English FAN: Lists of uniform tilings on the sphere, plane, and hyperbolic plane

Tuesday, July 2, 2019

Lists of uniform tilings on the sphere, plane, and hyperbolic plane

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Lists of uniform tilings on the sphere, plane, and hyperbolic plane

Tomruen: start, moved contents from [Wythoff symbol]]

In [[geometry]], many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by [[Wythoff construction]] within a fundamental triangle, (p q r), defined by internal angles as &pi/p, &pi/q, and &pi/r.

There are seven generator points with each set of <math>p, q, r</math> (and a few special forms):
{| class="wikitable"
!colspan=4|General
!colspan=5|Right triangle (r=2)
|-
!Description
!Wythoff symbol
![[Vertex configuration|Vertex configuration]]
![[Coxeter-Dynkin diagram|Coxeter diagram]] 
!Wythoff symbol
!Vertex configuration
!colspan=2|[[Schläfli symbol|Schläfli symbol]]
!Coxeter diagram 
|- align=center
|rowspan=3|[[Regular polyhedron|regular]] and [[Quasiregular polyhedron|quasiregular]]
|
|
|
|
|
|colspan=2| {p,q}
|
|- align=center
|
|
|
|
|
|colspan=2|{q,p}
|
|- align=center
|
|
|
|
|
|r{p,q}||t1{p,q}
|
|- align=center
|rowspan=3|[[Truncation (geometry)|truncated]] and [[Expansion (geometry)|expanded]]
|
|
|
|
|
|t{p,q}|| t0,1{p,q}
|
|- align=center
|
|
|
|
|
|t{q,p}|| t0,1{q,p}
|
|- align=center
|
|
|
|
|
| rr{p,q}|| t0,2{p,q}
|
|- align=center
|rowspan=2| [[Zonohedron|even-faced]]
|
|
|
|
|
| tr{p,q}||t0,1,2{p,q}
|
|- align=center
|
|
| -
|
|
|colspan=2|
| -
|- align=center
|rowspan=2| [[Snub (geometry)|snub]]
|
|
|
|
|
|colspan=2| sr{p,q}
|
|- align=center
|
|
| -
| -
| -
|colspan=2|
| -
|}

There are three special cases:
* <math>p\ q\ (r\ s)\ |</math> – This is a mixture of <math>p\ q\ r\ |</math> and <math>p\ q\ s\ |</math>, containing only the faces shared by both.
* <math>|\ p\ q\ r</math> – Snub forms (alternated) are given by this otherwise unused symbol.
* <math>|\ p\ q\ r\ s</math> – A unique snub form for [[Great dirhombicosidodecahedron|U75]] that isn't Wythoff-constructible.

== Symmetry triangles ==

There are 4 symmetry classes of reflection on the [[sphere]], and three in the [[Euclidean plane]]. A few of the [[List of regular polytopes#Hyperbolic tilings|infinitely many]] such patterns in the [[Hyperbolic space|hyperbolic plane]] are also listed. (Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.)

Point groups:
* (p 2 2) [[Dihedral symmetry in three dimensions|dihedral symmetry]], <math>p = 2, 3, 4 \dots </math> (order <math>4p</math>)
* (3 3 2) [[tetrahedral symmetry]] (order 24)
* (4 3 2) [[octahedral symmetry]] (order 48)
* (5 3 2) [[icosahedral symmetry]] (order 120)
Euclidean (affine) groups:
* (4 4 2) [[Square tiling|*442 symmetry]]: 45°-45°-90° triangle
* (6 3 2) *[[632 symmetry]]: 30°-60°-90° triangle
* (3 3 3) *[[triangular tiling|333 symmetry]]: 60°-60°-60° triangle
Hyperbolic groups:
* (7 3 2) *[[732 symmetry]]
* (8 3 2) *[[832 symmetry]]
* (4 3 3) *[[433 symmetry]]
* (4 4 3) *[[443 symmetry]]
* (4 4 4) *[[444 symmetry]]
* (5 4 2) *[[542 symmetry]]
* (6 4 2) *[[642 symmetry]]
* ...
{| class="wikitable"
!colspan=5|Dihedral spherical
!colspan=3|Spherical
|-
!D2h
!D3h
!D4h
!D5h
!D6h
!Td
!Oh
!Ih
|-
!*222
!*322
!*422
!*522
!*622
!*332
!*432
!*532
|- align=center
|[[Image:Spherical square bipyramid2.svg|100px]] (2 2 2)
|[[Image:Spherical hexagonal bipyramid2.png|100px]] (3 2 2)
|[[Image:Spherical octagonal bipyramid2.png|100px]] (4 2 2)
|[[Image:Spherical decagonal bipyramid2.png|100px]] (5 2 2)
|[[Image:Spherical dodecagonal bipyramid2.png|100px]] (6 2 2)
|[[Image:Tetrahedral reflection domains.png|100px]] (3 3 2)
|[[Image:Octahedral reflection domains.png|100px]] (4 3 2)
|[[Image:Icosahedral reflection domains.png|100px]] (5 3 2)
|}

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

{| class="wikitable"
|+ Euclidean plane
|-
!p4m
!p3m
!p6m
|-
!*442
!*333
!*632
|-
|[[Image:Tile V488 bicolor.svg|200px]] (4 4 2)
|[[Image:Tile 3,6.svg|200px]] (3 3 3)
|[[Image:Tile V46b.svg|200px]] (6 3 2)
|}

{| class="wikitable"
|+ Hyperbolic plane
|-
!*732
!*542
!*433
|-
|[[File:3-7 kisrhombille.svg|200px]] (7 3 2)
|[[Image:Order-4 bisected pentagonal tiling.png|200px]] (5 4 2)
|[[Image:Uniform dual tiling 433-t012.png|200px]] (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 ==

Selected tilings created by the Wythoff construction are given below.

