Wythoff symbol

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Example Wythoff construction triangles with the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the 3 nodes opposite them as associated by the Wythoff symbol.
The eight forms for the Wythoff constructions from a general triangle (p q r).

In geometry, the Wythoff symbol was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.

A Schwarz triangle is a triangle that, with its own reflections in its edges, covers the sphere or the plane a finite number of times. The usual representation for the triangle is three numbers – integers or fractions – such that π/x is the angle at one vertex. For example, the triangle (2 3 4) represents the symmetry of a cube, while (5/2 5/2 5/2) is the face of an icosahedron.

Wythoff's construction in three dimensions consists of choosing a point in the triangle whose distance from each of the sides, if nonzero, is equal, and dropping perpendiculars to each of the edges.

Each edge of the triangle is named for the opposite angle; thus an edge opposite a right angle is designated '2'. The symbol then corresponds to a representation of off | on. Each of the numbers p in the symbol becomes a polygon pn, where n is the number of other edges that appear before the bar. So in 3 | 4 2 the vertex – a point, being here a degenerate polygon with 3×0 sides – lies on the π/3 corner of the triangle, and the altitude from that corner can be considered as forming half of the boundary between a square (having 4×1 sides) and a digon (having 2×1 sides) of zero area.

The special case of the snub figures is done by using the symbol | p q r, which would normally put the vertex at the centre of the sphere. The faces of a snub alternate as p 3 q 3 r 3. This gives an antiprism when q=r=2.

Each symbol represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with Oh symmetry, and 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D2h symmetry.

It can be applied with a slight extension to all uniform polyhedra, but the construction methods do not lead to all uniform tilings in euclidean or hyperbolic space.

Summary table[edit]

There are seven generator points with each set of p,q,r (and a few special forms):

General Right triangle (r=2)
Description Wythoff
symbol
Vertex
configuration
Coxeter
diagram

CDel pqr.png
Wythoff
symbol
Vertex
configuration
Schläfli
symbol
Coxeter
diagram
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
regular and
quasiregular
q | p r (p.r)q CDel 3.pngCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.png q | p 2 pq {p,q} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
p | q r (q.r)p CDel 3.pngCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.png p | q 2 qp {q,p} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
r | p q (q.p)r CDel 3.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.png 2 | p q (q.p)² r{p,q} t1{p,q} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
truncated and
expanded
q r | p q.2p.r.2p CDel 3.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.png q 2 | p q.2p.2p t{p,q} t0,1{p,q} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
p r | q p.2q.r.2q CDel 3.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.png p 2 | q p. 2q.2q t{q,p} t0,1{q,p} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
p q | r 2r.q.2r.p CDel 3.pngCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.png p q | 2 4.q.4.p rr{p,q} t0,2{p,q} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
even-faced p q r | 2r.2q.2p CDel 3.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.png p q 2 | 4.2q.2p tr{p,q} t0,1,2{p,q} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
p q (r s) | 2p.2q.-2p.-2q - p 2 (r s) | 2p.4.-2p.4/3 -
snub | p q r 3.r.3.q.3.p CDel 3.pngCDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.png | p q 2 3.3.q.3.p sr{p,q} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
| p q r s (4.p.4.q.4.r.4.s)/2 - - - -

There are three special cases:

  • p q (r s) | – This is a mixture of p q r | and p q s |.
  • | p q r – Snub forms (alternated) are give this otherwise unused symbol.
  • | p q r s – A unique snub form for U75 that isn't Wythoff-constructible.

Description[edit]

The numbers p,q,r describe the fundamental triangle of the symmetry group: at its vertices, the generating mirrors meet in angles of π/p, π/q, π/r. On the sphere there are 3 main symmetry types: (3 3 2), (4 3 2), (5 3 2), and one infinite family (p 2 2), for any p. (All simple families have one right angle and so r=2.)

The position of the vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle. The generator point can either be on or off each mirror, activated or not. This distinction creates 8 (2³) possible forms, neglecting one where the generator point is on all the mirrors.

In this notation the mirrors are labeled by the reflection-order of the opposite vertex. The p,q,r values are listed before the bar if the corresponding mirror is active.

The one impossible symbol | p q r implies the generator point is on all mirrors, which is only possible if the triangle is degenerate, reduced to a point. This unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored. The resulting figure has rotational symmetry only.

This symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. (An arc representing a right angle is omitted.) A node is circled if the generator point is not on the mirror.

Symmetry triangles[edit]

There are 4 symmetry classes of reflection on the sphere, and two in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are also listed. (Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.)

Point groups:

Euclidean (affine) groups:

Hyperbolic groups:

Dihedral spherical Spherical
D2h D3h D4h D5h D6h Td Oh Ih
*222 *322 *422 *522 *622 *332 *432 *532
Spherical square bipyramid2.png
(2 2 2)
Spherical hexagonal bipyramid2.png
(3 2 2)
Spherical octagonal bipyramid2.png
(4 2 2)
Spherical decagonal bipyramid2.png
(5 2 2)
Spherical dodecagonal bipyramid2.png
(6 2 2)
Tetrahedral reflection domains.png
(3 3 2)
Octahedral reflection domains.png
(4 3 2)
Icosahedral reflection domains.png
(5 3 2)

The above symmetry groups only includes the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, and determine the full set of solutions of nonconvex uniform polyhedra.

