2}	1200 {3}	120 {5,3}	4	−480	H4 [5,3,3]	Icosahedral 120-cell
Great 120-cell			{5,5/2,5}	120 {5,5/2}	720 {5}	720 {5}	120 {5/2,5}	6	0	H4 [5,3,3]	Self-dual
Grand 120-cell			{5,3,5/2}	120 {5,3}	720 {5}	720 {5/2}	120 {3,5/2}	20	0	H4 [5,3,3]	Great stellated 120-cell
Great stellated 120-cell			{5/2,3,5}	120 {5/2,3}	720 {5/2}	720 {5}	120 {3,5}	20	0	H4 [5,3,3]	Grand 120-cell
Grand stellated 120-cell			{5/2,5,5/2}	120 {5/2,5}	720 {5/2}	720 {5/2}	120 {5,5/2}	66	0	H4 [5,3,3]	Self-dual
Great grand 120-cell			{5,5/2,3}	120 {5,5/2}	720 {5}	1200 {3}	120 {5/2,3}	76	−480	H4 [5,3,3]	Great icosahedral 120-cell
Great icosahedral 120-cell (great faceted 600-cell)			{3,5/2,5}	120 {3,5/2}	1200 {3}	720 {5}	120 {5/2,5}	76	480	H4 [5,3,3]	Great grand 120-cell
Grand 600-cell			{3,3,5/2}	600 {3,3}	1200 {3}	720 {5/2}	120 {3,5/2}	191	0	H4 [5,3,3]	Great grand stellated 120-cell
Great grand stellated 120-cell			{5/2,3,3}	120 {5/2,3}	720 {5/2}	1200 {3}	600 {3,3}	191	0	H4 [5,3,3]	Grand 600-cell

There are 4failed 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.

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In addition to the 16 planar 4-polytopes above there are 18 finite skew polytopes.^[12] 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.

5-polytopes can be given the symbol${\displaystyle \{p,q,r,s\}}$ where${\displaystyle \{p,q,r\}}$ is the 4-face type,${\displaystyle \{p,q\}}$ is the cell type,${\displaystyle \{p\}}$ is the face type, and${\displaystyle \{s\}}$ is the face figure,${\displaystyle \{r,s\}}$ is the edge figure, and${\displaystyle \{q,r,s\}}$ is the vertex figure.

Avertex figure (of a 5-polytope) is a 4-polytope, seen by the arrangement of neighboring vertices to each vertex.

Anedge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.

Aface figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.

A regular 5-polytope${\displaystyle \{p,q,r,s\}}$ exists only if${\displaystyle \{p,q,r\}}$ and${\displaystyle \{q,r,s\}}$ are regular 4-polytopes.

The space it fits in is based on the expression:

${\displaystyle \frac{{\cos }^2\left(\frac{\pi }{q}\right)}{{\sin }^2\left(\frac{\pi }{p}\right)}+\frac{{\cos }^2\left(\frac{\pi }{r}\right)}{{\sin }^2\left(\frac{\pi }{s}\right)}}$

 : Spherical 4-space tessellation or 5-space polytope

${\displaystyle =1}$ : Euclidean 4-space tessellation

${\displaystyle >1}$ : hyperbolic 4-space tessellation

Enumeration of these constraints produce 3 convex polytopes, no star polytopes, 3 tessellations of Euclidean 4-space, and5 tessellations of paracompact hyperbolic 4-space. The only non-convex regular polytopes for ranks 5 and higher are skews.

In dimensions 5 and higher, there are only three kinds of convex regular polytopes.^[13]

Name	Schläfli Symbol {p1,...,pn−1}	Coxeter	k-faces	Facet type	Vertex figure	Dual
n-simplex	{3n−1}	...	${\displaystyle (\genfrac{}{}{0ex}{}{n+1}{k+1})}$	{3n−2}	{3n−2}	Self-dual
n-cube	{4,3n−2}	...	${\displaystyle 2^{n-k}(\genfrac{}{}{0ex}{}{n}{k})}$	{4,3n−3}	{3n−2}	n-orthoplex
n-orthoplex	{3n−2,4}	...	${\displaystyle 2^{k+1}(\genfrac{}{}{0ex}{}{n}{k+1})}$	{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}.