=== Spherical tilings (''r'' = 2) ===
{| class="wikitable"
|-
!(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
|-
!rowspan=3|[[Schläfli symbol|Schläfli symbol]]
!<math>\begin{Bmatrix} p , q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p , q \end{Bmatrix}</math>
!<math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} q , p \end{Bmatrix}</math>
!<math>\begin{Bmatrix} q , p \end{Bmatrix}</math>
!<math>r\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>s\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
|-
!{p,q}
!t{p,q}
!r{p,q}
!t{q,p}
!{q,p}
!rr{p,q}
!tr{p,q}
!rowspan=2|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|Coxeter diagram]]
!
!
!
!
!
!
!
!
|-
![[Vertex configuration|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
|-
![[Image:Tetrahedral reflection domains.png|72px]] [[Tetrahedral symmetry|(3 3 2)]]
|[[Image:Uniform tiling 332-t0-1-.png|64px]] [[Tetrahedron|{3,3}]]
|[[File:Uniform tiling 332-t01-1-.png|64px]] [[Truncated tetrahedron|(3.6.6)]]
|[[Image:Uniform tiling 332-t1-1-.png|64px]] [[Octahedron|(3.3a.3.3a)]]
|[[Image:Uniform tiling 332-t12.png|64px]] [[Truncated tetrahedron|(3.6.6)]]
|[[Image:Uniform tiling 332-t2.png|64px]] [[Tetrahedron|{3,3}]]
| [[Image:Uniform tiling 332-t02.png|64px]] [[Cuboctahedron|(3a.4.3b.4)]]
|[[Image:Uniform tiling 332-t012.png|64px]] [[Truncated octahedron|(4.6a.6b)]]
|[[File:Spherical snub tetrahedron.png|64px]] [[Icosahedron|(3.3.3a.3.3b)]]
|-
![[Image:Octahedral reflection domains.png|73px]] [[Octahedral symmetry|(4 3 2)]]
|[[Image:Uniform tiling 432-t0.png|64px]] [[Cube|{4,3}]]
|[[Image:Uniform tiling 432-t01.png|64px]] [[Truncated cube|(3.8.8)]]
|[[Image:Uniform tiling 432-t1.png|64px]] [[Cuboctahedron|(3.4.3.4)]]
|[[Image:Uniform tiling 432-t12.png|64px]] [[Truncated octahedron|(4.6.6)]]
|[[Image:Uniform tiling 432-t2.png|64px]] [[Octahedron|{3,4}]]
|[[Image:Uniform tiling 432-t02.png|64px]] [[Rhombicuboctahedron|(3.4.4a.4)]]
|[[Image:Uniform tiling 432-t012.png|64px]] [[Truncated cuboctahedron|(4.6.8)]]
|[[File:Spherical snub cube.png|65px]] [[Snub cube|(3.3.3a.3.4)]]
|-
![[Image:Icosahedral reflection domains.png|72px]] [[Icosahedral symmetry|(5 3 2)]]
|[[Image:Uniform tiling 532-t0.png|64px]] [[Dodecahedron|{5,3}]]
|[[Image:Uniform tiling 532-t01.png|64px]] [[Truncated dodecahedron|(3.10.10)]]
|[[Image:Uniform tiling 532-t1.png|64px]] [[Icosidodecahedron|(3.5.3.5)]]
|[[Image:Uniform tiling 532-t12.png|64px]] [[Truncated icosahedron|(5.6.6)]]
|[[Image:Uniform tiling 532-t2.png|64px]] [[Icosahedron|{3,5}]]
|[[Image:Uniform tiling 532-t02.png|64px]] [[Rhombicosidodecahedron|(3.4.5.4)]]
|[[Image:Uniform tiling 532-t012.png|64px]] [[Truncated icosidodecahedron|(4.6.10)]]
|[[File:Spherical snub dodecahedron.png|64px]] [[Snub dodecahedron|(3.3.3a.3.5)]]
|}

==== Some overlapping spherical tilings (''r'' = 2) ====
:''For a more complete list, including cases where ''r'' ≠ 2, see [[List of uniform polyhedra by Schwarz triangle]].''

Tilings are shown as [[polyhedron|polyhedra]]. Some of the forms are degenerate, given with brackets for [[vertex figure]]s, with overlapping edges or vertices.
{| class="wikitable"
|-
!(p q 2)
!Fund. triangle
!Parent
!Truncated
!Rectified
!Bitruncated
!Birectified (dual)
!Cantellated
!Omnitruncated (Cantitruncated)
!Snub
|-
![[Wythoff construction|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
|-
!rowspan=3|[[Schläfli symbol]]
!rowspan=3|
!<math>\begin{Bmatrix} p , q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p , q \end{Bmatrix}</math>
!<math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} q , p \end{Bmatrix}</math>
!<math>\begin{Bmatrix} q , p \end{Bmatrix}</math>
!<math>r\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>s\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
|-
!{p,q}
!t{p,q}
!r{p,q}
!t{q,p}
!{q,p}
!rr{p,q}
!tr{p,q}
!rowspan=2|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]]
!
!
!
!
!
!
!
!
!
|-
![[Vertex configuration|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)
| 
|[[Image:Great icosahedron.png|64px]] [[Great icosahedron|{3,5/2}]]
|[[Image:Great truncated icosahedron.png|64px]] [[Truncated great icosahedron|(5/2.6.6)]]
|[[Image:Great icosidodecahedron.png|64px]] [[Great icosidodecahedron|(3.5/2)2]]
|[[Image:Icosahedron.png|64px]] [[Small complex icosidodecahedron|[3.10/2.10/2]]]
|[[Image:Great stellated dodecahedron.png|64px]] [[Great stellated dodecahedron|{5/2,3}]]
|[[Image:Cantellated great icosahedron.png|64px]] [[Small complex rhombicosidodecahedron|[3.4.5/2.4]]]
|[[Image:Omnitruncated great icosahedron.png|64px]] [[Rhombicosahedron|[4.10/2.6]]]
|[[Image:Great snub icosidodecahedron.png|64px]] [[Great snub icosidodecahedron|(3.3.3.3.5/2)]]
|-
|Icosahedral (5 5/2 2)
| 
|[[Image:Great dodecahedron.png|64px]] [[Great dodecahedron|{5,5/2}]]
|[[Image:Great truncated dodecahedron.png|64px]] [[Truncated great dodecahedron|(5/2.10.10)]]
|[[Image:Dodecadodecahedron.png|64px]] [[Dodecadodecahedron|(5/2.5)2]]
|[[Image:dodecahedron.png|64px]] [[dodecahedron|[5.10/2.10/2]]]
|[[Image:Small stellated dodecahedron.png|64px]] [[Small stellated dodecahedron|{5/2,5}]]
|[[Image:Cantellated great dodecahedron.png|64px]] [[Rhombidodecadodecahedron|(5/2.4.5.4)]]
|[[Image:Omnitruncated great dodecahedron.png|64px]] [[Small rhombidodecahedron|[4.10/2.10]]]
|[[Image:Snub dodecadodecahedron.png|64px]] [[Snub dodecadodecahedron|(3.3.5/2.3.5)]]
|}

==== Dihedral symmetry (''q'' = ''r'' = 2) ====

Spherical tilings with [[dihedral symmetry]] exist for all <math>p = 2, 3, 4, \dots </math> many with [[digon]] faces which become degenerate polyhedra. Two of the eight forms (Rectified and cantillated) are replications and are skipped in the table.