Euclidean plane
p4m p3m p6m
*442 *333 *632
Tile V488 bicolor.svg
(4 4 2)
Tile 3,6.svg
(3 3 3)
Tile V46b.svg
(6 3 2)
Hyperbolic plane
*732 *542 *433
Order-3 heptakis heptagonal tiling.png
(7 3 2)
Order-4 bisected pentagonal tiling.png
(5 4 2)
Uniform dual tiling 433-t012.png
(4 3 3)

In the tilings above, each triangle is a fundamental domain, colored by even and odd reflections.

Summary spherical, Euclidean and hyperbolic tilings[edit]

Selected tilings created by the Wythoff construction are given below.

Spherical tilings (r = 2)[edit]

(p q 2) Parent Truncated Rectified Bitruncated Birectified
(dual)
Cantellated Omnitruncated
(Cantitruncated)
Snub
Wythoff
symbol
q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
Schläfli
symbol
\begin{Bmatrix} p , q \end{Bmatrix} t\begin{Bmatrix} p , q \end{Bmatrix} \begin{Bmatrix} p \\ q \end{Bmatrix} t\begin{Bmatrix} q , p \end{Bmatrix} \begin{Bmatrix} q , p \end{Bmatrix} r\begin{Bmatrix} p \\ q \end{Bmatrix} t\begin{Bmatrix} p \\ q \end{Bmatrix} s\begin{Bmatrix} p \\ q \end{Bmatrix}
{p,q} t{p,q} r{p,q} t{q,p} {q,p} rr{p,q} tr{p,q} sr{p,q}
t0{p,q} t0,1{p,q} t1{p,q} t1,2{p,q} t2{p,q} t0,2{p,q} t0,1,2{p,q}
Coxeter
diagram
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Vertex figure pq q.2p.2p (p.q)2 p. 2q.2q qp p. 4.q.4 4.2p.2q 3.3.p. 3.q
Tetrahedral reflection domains.png
(3 3 2)
Uniform tiling 332-t0-1-.png
{3,3}
Uniform tiling 332-t01-1-.png
(3.6.6)
Uniform tiling 332-t1-1-.png
(3.3a.3.3a)
Uniform tiling 332-t12.png
(3.6.6)
Uniform tiling 332-t2.png
{3,3}
Uniform tiling 332-t02.png
(3a.4.3b.4)
Uniform tiling 332-t012.png
(4.6a.6b)
Spherical snub tetrahedron.png
(3.3.3a.3.3b)
Octahedral reflection domains.png
(4 3 2)
Uniform tiling 432-t0.png
{4,3}
Uniform tiling 432-t01.png
(3.8.8)
Uniform tiling 432-t1.png
(3.4.3.4)
Uniform tiling 432-t12.png
(4.6.6)
Uniform tiling 432-t2.png
{3,4}
Uniform tiling 432-t02.png
(3.4.4a.4)
Uniform tiling 432-t012.png
(4.6.8)
Spherical snub cube.png
(3.3.3a.3.4)
Icosahedral reflection domains.png
(5 3 2)
Uniform tiling 532-t0.png
{5,3}
Uniform tiling 532-t01.png
(3.10.10)
Uniform tiling 532-t1.png
(3.5.3.5)
Uniform tiling 532-t12.png
(5.6.6)
Uniform tiling 532-t2.png
{3,5}
Uniform tiling 532-t02.png
(3.4.5.4)
Uniform tiling 532-t012.png
(4.6.10)
Spherical snub dodecahedron.png
(3.3.3a.3.5)

Some overlapping spherical tilings (r = 2)[edit]

For a more complete list, including cases where r ≠ 2, see List of uniform polyhedra by Schwarz triangle.

Tilings are shown as polyhedra. Some of the forms are degenerate, given with brackets for vertex figures, with overlapping edges or verices.

