This page lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. The Schläfli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.

The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one-lower-dimensional Euclidean space.

Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with 7 equilateral triangles and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.

Overview

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

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

Tessellations

The classical convex polytopes may be considered tessellations, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.

One dimension

The closed line segment, bounded by its two endpoints, is sometimes treated as a one-dimensional polytope. It can be represented by Schläfli symbol { }Cite error: A <ref> tag is missing the closing </ref> (see the help page)., and Coxeter diagram, . Although trivial as a polytope, it appears as the edges of higher dimensional polytopes.^[1] It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon.^[2]

Two dimensions (polygons)

The two-dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol {p}.

Usually only convex polygons are considered regular, but star polygons, like the pentagram, can also be considered regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to complete.

Star polygons should be called nonconvex rather than concave because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary.

Convex

The Schläfli symbol {p} represents a regular p-gon.

Name	Triangle (2-simplex)	Square (2-orthoplex) (2-cube)	Pentagon	Hexagon	Heptagon	Octagon
Schläfli	{3}	{4}	{5}	{6}	{7}	{8}
Coxeter
Image
Name	Enneagon	Decagon	Hendecagon	Dodecagon	Tridecagon	Tetradecagon
Schläfli	{9}	{10}	{11}	{12}	{13}	{14}
Dynkin
Image
Name	Pentadecagon	Hexadecagon	Heptadecagon	Octadecagon	Enneadecagon	Icosagon	...p-gon
Schläfli	{15}	{16}	{17}	{18}	{19}	{20}	{p}
Dynkin
Image

Improper (spherical)

The regular monogon {1} and regular digon {2} can be considered degenerate regular polygons. They can exist in non-Euclidean spaces like on the surface of a sphere or torus.

Name	Monogon	Digon
Schläfli symbol	{1}	{2}
Coxeter diagram
Image

Non-convex

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

In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are coprime.

Name	Pentagram	Heptagrams		Octagram	Enneagrams		Decagram	...n-agrams
Schläfli	{5/2}	{7/2}	{7/3}	{8/3}	{9/2}	{9/4}	{10/3}	{p/q}
Coxeter
Image

Tessellations

There is one tessellation of the line, giving one polytope, the (two-dimensional) apeirogon (aze). This has infinitely many vertices and edges. Its Schläfli symbol is {∞}, and Coxeter diagram .

......

Three dimensions (polyhedra)

In three dimensions, polytopes are called polyhedra:

A regular polyhedron with Schläfli symbol {p,q}, Coxeter diagrams , has a regular face type {p}, and regular vertex figure {q}.

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 {p,q} is constrained by an inequality, related to the vertex figure's angle defect:

 : Polyhedron (existing in Euclidean 3-space)

 : Euclidean plane tiling

 : Hyperbolic plane tiling

By enumerating the permutations, we find 5 convex forms, 4 star forms and 3 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.

Convex

The convex regular polyhedra are called the 5 Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have an Euler characteristic (χ) of 2.

Name	Schläfli {p,q}	Coxeter	Image (transparent)	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

Improper (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}, their dual dihedra {n,2} and monohedron {1,1}. Coxeter calls these cases "improper" tessellations.^[3]

Some include:

Name	Schläfli {p,q}	Coxeter diagram	Image (sphere)	Faces {p}	Edges { }	Vertices {q}	Symmetry	Dual
Monohedron (half monogonal hosohedron)	{1,1} =h{2,1}			1 {1}	0 { }	1 {1}	C1v [ ] (*1)	Self
Monogonal hosohedron	{2,1}			1 {2}	1 { }	2 {1}	D1h [2,1] (*22)	Monogonal dihedron
Digonal hosohedron (Same as digonal dihedron)	{2,2}			2 {2}	2 { }	2 {2}	D2h [2,2] (*222)	Self
Trigonal hosohedron	{2,3}			3 {2}	3 { }	2 {3}	D3h [2,3] (*322)	Trigonal dihedron
Square hosohedron	{2,4}			4 {2}	4 { }	2 {4}	D4h [2,4] (*422)	Square dihedron
Pentagonal hosohedron	{2,5}			5 {2}	5 { }	2 {5}	D5h [2,5] (*522)	Pentagonal dihedron
Hexagonal hosohedron	{2,6}			6 {2}	6 { }	2 {6}	D6h [2,6] (*622)	Hexagonal dihedron
Monogonal dihedron (half monogonal hosohedron)	{1,2} =h{2,2}			2 {1}	1 { }	1 {2}	D1h [1,2] (*22)	Monogonal hosohedron
Digonal dihedron (Same as digonal hosohedron)	{2,2}			2 {2}	2 { }	2 {2}	D2h [2,2] (*222)	Self
Trigonal dihedron	{3,2}			2 {3}	3 { }	3 {2}	D3h [3,2] (*322)	Trigonal hosohedron
Square dihedron	{4,2}			2 {4}	4 { }	4 {2}	D4h [4,2] (*422)	Square hosohedron
Pentagonal dihedron	{5,2}			2 {5}	5 { }	5 {2}	D5h [5,2] (*522)	Pentagonal hosohedron
Hexagonal dihedron	{6,2}			2 {6}	6 { }	6 {2}	D6h [6,2] (*622)	Hexagonal hosohedron

Non-convex

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 nonconvex 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 (transparent)	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

Tessellations

Euclidean tilings

There are three regular tessellations of the plane. All three have an Euler characteristic (χ) of 0.