{| class="wikitable"
|-
!(p 2 2) Fundamental domain
!Parent
!Truncated
!Bitruncated (truncated dual)
!Birectified (dual)
!Omnitruncated (Cantitruncated)
!Snub
|-
!rowspan=3|Extended [[Schläfli symbol]]
!<math>\begin{Bmatrix} p , 2 \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p , 2 \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} 2 , p \end{Bmatrix}</math>
!<math>\begin{Bmatrix} 2 , p \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
!<math>s\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
|-
!{p,2}
!t{p,2}
!t{2,p}
!{2,p}
!tr{p,2}
!rowspan=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]]
!
!
!
!
!
!
|-
![[Vertex configuration|Vertex figure]]
!p²
!(2.2p.2p)
!(4.4.p)
!2p
!(4.2p.4)
!(3.3.p. 3)
|- align=center
| [[Image:Spherical square bipyramid2.png|64px]] (2 2 2) [[Octahedron|V2.2.2]]
| [[Image:Digonal dihedron.png|64px]] [[Dihedron|{2,2}]]
| [[Image:Tetragonal dihedron.png|64px]] [[Dihedron|2.4.4]]
| [[Dihedron|4.4.2]]
| [[Image:Digonal dihedron.png|64px]] [[Hosohedron|{2,2}]]
| [[Image:Spherical square prism2.png|64px]] [[Cube|4.4.4]]
| [[Image:Spherical digonal antiprism.png|64px]] [[Tetrahedron|3.3.3.2]]
|- align=center
| [[Image:Spherical hexagonal bipyramid2.png|64px]] (3 2 2) [[Hexagonal bipyramid|V3.2.2]]
| [[Image:Trigonal dihedron.png|64px]] [[Dihedron|{3,2}]]
| [[Image:Hexagonal dihedron.png|64px]] [[Dihedron|2.6.6]]
| [[Image:Spherical triangular prism.png|64px]] [[Triangular prism|4.4.3]]
| [[Image:Triangular hosohedron.png|64px]] [[Hosohedron|{2,3}]]
| [[Image:Spherical hexagonal prism2.png|64px]] [[Hexagonal prism|4.4.6]]
| [[Image:Spherical trigonal antiprism.png|64px]] [[Octahedron|3.3.3.3]]
|- align=center
| [[Image:Spherical octagonal bipyramid2.png|64px]] (4 2 2) [[Octagonal bipyramid|V4.2.2]]
| [[Image:Tetragonal dihedron.png|64px]] [[Dihedron|{4,2}]]
| [[Dihedron|2.8.8]]
| [[Image:Spherical square prism.png|64px]] 4.4.4
| [[Image:Spherical square hosohedron.png|64px]] [[Hosohedron|{2,4}]]
| [[Image:Spherical octagonal prism2.png|64px]] 4.4.8
| [[Image:Spherical square antiprism.png|64px]] 3.3.3.4
|- align=center
| [[Image:Spherical decagonal bipyramid2.png|64px]] (5 2 2) [[Decagonal bipyramid|V5.2.2]]
| [[Image:Pentagonal dihedron.png|64px]] [[Dihedron|{5,2}]]
| [[Dihedron|2.10.10]]
| [[Image:Spherical pentagonal prism.png|64px]] [[Pentagonal prism|4.4.5]]
| [[Image:Spherical pentagonal hosohedron.png|64px]] [[Hosohedron|{2,5}]]
| [[Image:Spherical decagonal prism2.png|64px]] [[Decagonal prism|4.4.10]]
| [[Image:Spherical pentagonal antiprism.png|64px]] [[Pentagonal antiprism|3.3.3.5]]
|- align=center
| [[Image:Spherical dodecagonal bipyramid2.png|64px]] (6 2 2) [[Dodecagonal bipyramid|V6.2.2]]
| [[Image:Hexagonal dihedron.png|64px]] [[Dihedron|{6,2}]]
| [[Image:Dodecagonal dihedron.png|64px]] [[Dihedron|2.12.12]]
| [[Image:Spherical hexagonal prism.png|64px]] [[Hexagonal prism|4.4.6]]
| [[Image:Spherical hexagonal hosohedron.png|64px]] [[Hosohedron|{2,6}]]
| [[Image:Spherical dodecagonal prism2.png|64px]] [[Dodecagonal prism|4.4.12]]
| [[Image:Spherical hexagonal antiprism.png|64px]] [[Hexagonal antiprism|3.3.3.6]]
|-
|colspan=10|...
|}

=== Euclidean and hyperbolic tilings (''r'' = 2) ===

Some representative hyperbolic tilings are given, and shown as a [[Poincaré disk model|Poincaré disk]] projection.