(p q 2) Fund.
triangle
Parent Truncated Rectified Bitruncated Birectified
(dual)
Cantellated Omnitruncated
(Cantitruncated)
Snub
Wythoff symbol q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
Schläfli symbol \begin{Bmatrix} p , q \end{Bmatrix} t\begin{Bmatrix} p , q \end{Bmatrix} \begin{Bmatrix} p \\ q \end{Bmatrix} t\begin{Bmatrix} q , p \end{Bmatrix} \begin{Bmatrix} q , p \end{Bmatrix} r\begin{Bmatrix} p \\ q \end{Bmatrix} t\begin{Bmatrix} p \\ q \end{Bmatrix} s\begin{Bmatrix} p \\ q \end{Bmatrix}
{p,q} t{p,q} r{p,q} t{q,p} {q,p} rr{p,q} tr{p,q} sr{p,q}
t0{p,q} t0,1{p,q} t1{p,q} t1,2{p,q} t2{p,q} t0,2{p,q} t0,1,2{p,q}
Coxeter–Dynkin diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Vertex figure pq (q.2p.2p) (p.q.p.q) (p. 2q.2q) qp (p. 4.q.4) (4.2p.2q) (3.3.p. 3.q)
Icosahedral
(5/2 3 2)
  Great icosahedron.png
{3,5/2}
Great truncated icosahedron.png
(5/2.6.6)
Great icosidodecahedron.png
(3.5/2)2
Icosahedron.png
[3.10/2.10/2]
Great stellated dodecahedron.png
{5/2,3}
Cantellated great icosahedron.png
[3.4.5/2.4]
Omnitruncated great icosahedron.png
[4.10/2.6]
Great snub icosidodecahedron.png
(3.3.3.3.5/2)
Icosahedral
(5 5/2 2)
  Great dodecahedron.png
{5,5/2}
Great truncated dodecahedron.png
(5/2.10.10)
Dodecadodecahedron.png
(5/2.5)2
Dodecahedron.png
[5.10/2.10/2]
Small stellated dodecahedron.png
{5/2,5}
Cantellated great dodecahedron.png
(5/2.4.5.4)
Omnitruncated great dodecahedron.png
[4.10/2.10]
Snub dodecadodecahedron.png
(3.3.5/2.3.5)

Dihedral symmetry (q = r = 2)[edit]

Spherical tilings with dihedral symmetry exist for all p = 2, 3, 4, ... many with digon faces which become degenerate polyhedra. Two of the eight forms (Rectified and cantellated) are replications and are skipped in the table.

(p 2 2)
Fundamental
domain
Parent Truncated Bitruncated
(truncated dual)
Birectified
(dual)
Omnitruncated
(Cantitruncated)
Snub
Extended
Schläfli symbol
\begin{Bmatrix} p , 2 \end{Bmatrix} t\begin{Bmatrix} p , 2 \end{Bmatrix} t\begin{Bmatrix} 2 , p \end{Bmatrix} \begin{Bmatrix} 2 , p \end{Bmatrix} t\begin{Bmatrix} p \\ 2 \end{Bmatrix} s\begin{Bmatrix} p \\ 2 \end{Bmatrix}
{p,2} t{p,2} t{2,p} {2,p} tr{p,2} sr{p,2}
t0{p,2} t0,1{p,2} t1,2{p,2} t2{p,2} t0,1,2{p,2}
Wythoff symbol 2 | p 2 2 2 | p 2 p | 2 p | 2 2 p 2 2 | | p 2 2
Coxeter–Dynkin diagram CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node h.pngCDel 2.pngCDel node h.png
Vertex figure (2.2p.2p) (4.4.p) 2p (4.2p.4) (3.3.p. 3)
Spherical square bipyramid2.png
(2 2 2)
V2.2.2
Sphere symmetry group cs.png
{2,2}
2.4.4 4.4.2 Sphere symmetry group cs.png
{2,2}
Spherical square prism2.png
4.4.4
Spherical digonal antiprism.png
3.3.3.2
Spherical hexagonal bipyramid2.png
(3 2 2)
V3.2.2
Trigonal dihedron.png
{3,2}
Hexagonal dihedron.png
2.6.6
Spherical triangular prism.png
4.4.3
Triangular hosohedron.png
{2,3}
Spherical hexagonal prism2.png
4.4.6
Spherical trigonal antiprism.png
3.3.3.3
Spherical octagonal bipyramid2.png
(4 2 2)
V4.2.2
{4,2} 2.8.8 Spherical square prism.png
4.4.4
Spherical square hosohedron.png
{2,4}
Spherical octagonal prism2.png
4.4.8
Spherical square antiprism.png
3.3.3.4
Spherical decagonal bipyramid2.png
(5 2 2)
V5.2.2
{5,2} 2.10.10 Spherical pentagonal prism.png
4.4.5
Spherical pentagonal hosohedron.png
{2,5}
Spherical decagonal prism2.png
4.4.10
Spherical pentagonal antiprism.png
3.3.3.5
Spherical dodecagonal bipyramid2.png
(6 2 2)
V6.2.2
Hexagonal dihedron.png
{6,2}
2.12.12 Spherical hexagonal prism.png
4.4.6
Spherical hexagonal hosohedron.png
{2,6}
Spherical dodecagonal prism2.png
4.4.12
Spherical hexagonal antiprism.png
3.3.3.6
...

Euclidean and hyperbolic tilings (r = 2)[edit]

Some representative hyperbolic tilings are given, and shown as a Poincaré disk projection.