Name	Schläfli Symbol {p,q,r,s} Coxeter	Facets {p,q,r}	Cells {p,q}	Faces {p}	Edges	Vertices	Face figure {s}	Edge figure {r,s}	Vertex figure {q,r,s}
5-simplex	{3,3,3,3}	6 {3,3,3}	15 {3,3}	20 {3}	15	6	{3}	{3,3}	{3,3,3}
5-cube	{4,3,3,3}	10 {4,3,3}	40 {4,3}	80 {4}	80	32	{3}	{3,3}	{3,3,3}
5-orthoplex	{3,3,3,4}	32 {3,3,3}	80 {3,3}	80 {3}	40	10	{4}	{3,4}	{3,3,4}

5-simplex

5-cube

5-orthoplex

6-simplex

6-cube

6-orthoplex

7-simplex

7-cube

7-orthoplex

8-simplex

8-cube

8-orthoplex

9-simplex

9-cube

9-orthoplex

10-simplex

10-cube

10-orthoplex

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.e.g. hosotopes and ditopes.

A projective regular(n+1)-polytope exists when an original regularn-spherical tessellation, {p,q,...}, iscentrally 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 withh as thecoxeter number.^[14]

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

There are 4 regularprojective polyhedra related to 4 of 5Platonic solids.

The hemi-cube and hemi-octahedron generalize as hemi-n-cubes and hemi-n-orthoplexes to any rank.

rank 3 regular hemi-polytopes

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

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.

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:

Anapeirotope orinfinite polytope is apolytope which has infinitely manyfacets. Ann-apeirotope is an infiniten-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:^[15]

• Regularhoneycombs inn dimensions, which completely fill ann-dimensional space.
• Regularskew apeirotopes, comprising ann-dimensional manifold in a higher space.

The straightapeirogon is a regular tessellation of the line, subdividing it into infinitely many equal segments. It has infinitely many vertices and edges. ItsSchläfli symbol is {∞}, and Coxeter diagram.

......

It exists as the limit of thep-gon asp 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	or
Image

Apeirogons in thehyperbolic plane, most notably theregular apeirogon, {∞}, can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed byhorocycles orhypercycles rather thancircles.

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}
Apeirogon onhorocycle	Apeirogon onhypercycle

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

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 regularskew apeirogon traces out a helical spiral and may be either left- or right-handed.

2 dimensions	3 dimensions
Zig-zag apeirogon	Helix apeirogon

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
Image

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

{∞,2},

{2,∞},

There are no regular plane tilings ofstar 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.

Tessellations ofhyperbolic 2-space arehyperbolic 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 generalSchwarz 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 thePoincaré 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 camerafisheye 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:

Regular hyperbolic tiling table v t e
Spherical(improper/Platonic)/Euclidean/hyperbolic (Poincaré disk:compact/paracompact/noncompact) tessellations with theirSchläfli symbol
p \ q	2	3	4	5	6	7	8	...	∞	...	iπ/λ
2	{2,2}	{2,3}	{2,4}	{2,5}	{2,6}	{2,7}	{2,8}		{2,∞}		{2,iπ/λ}
3	{3,2}	(tetrahedron) {3,3}	(octahedron) {3,4}	(icosahedron) {3,5}	(deltille) {3,6}	{3,7}	{3,8}		{3,∞}		{3,iπ/λ}
4	{4,2}	(cube) {4,3}	(quadrille) {4,4}	{4,5}	{4,6}	{4,7}	{4,8}		{4,∞}		{4,iπ/λ}
5	{5,2}	(dodecahedron) {5,3}	{5,4}	{5,5}	{5,6}	{5,7}	{5,8}		{5,∞}		{5,iπ/λ}
6	{6,2}	(hextille) {6,3}	{6,4}	{6,5}	{6,6}	{6,7}	{6,8}		{6,∞}		{6,iπ/λ}
7	{7,2}	{7,3}	{7,4}	{7,5}	{7,6}	{7,7}	{7,8}		{7,∞}		{7,iπ/λ}
8	{8,2}	{8,3}	{8,4}	{8,5}	{8,6}	{8,7}	{8,8}		{8,∞}		{8,iπ/λ}
...
∞	{∞,2}	{∞,3}	{∞,4}	{∞,5}	{∞,6}	{∞,7}	{∞,8}	{∞,∞}	{∞,iπ/λ}
...
iπ/λ	{iπ/λ,2}	{iπ/λ,3}	{iπ/λ,4}	{iπ/λ,5}	{iπ/λ,6}	{iπ/λ,7}	{iπ/λ,8}	{iπ/λ,∞}	{iπ/λ, iπ/λ}