{| class="wikitable"
|-
!(p q 2)
!Fund. triangles
!Parent
!Truncated
!Rectified
!Bitruncated
!Birectified (dual)
!Cantellated
!Omnitruncated (Cantitruncated)
!Snub
|-
![[Wythoff construction|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
|-
!rowspan=3|[[Schläfli symbol]]
!rowspan=3|
!<math>\begin{Bmatrix} p , q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p , q \end{Bmatrix}</math>
!<math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} q , p \end{Bmatrix}</math>
!<math>\begin{Bmatrix} q , p \end{Bmatrix}</math>
!<math>r\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>t\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
!<math>s\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
|-
!{p,q}
!t{p,q}
!r{p,q}
!t{q,p}
!{q,p}
!rr{p,q}
!tr{p,q}
!rowspan=2|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]]
!
!
!
!
!
!
!
!
!
|-
![[Vertex configuration|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)
|- align=center
|[[Hexagonal tiling]] (6 3 2)
|[[Image:Tile V46b.svg|64px]] [[Bisected hexagonal tiling|V4.6.12]]
|[[Image:Uniform tiling 63-t0.png|64px]] [[Hexagonal tiling|{6,3}]]
|[[Image:Uniform tiling 63-t01.png|64px]] [[Truncated hexagonal tiling|3.12.12]]
|[[Image:Uniform tiling 63-t1.png|64px]] [[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 63-t12.png|64px]] [[Hexagonal tiling|6.6.6]]
|[[Image:Uniform tiling 63-t2.png|64px]] [[Triangular tiling|{3,6}]]
|[[Image:Uniform tiling 63-t02.png|64px]] [[Rhombitrihexagonal tiling|3.4.6.4]]
|[[Image:Uniform tiling 63-t012.svg|64px]] [[Truncated trihexagonal tiling|4.6.12]]
|[[Image:Uniform tiling 63-snub.png|65px]] [[Snub hexagonal tiling|3.3.3.3.6]]
|- align=center
|(Hyperbolic plane) (7 3 2)
|[[File:Hyperbolic domains 732.png|72px]] [[Order 3-7 kisrhombille|V4.6.14]]
|[[Image:Heptagonal tiling.svg|64px]] [[Heptagonal tiling|{7,3}]]
|[[Image:Truncated heptagonal tiling.svg|64px]] [[Truncated heptagonal tiling|3.14.14]]
|[[Image:Triheptagonal tiling.svg|64px]] [[Triheptagonal tiling|3.7.3.7]]
|[[Image:Truncated order-7 triangular tiling.svg|64px]] [[Truncated order-7 triangular tiling|7.6.6]]
|[[Image:Order-7 triangular tiling.svg|64px]] [[Order-7 triangular tiling|{3,7}]]
|[[Image:Uniform tiling 73-t02.png|64px]] [[Rhombitriheptagonal tiling|3.4.7.4]]
|[[Image:Uniform tiling 73-t012.png|64px]] [[Truncated triheptagonal tiling|4.6.14]]
|[[Image:Uniform tiling 73-snub.png|65px]] [[Snub heptagonal tiling|3.3.3.3.7]]
|- align=center
|(Hyperbolic plane) (8 3 2)
|[[File:Hyperbolic domains 832.png|72px]] [[Order 3-8 kisrhombille|V4.6.16]]
|[[Image:Uniform tiling 83-t0.png|64px]] [[Octagonal tiling|{8,3}]]
|[[Image:Uniform tiling 83-t01.png|64px]] [[Truncated octagonal tiling|3.16.16]]
|[[Image:Uniform tiling 83-t1.png|64px]] [[Trioctagonal tiling|3.8.3.8]]
|[[Image:Uniform tiling 83-t12.png|64px]] [[Truncated order-8 triangular tiling|8.6.6]]
|[[Image:Uniform tiling 83-t2.png|64px]] [[Order-8 triangular tiling|{3,8}]]
|[[Image:Uniform tiling 83-t02.png|64px]] [[Rhombitrioctagonal tiling|3.4.8.4]]
|[[Image:Uniform tiling 83-t012.png|64px]] [[Truncated trioctagonal tiling|4.6.16]]
|[[Image:Uniform tiling 83-snub.png|65px]] [[Snub octagonal tiling|3.3.3.3.8]]
|- align=center
|[[Square tiling]] (4 4 2)
|[[Image:Tiling Dual Semiregular V4-8-8 Tetrakis Square-2-color-zoom.svg|64px]] [[Tetrakis square tiling|V4.8.8]]
|[[Image:Uniform tiling 44-t0.svg|64px]] [[Square tiling|{4,4}]]
|[[Image:Uniform tiling 44-t01.png|64px]] [[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-t1.png|64px]] [[Square tiling|4.4a.4.4a]]
|[[Image:Uniform tiling 44-t12.png|64px]] [[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-t2.png|64px]] [[Square tiling|{4,4}]]
|[[Image:Uniform tiling 44-t02.png|64px]] [[Square tiling|4.4a.4b.4a]]
|[[Image:Uniform tiling 44-t012.png|64px]] [[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-snub.png|64px]] [[Snub square tiling|3.3.4a.3.4b]]
|- align=center
|(Hyperbolic plane) (5 4 2)
|[[File:Hyperbolic domains 542.png|72px]] V4.8.10
|[[Image:Uniform tiling 54-t0.png|64px]] [[Order-4 pentagonal tiling|{5,4}]]
|[[Image:Uniform tiling 54-t01.png|64px]] [[Truncated order-4 pentagonal tiling|4.10.10]]
|[[Image:Uniform tiling 54-t1.png|64px]] [[Tetrapentagonal tiling|4.5.4.5]]
|[[Image:Uniform tiling 54-t12.png|64px]] [[Truncated order-5 square tiling|5.8.8]]
|[[Image:Uniform tiling 54-t2.png|64px]] [[Order-5 square tiling|{4,5}]]
|[[Image:Uniform tiling 54-t02.png|64px]] [[Rhombitetrapentagonal tiling|4.4.5.4]]
|[[Image:Uniform tiling 54-t012.png|64px]] [[Truncated tetrapentagonal tiling|4.8.10]]
|[[Image:Uniform tiling 54-snub.png|64px]] [[Snub tetrapentagonal tiling|3.3.4.3.5]]
|- align=center
|(Hyperbolic plane) (6 4 2)
|[[File:Hyperbolic domains 642.png|72px]] V4.8.12