(p q 2) Fund.
triangles
Parent Truncated Rectified Bitruncated Birectified
(dual)
Cantellated Omnitruncated
(Cantitruncated)
Snub
Wythoff symbol q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
Schläfli symbol \begin{Bmatrix} p , q \end{Bmatrix} t\begin{Bmatrix} p , q \end{Bmatrix} \begin{Bmatrix} p \\ q \end{Bmatrix} t\begin{Bmatrix} q , p \end{Bmatrix} \begin{Bmatrix} q , p \end{Bmatrix} r\begin{Bmatrix} p \\ q \end{Bmatrix} t\begin{Bmatrix} p \\ q \end{Bmatrix} s\begin{Bmatrix} p \\ q \end{Bmatrix}
{p,q} t{p,q} r{p,q} t{q,p} {q,p} rr{p,q} tr{p,q} sr{p,q}
t0{p,q} t0,1{p,q} t1{p,q} t1,2{p,q} t2{p,q} t0,2{p,q} t0,1,2{p,q}
Coxeter–Dynkin diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Vertex figure pq (q.2p.2p) (p.q.p.q) (p. 2q.2q) qp (p. 4.q.4) (4.2p.2q) (3.3.p. 3.q)
Hexagonal tiling
(6 3 2)
Tile V46b.svg
V4.6.12
Uniform tiling 63-t0.png
{6,3}
Uniform tiling 63-t01.png
3.12.12
Uniform tiling 63-t1.png
3.6.3.6
Uniform tiling 63-t12.png
6.6.6
Uniform tiling 63-t2.png
{3,6}
Uniform tiling 63-t02.png
3.4.6.4
Uniform tiling 63-t012.png
4.6.12
Uniform tiling 63-snub.png
3.3.3.3.6
(Hyperbolic plane)
(7 3 2)
Hyperbolic domains 732.png
V4.6.14
Uniform tiling 73-t0.png
{7,3}
Uniform tiling 73-t01.png
3.14.14
Uniform tiling 73-t1.png
3.7.3.7
Uniform tiling 73-t12.png
7.6.6
Uniform tiling 73-t2.png
{3,7}
Uniform tiling 73-t02.png
3.4.7.4
Uniform tiling 73-t012.png
4.6.14
Uniform tiling 73-snub.png
3.3.3.3.7
(Hyperbolic plane)
(8 3 2)
Hyperbolic domains 832.png
V4.6.16
Uniform tiling 83-t0.png
{8,3}
Uniform tiling 83-t01.png
3.16.16
Uniform tiling 83-t1.png
3.8.3.8
Uniform tiling 83-t12.png
8.6.6
Uniform tiling 83-t2.png
{3,8}
Uniform tiling 83-t02.png
3.4.8.4
Uniform tiling 83-t012.png
4.6.16
Uniform tiling 83-snub.png
3.3.3.3.8
Square tiling
(4 4 2)
Tiling Dual Semiregular V4-8-8 Tetrakis Square-2-color-zoom.svg
V4.8.8
Uniform tiling 44-t0.png
{4,4}
Uniform tiling 44-t01.png
4.8.8
Uniform tiling 44-t1.png
4.4a.4.4a
Uniform tiling 44-t12.png
4.8.8
Uniform tiling 44-t2.png
{4,4}
Uniform tiling 44-t02.png
4.4a.4b.4a
Uniform tiling 44-t012.png
4.8.8
Uniform tiling 44-snub.png
3.3.4a.3.4b
(Hyperbolic plane)
(5 4 2)
Hyperbolic domains 542.png
V4.8.10
Uniform tiling 54-t0.png
{5,4}
Uniform tiling 54-t01.png
4.10.10
Uniform tiling 54-t1.png
4.5.4.5
Uniform tiling 54-t12.png
5.8.8
Uniform tiling 54-t2.png
{4,5}
Uniform tiling 54-t02.png
4.4.5.4
Uniform tiling 54-t012.png
4.8.10
Uniform tiling 54-snub.png
3.3.4.3.5
(Hyperbolic plane)
(6 4 2)
Hyperbolic domains 642.png
V4.8.12
Uniform tiling 64-t0.png
{6,4}
Uniform tiling 64-t01.png
4.12.12
Uniform tiling 64-t1.png
4.6.4.6
Uniform tiling 64-t12.png
6.8.8
Uniform tiling 64-t2.png
{4,6}
Uniform tiling 64-t02.png
4.4.6.4
Uniform tiling 64-t012.png
4.8.12
Uniform tiling 64-snub.png
3.3.4.3.6