Name	Square tiling (quadrille)	Triangular tiling (deltille)	Hexagonal tiling (hextille)
Symmetry	p4m, [4,4]	p6m, [6,3]
Schläfli {p,q}	{4,4}	{3,6}	{6,3}
Coxeter diagram
Image
Symmetry	*442 (p4m)	*632 (p6m)

There is one degenerate regular tiling, {∞,2} (azat), made from two apeirogons, each filling half the plane. This tiling is related to a 2-faced dihedron, {p,2}, on the sphere.

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 can be called 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.

A sampling:

Regular hyperbolic tiling table
Spherical (improper/Platonic)/Euclidean/hyperbolic (Poincaré disc: compact/paracompact/noncompact) tessellations with their Schläfli 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}

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, .... 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 greatenings 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, and when m = 3, we obtain the tetrahedron. 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}			{7/2}	{7}	3	*732	Heptagrammic-order heptagonal tiling
Heptagrammic-order heptagonal tiling	{7,7/2}			{7}	{7/2}	3	*732	Order-7 heptagrammic tiling
Order-9 enneagrammic tiling	{9/2,9}			{9/2}	{9}	3	*932	Enneagrammic-order enneagonal tiling
Enneagrammic-order enneagonal tiling	{9,9/2}			{9}	{9/2}	3	*932	Order-9 enneagrammic tiling
Order-p p-grammic tiling	{p/2,p}			{p/2}	{p}	3	*p32	p-grammic-order p-gonal tiling
p-grammic-order p-gonal tiling	{p,p/2}			{p}	{p/2}	3	*p32	Order-p p-grammic tiling

Four dimensions

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

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

The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra .

Each will exist in a space dependent upon this expression:

 : Hyperspherical 3-space honeycomb or 4-polytope

 : 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.

The Euler characteristic for polychora is and is zero for all forms.

Convex

The 6 convex regular 4-polytopes are shown in the table below. All these polychora have an Euler 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) (Pentachoron) (pen)	{3,3,3}		5 {3,3}	10 {3}	10 {3}	5 {3,3}	(self)
8-cell (4-cube) (Tesseract) (tes)	{4,3,3}		8 {4,3}	24 {4}	32 {3}	16 {3,3}	16-cell
16-cell (4-orthoplex) (hex)	{3,3,4}		16 {3,3}	32 {3}	24 {4}	8 {3,4}	Tesseract
24-cell (ico)	{3,4,3}		24 {3,4}	96 {3}	96 {3}	24 {4,3}	(self)
120-cell (hi)	{5,3,3}		120 {5,3}	720 {5}	1200 {3}	600 {3,3}	600-cell
600-cell (ex)	{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) skew orthographic projections
Solid orthographic 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)
Wireframe Schlegel diagrams (Perspective projection)
(Cell-centered)	(Cell-centered)	(Cell-centered)	(Cell-centered)	(Cell-centered)	(Vertex-centered)
Wireframe stereographic projections (Hyperspherical)

Improper (spherical)

Dichora and hosochora exist as regular tessellations of the 3-sphere.

Regular dichora (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their hosochora duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,p}. Polychora of the form {2,p,2} are both dichora and hosochora.

Non-convex

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 nonconvex polychora, 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) (fix)			{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 (sishi)			{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 (gohi)			{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 (gahi)			{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 (gishi)			{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 (gashi)			{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 (gaghi)			{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) (gofix)			{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 (gax)			{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 (gogishi)			{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 4 failed potential nonconvex regular polychora 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.

Tessellations of Euclidean 3-space

There is only one non-degenerate regular tessellation of 3-space (honeycombs):

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

Improper tessellations of Euclidean 3-space

There are six improper regular tessellations, pairs based on the three regular Euclidean tilings. Their cells and vertex figures are all regular hosohedra {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}		{2,4}	{2}	{4}	{4,4}
{2,3,6}		{2,3}	{2}	{6}	{3,6}
{2,6,3}		{2,6}	{2}	{3}	{6,3}
{4,4,2}		{4,4}	{4}	{2}	{4,2}
{3,6,2}		{3,6}	{3}	{2}	{6,2}
{6,3,2}		{6,3}	{6}	{2}	{3,2}

Tessellations of hyperbolic 3-space

4 compact regular honeycombs {5,3,4} {5,3,5} {4,3,5} {3,5,3}

4 of 11 paracompact regular honeycombs {3,4,4} {3,6,3} {4,4,3} {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.

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

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

A subset of uniform polychora and honeycombs are contained in the form {p,3,q} for values 3 to 6. Noncompact solutions exist as Lorentzian Coxeter groups for p or q greater than 6, and can be visualized with open domains in hyperbolic space, and a few are drawn below as tilings on the ideal half-space plane.