|[[Image:Uniform tiling 64-t0.png|64px]] [[Order-4 hexagonal tiling|{6,4}]]
|[[Image:Uniform tiling 64-t01.png|64px]] [[Truncated order-4 hexagonal tiling|4.12.12]]
|[[Image:Uniform tiling 64-t1.png|64px]] [[Tetrahexagonal tiling|4.6.4.6]]
|[[Image:Uniform tiling 64-t12.png|64px]] [[Truncated order-6 square tiling|6.8.8]]
|[[Image:Uniform tiling 64-t2.png|64px]] [[Order-6 square tiling|{4,6}]]
|[[Image:Uniform tiling 64-t02.png|64px]] [[Rhombitetrahexagonal tiling|4.4.6.4]]
|[[Image:Uniform tiling 64-t012.png|64px]] [[Truncated tetrahexagonal tiling|4.8.12]]
|[[Image:Uniform tiling 64-snub.png|64px]] [[Snub tetrahexagonal tiling|3.3.4.3.6]]
|- align=center
|(Hyperbolic plane) (7 4 2)
|[[File:Hyperbolic domains 742.png|72px]] V4.8.14
|[[Image:Uniform tiling 74-t0.png|64px]] [[Order-4 heptagonal tiling|{7,4}]]
|[[Image:Uniform tiling 74-t01.png|64px]] [[Truncated order-4 heptagonal tiling|4.14.14]]
|[[Image:Uniform tiling 74-t1.png|64px]] [[Tetraheptagonal tiling|4.7.4.7]]
|[[Image:Uniform tiling 74-t12.png|64px]] [[Truncated order-7 square tiling|7.8.8]]
|[[Image:Uniform tiling 74-t2.png|64px]] [[Order-7 square tiling|{4,7}]]
|[[Image:Uniform tiling 74-t02.png|64px]] [[Rhombitetraheptagonal tiling|4.4.7.4]]
|[[Image:Uniform tiling 74-t012.png|64px]] [[Truncated tetraheptagonal tiling|4.8.14]]
|[[Image:Uniform tiling 74-snub.png|64px]] [[Snub tetraheptagonal tiling|3.3.4.3.7]]
|- align=center
|(Hyperbolic plane) (8 4 2)
|[[File:Hyperbolic domains 842.png|72px]] V4.8.16
|[[Image:Uniform tiling 84-t0.png|64px]] [[Order-4 octagonal tiling|{8,4}]]
|[[Image:Uniform tiling 84-t01.png|64px]] [[Truncated order-4 octagonal tiling|4.16.16]]
|[[Image:Uniform tiling 84-t1.png|64px]] [[Tetraoctagonal tiling|4.8.4.8]]
|[[Image:Uniform tiling 84-t12.png|64px]] [[Octagonal tiling|8.8.8]]
|[[Image:Uniform tiling 84-t2.png|64px]] [[Order-8 square tiling|{4,8}]]
|[[Image:Uniform tiling 84-t02.png|64px]] [[Rhombitetraoctagonal tiling|4.4.8.4]]
|[[Image:Uniform tiling 84-t012.png|64px]] [[Truncated tetraoctagonal tiling|4.8.16]]
|[[Image:Uniform tiling 84-snub.png|64px]] [[Snub tetraoctagonal tiling|3.3.4.3.8]]
|- align=center
|(Hyperbolic plane) (5 5 2)
|[[File:Hyperbolic domains 552.png|72px]] V4.10.10
|[[Image:Uniform tiling 552-t0.png|64px]] [[Order-5 pentagonal tiling|{5,5}]]
|[[Image:Uniform tiling 552-t01.png|64px]] [[Truncated order-5 pentagonal tiling|5.10.10]]
|[[Image:Uniform tiling 552-t1.png|64px]] [[Order-4 pentagonal tiling|5.5.5.5]]
|[[Image:Uniform tiling 552-t12.png|64px]] [[Truncated order-5 pentagonal tiling|5.10.10]]
|[[Image:Uniform tiling 552-t2.png|64px]] [[Order-5 pentagonal tiling|{5,5}]]
|[[Image:Uniform tiling 552-t02.png|64px]] [[Tetrapentagonal tiling|5.4.5.4]]
|[[Image:Uniform tiling 552-t012.png|64px]] [[Truncated order-4 pentagonal tiling|4.10.10]]
|[[Image:Uniform tiling 552-snub.png|64px]] [[Snub pentapentagonal tiling|3.3.5.3.5]]
|- align=center
|(Hyperbolic plane) (6 5 2)
|[[File:Hyperbolic domains 652.png|72px]] V4.10.12
|[[File:H2 tiling 256-1.png|64px]] [[Order-5 hexagonal tiling|{6,5}]]
|[[File:H2 tiling 256-3.png|64px]] [[Truncated order-5 hexagonal tiling|5.12.12]]
|[[File:H2 tiling 256-2.png|64px]] [[Pentahexagonal tiling|5.6.5.6]]
|[[File:H2 tiling 256-6.png|64px]] [[Truncated order-6 pentagonal tiling|6.10.10]]
|[[File:H2 tiling 256-4.png|64px]] [[Order-6 pentagonal tiling|{5,6}]]
|[[File:H2 tiling 256-5.png|64px]] [[Rhombipentahexagonal tiling|5.4.6.4]]
|[[File:H2 tiling 256-7.png|64px]] [[Truncated pentahexagonal tiling|4.10.12]]
|[[File:Uniform tiling 65-snub.png|64px]] [[Snub pentahexagonal tiling|3.3.5.3.6]]
|- align=center
|(Hyperbolic plane) (7 5 2)
|[[File:Hyperbolic domains 752.png|72px]] V4.10.14
|[[File:H2 tiling 257-1.png|64px]] [[Order-5 heptagonal tiling|{7,5}]]
|[[File:H2 tiling 257-3.png|64px]] [[Truncated order-5 heptagonal tiling|5.14.14]]
|[[File:H2 tiling 257-2.png|64px]] [[Pentaheptagonal tiling|5.7.5.7]]
|[[File:H2 tiling 257-6.png|64px]] [[Truncated order-7 pentagonal tiling|7.10.10]]
|[[File:H2 tiling 257-4.png|64px]] [[Order-7 pentagonal tiling|{5,7}]]
|[[File:H2 tiling 257-5.png|64px]] [[Rhombipentaheptagonal tiling|5.4.7.4]]
|[[File:H2 tiling 257-7.png|64px]] [[Truncated pentaheptagonal tiling|4.10.14]]
|[[File:Uniform tiling 75-snub.png|64px]] [[Snub pentaheptagonal tiling|3.3.5.3.7]]
|- align=center
|(Hyperbolic plane) (8 5 2)
|[[File:Hyperbolic domains 852.png|72px]] V4.10.16
|[[File:H2 tiling 258-1.png|64px]] [[Order-5 octagonal tiling|{8,5}]]
|[[File:H2 tiling 258-3.png|64px]] [[Truncated order-5 octagonal tiling|5.16.16]]
|[[File:H2 tiling 258-2.png|64px]] [[Pentaoctagonal tiling|5.8.5.8]]
|[[File:H2 tiling 258-6.png|64px]] [[Truncated order-8 pentagonal tiling|8.10.10]]
|[[File:H2 tiling 258-4.png|64px]] [[Order-8 pentagonal tiling|{5,8}]]