(Hyperbolic plane)
(7 4 2)
Hyperbolic domains 742.png
V4.8.14
Uniform tiling 74-t0.png
{7,4}
Uniform tiling 74-t01.png
4.14.14
Uniform tiling 74-t1.png
4.7.4.7
Uniform tiling 74-t12.png
7.8.8
Uniform tiling 74-t2.png
{4,7}
Uniform tiling 74-t02.png
4.4.7.4
Uniform tiling 74-t012.png
4.8.14
Uniform tiling 74-snub.png
3.3.4.3.7
(Hyperbolic plane)
(8 4 2)
Hyperbolic domains 842.png
V4.8.16
Uniform tiling 84-t0.png
{8,4}
Uniform tiling 84-t01.png
4.16.16
Uniform tiling 84-t1.png
4.8.4.8
Uniform tiling 84-t12.png
8.8.8
Uniform tiling 84-t2.png
{4,8}
Uniform tiling 84-t02.png
4.4.8.4
Uniform tiling 84-t012.png
4.8.16
Uniform tiling 84-snub.png
3.3.4.3.8
(Hyperbolic plane)
(5 5 2)
Hyperbolic domains 552.png
V4.10.10
Uniform tiling 552-t0.png
{5,5}
Uniform tiling 552-t01.png
5.10.10
Uniform tiling 552-t1.png
5.5.5.5
Uniform tiling 552-t12.png
5.10.10
Uniform tiling 552-t2.png
{5,5}
Uniform tiling 552-t02.png
5.4.5.4
Uniform tiling 552-t012.png
4.10.10
Uniform tiling 552-snub.png
3.3.5.3.5
(Hyperbolic plane)
(6 5 2)
Hyperbolic domains 652.png
V4.10.12
H2 tiling 256-1.png
{6,5}
H2 tiling 256-3.png
5.12.12
H2 tiling 256-2.png
5.6.5.6
H2 tiling 256-6.png
6.10.10
H2 tiling 256-4.png
{5,6}
H2 tiling 256-5.png
5.4.6.4
H2 tiling 256-7.png
4.10.12
Uniform tiling 65-snub.png
3.3.5.3.6
(Hyperbolic plane)
(7 5 2)
Hyperbolic domains 752.png
V4.10.14
H2 tiling 257-1.png
{7,5}
H2 tiling 257-3.png
5.14.14
H2 tiling 257-2.png
5.7.5.7
H2 tiling 257-6.png
7.10.10
H2 tiling 257-4.png
{5,7}
H2 tiling 257-5.png
5.4.7.4
H2 tiling 257-7.png
4.10.14
Uniform tiling 75-snub.png
3.3.5.3.7
(Hyperbolic plane)
(8 5 2)
Hyperbolic domains 852.png
V4.10.16
H2 tiling 258-1.png
{8,5}
H2 tiling 258-3.png
5.16.16
H2 tiling 258-2.png
5.8.5.8
H2 tiling 258-6.png
8.10.10
H2 tiling 258-4.png
{5,8}
H2 tiling 258-5.png
5.4.8.4
H2 tiling 258-7.png
4.10.16
3.3.5.3.8
(Hyperbolic plane)
(6 6 2)
Hyperbolic domains 662.png
V4.12.12
Uniform tiling 66-t2.png
{6,6}
Uniform tiling 66-t12.png
6.12.12
Uniform tiling 66-t1.png
6.6.6.6
Uniform tiling 66-t01.png
6.12.12
Uniform tiling 66-t0.png
{6,6}
Uniform tiling 66-t02.png
6.4.6.4
Uniform tiling 66-t012.png
4.12.12
Uniform tiling 66-snub.png
3.3.6.3.6
(Hyperbolic plane)
(7 6 2)
Hyperbolic domains 762.png
V4.12.14
H2 tiling 267-1.png
{7,6}
H2 tiling 267-3.png
6.14.14
H2 tiling 267-2.png
6.7.6.7
H2 tiling 267-6.png
7.12.12
H2 tiling 267-4.png
{6,7}
H2 tiling 267-5.png
6.4.7.4
H2 tiling 267-7.png
4.12.14
3.3.6.3.7
(Hyperbolic plane)
(8 6 2)
Hyperbolic domains 862.png
V4.12.16
H2 tiling 268-1.png
{8,6}
H2 tiling 268-3.png
6.16.16
H2 tiling 268-2.png
6.8.6.8
H2 tiling 268-6.png
8.12.12
H2 tiling 268-4.png
{6,8}
H2 tiling 268-5.png
6.4.8.4
H2 tiling 268-7.png
4.12.16
Uniform tiling 86-snub.png
3.3.6.3.8
(Hyperbolic plane)
(7 7 2)
Hyperbolic domains 772.png