Spherical (improper/Platonic)/Euclidean/hyperbolic(compact/paracompact/noncompact) honeycombs {p,3,r}

{p,3} \ r	2	3	4	5	6	7	8	... ∞
{2,3}	{2,3,2}	{2,3,3}	{2,3,4}	{2,3,5}	{2,3,6}	{2,3,7}	{2,3,8}	{2,3,∞}
{3,3}	{3,3,2}	{3,3,3}	{3,3,4}	{3,3,5}	{3,3,6}	{3,3,7}	{3,3,8}	{3,3,∞}
{4,3}	{4,3,2}	{4,3,3}	{4,3,4}	{4,3,5}	{4,3,6}	{4,3,7}	{4,3,8}	{4,3,∞}
{5,3}	{5,3,2}	{5,3,3}	{5,3,4}	{5,3,5}	{5,3,6}	{5,3,7}	{5,3,8}	{5,3,∞}
{6,3}	{6,3,2}	{6,3,3}	{6,3,4}	{6,3,5}	{6,3,6}	{6,3,7}	{6,3,8}	{6,3,∞}
{7,3}	{7,3,2}	{7,3,3}	{7,3,4}	{7,3,5}	{7,3,6}	{7,3,7}	{7,3,8}	{7,3,∞}
{8,3}	{8,3,2}	{8,3,3}	{8,3,4}	{8,3,5}	{8,3,6}	{8,3,7}	{8,3,8}	{8,3,∞}
... {∞,3}	{∞,3,2}	{∞,3,3}	{∞,3,4}	{∞,3,5}	{∞,3,6}	{∞,3,7}	{∞,3,8}	{∞,3,∞}

{p,4,r} {p,4} \ r 2 3 4 5 6 ∞ {2,4} {2,4,2} {2,4,3} {2,4,4} {2,4,5} {2,4,6} {2,4,∞} {3,4} {3,4,2} {3,4,3} {3,4,4} {3,4,5} {3,4,6} {3,4,∞} {4,4} {4,4,2} {4,4,3} {4,4,4} {4,4,5} {4,4,6} {4,4,∞} {5,4} {5,4,2} {5,4,3} {5,4,4} {5,4,5} {5,4,6} {5,4,∞} {6,4} {6,4,2} {6,4,3} {6,4,4} {6,4,5} {6,4,6} {6,4,∞} {∞,4} {∞,4,2} {∞,4,3} {∞,4,4} {∞,4,5} {∞,4,6} {∞,4,∞}

{p,5,r} {p,5} \ r 2 3 4 5 6 ∞ {2,5} {2,5,2} {2,5,3} {2,5,4} {2,5,5} {2,5,6} {2,5,∞} {3,5} {3,5,2} {3,5,3} {3,5,4} {3,5,5} {3,5,6} {3,5,∞} {4,5} {4,5,2} {4,5,3} {4,5,4} {4,5,5} {4,5,6} {4,5,∞} {5,5} {5,5,2} {5,5,3} {5,5,4} {5,5,5} {5,5,6} {5,5,∞} {6,5} {6,5,2} {6,5,3} {6,5,4} {6,5,5} {6,5,6} {6,5,∞} {∞,5} {∞,5,2} {∞,5,3} {∞,5,4} {∞,5,5} {∞,5,6} {∞,5,∞}

{p,6,r} {p,6} \ r 2 3 4 5 6 ∞ {2,6} {2,6,2} {2,6,3} {2,6,4} {2,6,5} {2,6,6} {2,6,∞} {3,6} {3,6,2} {3,6,3} {3,6,4} {3,6,5} {3,6,6} {3,6,∞} {4,6} {4,6,2} {4,6,3} {4,6,4} {4,6,5} {4,6,6} {4,6,∞} {5,6} {5,6,2} {5,6,3} {5,6,4} {5,6,5} {5,6,6} {5,6,∞} {6,6} {6,6,2} {6,6,3} {6,6,4} {6,6,5} {6,6,6} {6,6,∞} {∞,6} {∞,6,2} {∞,6,3} {∞,6,4} {∞,6,5} {∞,6,6} {∞,6,∞}

{p,7,r} {p,7} \ r 2 3 4 5 6 ∞ {2,7} {2,7,2} {2,7,3} {2,7,4} {2,7,5} {2,7,6} {2,7,∞} {3,7} {3,7,2} {3,7,3} {3,7,4} {3,7,5} {3,7,6} {3,7,∞} {4,7} {4,7,2} {4,7,3} {4,7,4} {4,7,5} {4,7,6} {4,7,∞} {5,7} {5,7,2} {5,7,3} {5,7,4} {5,7,5} {5,7,6} {5,7,∞} {6,7} {6,7,2} {6,7,3} {6,7,4} {6,7,5} {6,7,6} {6,7,∞} {∞,7} {∞,7,2} {∞,7,3} {∞,7,4} {∞,7,5} {∞,7,6} {∞,7,∞}