|[[File:H2 tiling 258-5.png|64px]] [[Rhombipentaoctagonal tiling|5.4.8.4]]
|[[File:H2 tiling 258-7.png|64px]] [[Truncated pentaoctagonal tiling|4.10.16]]
|valign=bottom|[[Snub pentaoctagonal tiling|3.3.5.3.8]]
|- align=center
|(Hyperbolic plane) (6 6 2)
|[[File:Hyperbolic domains 662.png|72px]] V4.12.12
|[[Image:Uniform tiling 66-t2.png|64px]] [[Order-6 hexagonal tiling|{6,6}]]
|[[Image:Uniform tiling 66-t12.png|64px]] [[Truncated order-6 hexagonal tiling|6.12.12]]
|[[Image:Uniform tiling 66-t1.png|64px]] [[Order-4 hexagonal tiling|6.6.6.6]]
|[[Image:Uniform tiling 66-t01.png|64px]] [[Truncated order-6 hexagonal tiling|6.12.12]]
|[[Image:Uniform tiling 66-t0.png|64px]] [[Order-6 hexagonal tiling|{6,6}]]
|[[Image:Uniform tiling 66-t02.png|64px]] [[Tetrahexagonal tiling|6.4.6.4]]
|[[Image:Uniform tiling 66-t012.png|64px]] [[Truncated order-4 hexagonal tiling|4.12.12]]
|[[Image:Uniform tiling 66-snub.png|64px]] [[Snub hexahexagonal tiling|3.3.6.3.6]]
|- align=center
|(Hyperbolic plane) (7 6 2)
|[[File:Hyperbolic domains 762.png|72px]] V4.12.14
|[[File:H2 tiling 267-1.png|64px]] [[Order-6 heptagonal tiling|{7,6}]]
|[[File:H2 tiling 267-3.png|64px]] [[Truncated order-6 heptagonal tiling|6.14.14]]
|[[File:H2 tiling 267-2.png|64px]] [[Hexaheptagonal tiling|6.7.6.7]]
|[[File:H2 tiling 267-6.png|64px]] [[Truncated order-7 hexagonal tiling|7.12.12]]
|[[File:H2 tiling 267-4.png|64px]] [[Order-7 hexagonal tiling|{6,7}]]
|[[File:H2 tiling 267-5.png|64px]] [[Rhombihexaheptagonal tiling|6.4.7.4]]
|[[File:H2 tiling 267-7.png|64px]] [[Truncated hexaheptagonal tiling|4.12.14]]
|valign=bottom|[[Snub hexaheptagonal tiling|3.3.6.3.7]]
|- align=center
|(Hyperbolic plane) (8 6 2)
|[[File:Hyperbolic domains 862.png|72px]] V4.12.16
|[[File:H2 tiling 268-1.png|64px]] [[Order-6 octagonal tiling|{8,6}]]
|[[File:H2 tiling 268-3.png|64px]] [[Truncated order-6 octagonal tiling|6.16.16]]
|[[File:H2 tiling 268-2.png|64px]] [[Hexaoctagonal tiling|6.8.6.8]]
|[[File:H2 tiling 268-6.png|64px]] [[Truncated order-8 hexagonal tiling|8.12.12]]
|[[File:H2 tiling 268-4.png|64px]] [[Order-8 hexagonal tiling|{6,8}]]
|[[File:H2 tiling 268-5.png|64px]] [[Rhombihexaoctagonal tiling|6.4.8.4]]
|[[File:H2 tiling 268-7.png|64px]] [[Truncated hexaoctagonal tiling|4.12.16]]
|[[File:Uniform tiling 86-snub.png|64px]] [[Snub hexaoctagonal tiling|3.3.6.3.8]]
|- align=center
|(Hyperbolic plane) (7 7 2)
|[[File:Hyperbolic domains 772.png|72px]] V4.14.14
|[[Image:Uniform tiling 77-t2.png|64px]] [[Order-7 heptagonal tiling|{7,7}]]
|[[Image:Uniform tiling 77-t12.png|64px]] [[Truncated order-7 heptagonal tiling|7.14.14]]
|[[Image:Uniform tiling 77-t1.png|64px]] [[Order-4 heptagonal tiling|7.7.7.7]]
|[[Image:Uniform tiling 77-t01.png|64px]] [[Truncated order-7 heptagonal tiling|7.14.14]]
|[[Image:Uniform tiling 77-t0.png|64px]] [[Order-7 heptagonal tiling|{7,7}]]
|[[Image:Uniform tiling 77-t02.png|64px]] [[Tetraheptagonal tiling|7.4.7.4]]
|[[Image:Uniform tiling 77-t012.png|64px]] [[Truncated order-4 heptagonal tiling|4.14.14]]
|[[Image:Uniform tiling 77-snub.png|64px]] [[Snub heptaheptagonal tiling|3.3.7.3.7]]
|- align=center
|(Hyperbolic plane) (8 7 2)
|[[File:Hyperbolic domains 872.png|72px]] V4.14.16
|[[File:H2 tiling 278-1.png|64px]] [[Order-7 octagonal tiling|{8,7}]]
|[[File:H2 tiling 278-3.png|64px]] [[Truncated order-7 octagonal tiling|7.16.16]]
|[[File:H2 tiling 278-2.png|64px]] [[Heptaoctagonal tiling|7.8.7.8]]
|[[File:H2 tiling 278-6.png|64px]] [[Truncated order-8 heptagonal tiling|8.14.14]]
|[[File:H2 tiling 278-4.png|64px]] [[Order-8 heptagonal tiling|{7,8}]]
|[[File:H2 tiling 278-5.png|64px]] [[Rhombiheptaoctagonal tiling|7.4.8.4]]
|[[File:H2 tiling 278-7.png|64px]] [[Truncated heptaoctagonal tiling|4.14.16]]
|valign=bottom|[[Snub heptaoctagonal tiling|3.3.7.3.8]]
|- align=center
|(Hyperbolic plane) (8 8 2)
|[[File:Hyperbolic domains 882.png|72px]] V4.16.16
|[[Image:Uniform tiling 88-t2.png|64px]] [[Order-8 octagonal tiling|{8,8}]]
|[[Image:Uniform tiling 88-t12.png|64px]] [[Truncated order-8 octagonal tiling|8.16.16]]
|[[Image:Uniform tiling 88-t1.png|64px]] [[Order-4 octagonal tiling|8.8.8.8]]
|[[Image:Uniform tiling 88-t01.png|64px]] [[Truncated order-8 octagonal tiling|8.16.16]]
|[[Image:Uniform tiling 88-t0.png|64px]] [[Order-8 octagonal tiling|{8,8}]]
|[[Image:Uniform tiling 88-t02.png|64px]] [[Tetraoctagonal tiling|8.4.8.4]]
|[[Image:Uniform tiling 88-t012.png|64px]] [[Truncated order-4 octagonal tiling|4.16.16]]
|[[Image:Uniform tiling 88-snub.png|64px]] [[Snub octaoctagonal tiling|3.3.8.3.8]]
|- align=center
|(Hyperbolic plane) (∞ 3 2)
|[[File:H2checkers 23i.png|72px]] V4.6.∞
|[[File:H2 tiling 23i-1.png|65px]] [[Order-3 apeirogonal tiling|{∞,3}]]