V4.14.14
Uniform tiling 77-t2.png
{7,7}
Uniform tiling 77-t12.png
7.14.14
Uniform tiling 77-t1.png
7.7.7.7
Uniform tiling 77-t01.png
7.14.14
Uniform tiling 77-t0.png
{7,7}
Uniform tiling 77-t02.png
7.4.7.4
Uniform tiling 77-t012.png
4.14.14
Uniform tiling 77-snub.png
3.3.7.3.7
(Hyperbolic plane)
(8 7 2)
Hyperbolic domains 872.png
V4.14.16
H2 tiling 278-1.png
{8,7}
H2 tiling 278-3.png
7.16.16
H2 tiling 278-2.png
7.8.7.8
H2 tiling 278-6.png
8.14.14
H2 tiling 278-4.png
{7,8}
H2 tiling 278-5.png
7.4.8.4
H2 tiling 278-7.png
4.14.16
3.3.7.3.8
(Hyperbolic plane)
(8 8 2)
Hyperbolic domains 882.png
V4.16.16
Uniform tiling 88-t2.png
{8,8}
Uniform tiling 88-t12.png
8.16.16
Uniform tiling 88-t1.png
8.8.8.8
Uniform tiling 88-t01.png
8.16.16
Uniform tiling 88-t0.png
{8,8}
Uniform tiling 88-t02.png
8.4.8.4
Uniform tiling 88-t012.png
4.16.16
Uniform tiling 88-snub.png
3.3.8.3.8
(Hyperbolic plane)
(∞ 3 2)
H2checkers 23i.png
V4.6.∞
H2 tiling 23i-1.png
{∞,3}
H2 tiling 23i-3.png
3.∞.∞
H2 tiling 23i-2.png
3.∞.3.∞
H2 tiling 23i-6.png
∞.6.6
H2 tiling 23i-4.png
{3,∞}
H2 tiling 23i-5.png
3.4.∞.4
H2 tiling 23i-7.png
4.6.∞
Uniform tiling i32-snub.png
3.3.3.3.∞
(Hyperbolic plane)
(∞ 4 2)
H2checkers 24i.png
V4.8.∞
H2 tiling 24i-1.png
{∞,4}
H2 tiling 24i-3.png
4.∞.∞
H2 tiling 24i-2.png
4.∞.4.∞
H2 tiling 24i-6.png
∞.8.8
H2 tiling 24i-4.png
{4,∞}
H2 tiling 24i-5.png
4.4.∞.4
H2 tiling 24i-7.png
4.8.∞
Uniform tiling i42-snub.png
3.3.4.3.∞
(Hyperbolic plane)
(∞ 5 2)
H2checkers 25i.png
V4.10.∞
H2 tiling 25i-1.png
{∞,5}
H2 tiling 25i-3.png
5.∞.∞
H2 tiling 25i-2.png
5.∞.5.∞
H2 tiling 25i-6.png
∞.10.10
H2 tiling 25i-4.png
{5,∞}
H2 tiling 25i-5.png
5.4.∞.4
H2 tiling 25i-7.png
4.10.∞
Uniform tiling i52-snub.png
3.3.5.3.∞
(Hyperbolic plane)
(∞ 6 2)
H2checkers 26i.png
V4.12.∞
H2 tiling 26i-1.png
{∞,6}
H2 tiling 26i-3.png
6.∞.∞
H2 tiling 26i-2.png
6.∞.6.∞
H2 tiling 26i-6.png
∞.12.12
H2 tiling 26i-4.png
{6,∞}
H2 tiling 26i-5.png
6.4.∞.4
H2 tiling 26i-7.png
4.12.∞
Uniform tiling i62-snub.png
3.3.6.3.∞
(Hyperbolic plane)
(∞ 7 2)
H2checkers 27i.png
V4.14.∞
H2 tiling 27i-1.png
{∞,7}
H2 tiling 27i-3.png
7.∞.∞
H2 tiling 27i-2.png
7.∞.7.∞
H2 tiling 27i-6.png
∞.14.14
H2 tiling 27i-4.png
{7,∞}
H2 tiling 27i-5.png
7.4.∞.4
H2 tiling 27i-7.png
4.14.∞
3.3.7.3.∞
(Hyperbolic plane)
(∞ 8 2)
H2checkers 28i.png
V4.16.∞
H2 tiling 28i-1.png
{∞,8}
H2 tiling 28i-3.png
8.∞.∞
H2 tiling 28i-2.png
8.∞.8.∞
H2 tiling 28i-6.png
∞.16.16
H2 tiling 28i-4.png
{8,∞}
H2 tiling 28i-5.png
8.4.∞.4
H2 tiling 28i-7.png
4.16.∞
3.3.8.3.∞
(Hyperbolic plane)
(∞ ∞ 2)
H2checkers 2ii.png
V4.∞.∞
H2 tiling 2ii-1.png
{∞,∞}
H2 tiling 2ii-3.png
∞.∞.∞
H2 tiling 2ii-2.png
∞.∞.∞.∞
H2 tiling 2ii-6.png
∞.∞.∞
H2 tiling 2ii-4.png
{∞,∞}
H2 tiling 2ii-5.png
∞.4.∞.4
H2 tiling 2ii-7.png
4.∞.∞
Uniform tiling ii2-snub.png
3.3.∞.3.∞

Euclidean and hyperbolic tilings (r > 2)[edit]

The Coxeter–Dynkin diagram is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.

Wythoff symbol
(p q r)
Fund.
triangles
q | p r r q | p r | p q r p | q p | q r p q | r p q r | | p q r
Schläfli symbol (p,q,r) r(r,q,p) (q,r,p) r(p,q,r) (q,p,r) r(p,r,q) tr(p,q,r) s(p,q,r)
t0(p,q,r) t0,1(p,q,r) t1(p,q,r) t1,2(p,q,r) t2(p,q,r) t0,2(p,q,r) t0,1,2(p,q,r)
Coxeter diagram CDel 3.pngCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.png CDel 3.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.png CDel 3.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.png CDel 3.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.png CDel 3.pngCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.png CDel 3.pngCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.png CDel 3.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.png CDel 3.pngCDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.png
Vertex figure (p.r)q (r.2p.q.2p) (p.q)r (q.2r.p. 2r) (q.r)p (p. 2r.q.2r) (2p.2q.2r) (3.r.3.q.3.p)
Euclidean
(3 3 3)
CDel branch.pngCDel split2.pngCDel node.png
Tile 3,6.svg
V6.6.6
Uniform tiling 333-t0.png
(3.3)3
Uniform tiling 333-t01.png
3.6.3.6
Uniform tiling 333-t1.png
(3.3)3
Uniform tiling 333-t12.png
3.6.3.6
Uniform tiling 333-t2.png
(3.3)3
Uniform tiling 333-t02.png
3.6.3.6
Uniform tiling 333-t012.png
6.6.6
Uniform tiling 333-snub.png
3.3.3.3.3.3
Hyperbolic
(4 3 3)
CDel label4.pngCDel branch.pngCDel split2.pngCDel node.png
Hyperbolic domains 433.png
V6.6.8
Uniform tiling 433-t0.png
(3.4)3
Uniform tiling 433-t01.png
3.8.3.8
Uniform tiling 433-t1.png
(3.4)3
Uniform tiling 433-t12.png
3.6.4.6
Uniform tiling 433-t2.png
(3.3)4
Uniform tiling 433-t02.png
3.6.4.6
Uniform tiling 433-t012.png
6.6.8
Uniform tiling 433-snub2.png
3.3.3.3.3.4
Hyperbolic
(4 4 3)
CDel branch.pngCDel split2-44.pngCDel node.png
Hyperbolic domains 443.png
V6.8.8
Uniform tiling 443-t0.png
(3.4)4
Uniform tiling 443-t01.png
3.8.4.8
Uniform tiling 443-t1.png
(4.4)3
Uniform tiling 443-t12.png
3.8.4.8
Uniform tiling 443-t2.png
(3.4)4
Uniform tiling 443-t02.png
4.6.4.6
Uniform tiling 443-t012.png
6.8.8
Uniform tiling 443-snub1.png
3.3.3.4.3.4
Hyperbolic
(4 4 4)
CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node.png
Hyperbolic domains 444.png
V8.8.8
Uniform tiling 444-t0.png
(4.4)4
Uniform tiling 444-t01.png
4.8.4.8
Uniform tiling 444-t1.png
(4.4)4
Uniform tiling 444-t12.png
4.8.4.8
Uniform tiling 444-t2.png
(4.4)4
Uniform tiling 444-t02.png
4.8.4.8
Uniform tiling 444-t012.png
8.8.8
Uniform tiling 444-snub.png
3.4.3.4.3.4
Hyperbolic
(5 3 3)
CDel label5.pngCDel branch.pngCDel split2.pngCDel node.png
Hyperbolic domains 533.png
V6.6.10
H2 tiling 335-1.png
(3.5)3
H2 tiling 335-3.png
3.10.3.10
H2 tiling 335-2.png
(3.5)3
H2 tiling 335-6.png
3.6.5.6
H2 tiling 335-4.png
(3.3)5
H2 tiling 335-5.png
3.6.5.6
H2 tiling 335-7.png
6.6.10
3.3.3.3.3.5
Hyperbolic
(5 4 3)
CDel label5.pngCDel branch.pngCDel split2-43.pngCDel node.png
Hyperbolic domains 543.png
V6.8.10
H2 tiling 345-1.png
(3.5)4
H2 tiling 345-3.png
3.10.4.10
H2 tiling 345-2.png
(4.5)3
H2 tiling 345-6.png
3.8.5.8
H2 tiling 345-4.png
(3.4)5
H2 tiling 345-5.png
4.6.5.6
H2 tiling 345-7.png
6.8.10
Uniform tiling 543-snub.png