|[[File:H2 tiling 23i-3.png|65px]] [[Truncated order-3 apeirogonal tiling|3.∞.∞]]
|[[File:H2 tiling 23i-2.png|65px]] [[Triapeirogonal tiling|3.∞.3.∞]]
|[[File:H2 tiling 23i-6.png|65px]] [[Truncated infinite-order triangular tiling|∞.6.6]]
|[[File:H2 tiling 23i-4.png|65px]] [[Infinite-order triangular tiling|{3,∞}]]
|[[File:H2 tiling 23i-5.png|65px]] [[Rhombitriapeirogonal tiling|3.4.∞.4]]
|[[File:H2 tiling 23i-7.png|65px]] [[Truncated triapeirogonal tiling|4.6.∞]]
|[[File:Uniform tiling i32-snub.png|65px]] [[Snub triapeirogonal tiling|3.3.3.3.∞]]
|- align=center
|(Hyperbolic plane) (∞ 4 2)
|[[File:H2checkers 24i.png|72px]] V4.8.∞
|[[File:H2 tiling 24i-1.png|65px]] [[Order-4 apeirogonal tiling|{∞,4}]]
|[[File:H2 tiling 24i-3.png|65px]] [[Truncated order-4 apeirogonal tiling|4.∞.∞]]
|[[File:H2 tiling 24i-2.png|65px]] [[Tetraapeirogonal tiling|4.∞.4.∞]]
|[[File:H2 tiling 24i-6.png|65px]] [[Truncated infinite-order square tiling|∞.8.8]]
|[[File:H2 tiling 24i-4.png|65px]] [[Infinite-order square tiling|{4,∞}]]
|[[File:H2 tiling 24i-5.png|65px]] [[Rhombitetraapeirogonal tiling|4.4.∞.4]]
|[[File:H2 tiling 24i-7.png|65px]] [[Truncated tetraapeirogonal tiling|4.8.∞]]
|[[File:Uniform tiling i42-snub.png|65px]] [[Snub tetraapeirogonal tiling|3.3.4.3.∞]]
|- align=center
|(Hyperbolic plane) (∞ 5 2)
|[[File:H2checkers 25i.png|72px]] V4.10.∞
|[[File:H2 tiling 25i-1.png|65px]] [[Order-5 apeirogonal tiling|{∞,5}]]
|[[File:H2 tiling 25i-3.png|65px]] [[Truncated order-5 apeirogonal tiling|5.∞.∞]]
|[[File:H2 tiling 25i-2.png|65px]] [[Pentaapeirogonal tiling|5.∞.5.∞]]
|[[File:H2 tiling 25i-6.png|65px]] [[Truncated infinite-order pentagonal tiling|∞.10.10]]
|[[File:H2 tiling 25i-4.png|65px]] [[Infinite-order pentagonal tiling|{5,∞}]]
|[[File:H2 tiling 25i-5.png|65px]] [[Rhombipentaapeirogonal tiling|5.4.∞.4]]
|[[File:H2 tiling 25i-7.png|65px]] [[Truncated pentaapeirogonal tiling|4.10.∞]]
|[[File:Uniform tiling i52-snub.png|65px]] [[Snub pentaapeirogonal tiling|3.3.5.3.∞]]
|- align=center
|(Hyperbolic plane) (∞ 6 2)
|[[File:H2checkers 26i.png|72px]] V4.12.∞
|[[File:H2 tiling 26i-1.png|65px]] [[Order-6 apeirogonal tiling|{∞,6}]]
|[[File:H2 tiling 26i-3.png|65px]] [[Truncated order-6 apeirogonal tiling|6.∞.∞]]
|[[File:H2 tiling 26i-2.png|65px]] [[Hexaapeirogonal tiling|6.∞.6.∞]]
|[[File:H2 tiling 26i-6.png|65px]] [[Truncated infinite-order hexagonal tiling|∞.12.12]]
|[[File:H2 tiling 26i-4.png|65px]] [[Infinite-order hexagonal tiling|{6,∞}]]
|[[File:H2 tiling 26i-5.png|65px]] [[Rhombihexaapeirogonal tiling|6.4.∞.4]]
|[[File:H2 tiling 26i-7.png|65px]] [[Truncated hexaapeirogonal tiling|4.12.∞]]
|[[File:Uniform tiling i62-snub.png|65px]] [[Snub hexaapeirogonal tiling|3.3.6.3.∞]]
|- align=center
|(Hyperbolic plane) (∞ 7 2)
|[[File:H2checkers 27i.png|72px]] V4.14.∞
|[[File:H2 tiling 27i-1.png|65px]] [[Order-7 apeirogonal tiling|{∞,7}]]
|[[File:H2 tiling 27i-3.png|65px]] [[Truncated order-7 apeirogonal tiling|7.∞.∞]]
|[[File:H2 tiling 27i-2.png|65px]] [[Heptaapeirogonal tiling|7.∞.7.∞]]
|[[File:H2 tiling 27i-6.png|65px]] [[Truncated infinite-order heptagonal tiling|∞.14.14]]
|[[File:H2 tiling 27i-4.png|65px]] [[Infinite-order heptagonal tiling|{7,∞}]]
|[[File:H2 tiling 27i-5.png|65px]] [[Rhombiheptaapeirogonal tiling|7.4.∞.4]]
|[[File:H2 tiling 27i-7.png|65px]] [[Truncated heptaapeirogonal tiling|4.14.∞]]
|valign=bottom|[[Snub heptaapeirogonal tiling|3.3.7.3.∞]]
|- align=center
|(Hyperbolic plane) (∞ 8 2)
|[[File:H2checkers 28i.png|72px]] V4.16.∞
|[[File:H2 tiling 28i-1.png|65px]] [[Order-8 apeirogonal tiling|{∞,8}]]
|[[File:H2 tiling 28i-3.png|65px]] [[Truncated order-8 apeirogonal tiling|8.∞.∞]]
|[[File:H2 tiling 28i-2.png|65px]] [[Octaapeirogonal tiling|8.∞.8.∞]]
|[[File:H2 tiling 28i-6.png|65px]] [[Truncated infinite-order octagonal tiling|∞.16.16]]
|[[File:H2 tiling 28i-4.png|65px]] [[Infinite-order octagonal tiling|{8,∞}]]
|[[File:H2 tiling 28i-5.png|65px]] [[Rhombioctaapeirogonal tiling|8.4.∞.4]]
|[[File:H2 tiling 28i-7.png|65px]] [[Truncated octaapeirogonal tiling|4.16.∞]]
|valign=bottom|[[Snub octaapeirogonal tiling|3.3.8.3.∞]]
|- align=center
|(Hyperbolic plane) (∞ ∞ 2)
|[[File:H2checkers 2ii.png|72px]] V4.∞.∞
|[[File:H2 tiling 2ii-1.png|65px]] [[Infinite-order apeirogonal tiling|{∞,∞}]]
|[[File:H2 tiling 2ii-3.png|65px]] [[Order-3 apeirogonal tiling|∞.∞.∞]]
|[[File:H2 tiling 2ii-2.png|65px]] [[Order-4 apeirogonal tiling|∞.∞.∞.∞]]
|[[File:H2 tiling 2ii-6.png|65px]] [[Order-3 apeirogonal tiling|∞.∞.∞]]
|[[File:H2 tiling 2ii-4.png|65px]] [[Infinite-order apeirogonal tiling|{∞,∞}]]
|[[File:H2 tiling 2ii-5.png|65px]] [[Tetraapeirogonal tiling|∞.4.∞.4]]
|[[File:H2 tiling 2ii-7.png|65px]] [[Truncated order-4 apeirogonal tiling|4.∞.∞]]
|[[File:Uniform tiling ii2-snub.png|65px]] [[Snub apeiroapeirogonal tiling|3.3.∞.3.∞]]
|}