3.5.3.4.3.3
Hyperbolic
(5 4 4)
CDel label5.pngCDel branch.pngCDel split2-44.pngCDel node.png
Hyperbolic domains 544.png
V8.8.10
H2 tiling 445-1.png
(4.5)4
H2 tiling 445-3.png
4.10.4.10
H2 tiling 445-2.png
(4.5)4
H2 tiling 445-6.png
4.8.5.8
H2 tiling 445-4.png
(4.4)5
H2 tiling 445-5.png
4.8.5.8
H2 tiling 445-7.png
8.8.10
3.4.3.4.3.5
Hyperbolic
(6 3 3)
CDel label6.pngCDel branch.pngCDel split2.pngCDel node.png
Hyperbolic domains 633.png
V6.6.12
H2 tiling 336-1.png
(3.6)3
H2 tiling 336-3.png
3.12.3.12
H2 tiling 336-2.png
(3.6)3
H2 tiling 336-6.png
3.6.6.6
H2 tiling 336-4.png
(3.3)6
H2 tiling 336-5.png
3.6.6.6
H2 tiling 336-7.png
6.6.12
3.3.3.3.3.6
Hyperbolic
(6 4 3)
CDel label6.pngCDel branch.pngCDel split2-43.pngCDel node.png
Hyperbolic domains 643.png
V6.8.12
H2 tiling 346-1.png
(3.6)4
H2 tiling 346-3.png
3.12.4.12
H2 tiling 346-2.png
(4.6)3
H2 tiling 346-6.png
3.8.6.8
H2 tiling 346-4.png
(3.4)6
H2 tiling 346-5.png
4.6.6.6
H2 tiling 346-7.png
6.8.12
3.6.3.4.3.3
Hyperbolic
(6 4 4)
CDel label6.pngCDel branch.pngCDel split2-44.pngCDel node.png
Hyperbolic domains 644.png
V8.8.12
H2 tiling 446-1.png
(4.6)4
H2 tiling 446-3.png
4.12.4.12
H2 tiling 446-2.png
(4.6)4
H2 tiling 446-6.png
4.8.6.8
H2 tiling 446-4.png
(4.4)6
H2 tiling 446-5.png
4.8.6.8
H2 tiling 446-7.png
8.8.12
3.6.3.4.3.4
Hyperbolic
(∞ 3 3)
CDel labelinfin.pngCDel branch.pngCDel split2.pngCDel node.png
H2checkers 33i.png
V6.6.∞
H2 tiling 33i-1.png
(3.∞)3
H2 tiling 33i-3.png
3.∞.3.∞
H2 tiling 33i-2.png
(3.∞)3
H2 tiling 33i-6.png
3.6.∞.6
H2 tiling 33i-4.png
(3.3)
H2 tiling 33i-5.png
3.6.∞.6
H2 tiling 33i-7.png
6.6.∞
3.3.3.3.3.∞
Hyperbolic
(∞ 4 3)
CDel labelinfin.pngCDel branch.pngCDel split2-43.pngCDel node.png
H2checkers 34i.png
V6.8.∞
H2 tiling 34i-1.png
(3.∞)4
H2 tiling 34i-3.png
3.∞.4.∞
H2 tiling 34i-2.png
(4.∞)3
H2 tiling 34i-6.png
3.8.∞.8
H2 tiling 34i-4.png
(3.4)
H2 tiling 34i-5.png
4.6.∞.6
H2 tiling 34i-7.png
6.8.∞
3.∞.3.4.3.3
Hyperbolic
(∞ 4 4)
CDel labelinfin.pngCDel branch.pngCDel split2-44.pngCDel node.png
H2checkers 44i.png
V8.8.∞
H2 tiling 44i-1.png
(4.∞)4
H2 tiling 44i-3.png
4.∞.4.∞
H2 tiling 44i-2.png
(4.∞)4
H2 tiling 44i-6.png
4.8.∞.8
H2 tiling 44i-4.png
(4.4)
H2 tiling 44i-5.png
4.8.∞.8
H2 tiling 44i-7.png
8.8.∞
3.∞.3.4.3.4
Hyperbolic
(∞ ∞ 3)
CDel branch.pngCDel split2-ii.pngCDel node.png
H2checkers 3ii.png
V6.∞.∞
H2 tiling 3ii-1.png
(3.∞)
H2 tiling 3ii-3.png
3.∞.∞.∞
H2 tiling 3ii-2.png
(∞.∞)3
H2 tiling 3ii-6.png
3.∞.∞.∞
H2 tiling 3ii-4.png
(3.∞)
H2 tiling 3ii-5.png
∞.6.∞.6
H2 tiling 3ii-7.png
6.∞.∞
3.∞.3.∞.3.3
Hyperbolic
(∞ ∞ 4)
CDel label4.pngCDel branch.pngCDel split2-ii.pngCDel node.png
H2checkers 4ii.png
V8.∞.∞
H2 tiling 4ii-1.png
(4.∞)
H2 tiling 4ii-3.png
4.∞.∞.∞
H2 tiling 4ii-2.png
(∞.∞)4
H2 tiling 4ii-6.png
4.∞.∞.∞
H2 tiling 4ii-4.png
(4.∞)
H2 tiling 4ii-5.png
∞.8.∞.8
H2 tiling 4ii-7.png
8.∞.∞
3.∞.3.∞.3.4
Hyperbolic
(∞ ∞ ∞)
CDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node.png
H2checkers iii.png
V∞.∞.∞
H2 tiling iii-1.png
(∞.∞)
H2 tiling iii-3.png
∞.∞.∞.∞
H2 tiling iii-2.png
(∞.∞)
H2 tiling iii-6.png
∞.∞.∞.∞
H2 tiling iii-4.png
(∞.∞)
H2 tiling iii-5.png
∞.∞.∞.∞
H2 tiling iii-7.png
∞.∞.∞
Uniform tiling iii-snub.png
3.∞.3.∞.3.∞

See also[edit]

References[edit]

External links[edit]