=== Euclidean and hyperbolic tilings (''r'' > 2) ===

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.

{| class="wikitable"
![[Wythoff construction|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
|-
!rowspan=2|[[Schläfli symbol]]
!rowspan=2|
!(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)
!rowspan=2|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]]
!
!
!
!
!
!
!
!
!
|-
![[Vertex configuration|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)
|- align=center
|Euclidean (3 3 3) 
|[[Image:Tile 3,6.svg|72px]] V6.6.6
|[[Image:Uniform tiling 333-t0.png|64px]] [[Triangular tiling|(3.3)3]]
|[[Image:Uniform tiling 333-t01.png|64px]] [[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t1.png|64px]] [[Triangular tiling|(3.3)3]]
|[[Image:Uniform tiling 333-t12.png|64px]] [[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t2.png|64px]] [[Triangular tiling|(3.3)3]]
|[[Image:Uniform tiling 333-t02.png|64px]] [[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t012.png|64px]] [[Hexagonal tiling|6.6.6]]
|[[Image:Uniform tiling 333-snub.png|64px]] [[Triangular tiling|3.3.3.3.3.3]]
|- align=center
|Hyperbolic (4 3 3) 
|[[Image:Hyperbolic domains 433.png|72px]] V6.6.8
|[[Image:Uniform tiling 433-t0.png|64px]] [[Tritetragonal tiling|(3.4)3]]
|[[Image:Uniform tiling 433-t01.png|64px]] [[Trioctagonal tiling|3.8.3.8]]
|[[Image:Uniform tiling 433-t1.png|64px]] [[Tritetragonal tiling|(3.4)3]]
|[[Image:Uniform tiling 433-t12.png|64px]] [[Tritetratrigonal tiling|3.6.4.6]]
|[[Image:Uniform tiling 433-t2.png|64px]] [[Order-8 triangular tiling|(3.3)4]]
|[[Image:Uniform tiling 433-t02.png|64px]] [[Tritetratrigonal tiling|3.6.4.6]]
|[[Image:Uniform tiling 433-t012.png|64px]] [[Truncated order-8 triangular tiling|6.6.8]]
|[[Image:Uniform tiling 433-snub2.png|64px]] [[Snub tritetratrigonal tiling|3.3.3.3.3.4]]
|- align=center
|Hyperbolic (4 4 3) 
|[[Image:Hyperbolic domains 443.png|72px]] V6.8.8
|[[Image:Uniform tiling 443-t0.png|64px]] [[Ditetragonal tritetragonal tiling|(3.4)4]]
|[[Image:Uniform tiling 443-t01.png|64px]] [[Tritetratetragonal tiling|3.8.4.8]]
|[[Image:Uniform tiling 443-t1.png|64px]] [[Order-6 square tiling|(4.4)3]]
|[[Image:Uniform tiling 443-t12.png|64px]] [[Tritetratetragonal tiling|3.8.4.8]]
|[[Image:Uniform tiling 443-t2.png|64px]] [[Ditetragonal tritetragonal tiling|(3.4)4]]
|[[Image:Uniform tiling 443-t02.png|64px]] [[Tetrahexagonal tiling|4.6.4.6]]
|[[Image:Uniform tiling 443-t012.png|64px]] [[Truncated order-6 square tiling|6.8.8]]
|[[Image:Uniform tiling 443-snub1.png|64px]] [[Snub tetratritetragonal tiling|3.3.3.4.3.4]]
|- align=center
|Hyperbolic (4 4 4) 
|[[Image:Hyperbolic domains 444.png|72px]] [[Order-8 triangular tiling|V8.8.8]]
|[[Image:Uniform tiling 444-t0.png|64px]] [[Order-8 square tiling|(4.4)4]]
|[[Image:Uniform tiling 444-t01.png|64px]] [[Tetraoctagonal tiling|4.8.4.8]]
|[[Image:Uniform tiling 444-t1.png|64px]] [[Order-8 square tiling|(4.4)4]]
|[[Image:Uniform tiling 444-t12.png|64px]] [[Tetraoctagonal tiling|4.8.4.8]]
|[[Image:Uniform tiling 444-t2.png|64px]] [[Order-8 square tiling|(4.4)4]]
|[[Image:Uniform tiling 444-t02.png|64px]] [[Tetraoctagonal tiling|4.8.4.8]]
|[[Image:Uniform tiling 444-t012.png|64px]] [[Octagonal tiling|8.8.8]]
|[[Image:Uniform tiling 444-snub.png|64px]] [[Tritetragonal tiling|3.4.3.4.3.4]]
|- align=center
|Hyperbolic (5 3 3) 
|[[File:Hyperbolic domains 533.png|72px]] V6.6.10
|[[File:H2 tiling 335-1.png|65px]] [[Tripentagonal tiling|(3.5)3]]
|[[File:H2 tiling 335-3.png|65px]] [[Tridecagonal tiling|3.10.3.10]]
|[[File:H2 tiling 335-2.png|65px]] [[Tripentagonal tiling|(3.5)3]]
|[[File:H2 tiling 335-6.png|65px]] [[Tripentatrigonal tiling|3.6.5.6]]
|[[File:H2 tiling 335-4.png|65px]] [[Order-10 triangular tiling|(3.3)5]]
|[[File:H2 tiling 335-5.png|65px]] [[Tripentatrigonal tiling|3.6.5.6]]
|[[File:H2 tiling 335-7.png|65px]] [[Truncated order-10 triangular tiling|6.6.10]]
|valign=bottom|[[Snub tripentatrigonal tiling|3.3.3.3.3.5]]
|- align=center
|Hyperbolic (5 4 3) 
|[[File:Hyperbolic domains 543.png|72px]] V6.8.10
|[[File:H2 tiling 345-1.png|65px]] [[Ditetragonal tripentagonal tiling|(3.5)4]]
|[[File:H2 tiling 345-3.png|65px]] [[Tritetrapentagonal tiling|3.10.4.10]]
|[[File:H2 tiling 345-2.png|65px]] [[Ditrigonal tetrapentagonal tiling|(4.5)3]]
|[[File:H2 tiling 345-6.png|65px]] [[Tripentatetragonal tiling|3.8.5.8]]
|[[File:H2 tiling 345-4.png|65px]] [[Dipentagonal tritetragonal tiling|(3.4)5]]
|[[File:H2 tiling 345-5.png|65px]] [[Tetrapentatrigonal tiling|4.6.5.6]]
|[[File:H2 tiling 345-7.png|65px]] [[Trigonally truncated tetrapentagonal tiling|6.8.10]]
|[[File:Uniform tiling 543-snub.png|65px]] [[Snub tritetrapentagonal tiling|3.5.3.4.3.3]]
|- align=center
|Hyperbolic (5 4 4) 
|[[Image:Hyperbolic domains 544.png|72px]] V8.8.10
|[[Image:H2 tiling 445-1.png|64px]] [[Ditetragonal tetrapentagonal tiling|(4.5)4]]
|[[Image:H2 tiling 445-3.png|64px]] [[Tetradecagonal tiling|4.10.4.10]]
|[[Image:H2 tiling 445-2.png|64px]] [[Ditetragonal tetrapentagonal tiling|(4.5)4]]
|[[Image:H2 tiling 445-6.png|64px]] [[Tetrapentatetragonal tiling|4.8.5.8]]
|[[Image:H2 tiling 445-4.png|64px]] [[Order-10 square tiling|(4.4)5]]
|[[Image:H2 tiling 445-5.png|64px]] [[Tetrapentatetragonal tiling|4.8.5.8]]
|[[Image:H2 tiling 445-7.png|64px]] [[Truncated order-10 square tiling|8.8.10]]
|valign=bottom|[[Snub tetrapentatetragonal tiling|3.4.3.4.3.5]]
|- align=center
|Hyperbolic (6 3 3) 
|[[File:Hyperbolic domains 633.png|72px]] V6.6.12
|[[File:H2 tiling 336-1.png|65px]] [[Ditrigonal trihexagonal tiling|(3.6)3]]
|[[File:H2 tiling 336-3.png|65px]] [[Tridodecagonal tiling|3.12.3.12]]
|[[File:H2 tiling 336-2.png|65px]] [[Ditrigonal trihexagonal tiling|(3.6)3]]
|[[File:H2 tiling 336-6.png|65px]] [[Trihexatrigonal tiling|3.6.6.6]]
|[[File:H2 tiling 336-4.png|65px]] [[Order-12 triangular tiling|(3.3)6]]
|[[File:H2 tiling 336-5.png|65px]] [[Trihexatrigonal tiling|3.6.6.6]]
|[[File:H2 tiling 336-7.png|65px]] [[Truncated order-12 triangular tiling|6.6.12]]
|valign=bottom|[[Snub trihexatrigonal tiling|3.3.3.3.3.6]]
|- align=center
|Hyperbolic (6 4 3) 
|[[File:Hyperbolic domains 643.png|72px]] V6.8.12
|[[File:H2 tiling 346-1.png|65px]] [[Ditetragonal trihexagonal tiling|(3.6)4]]
|[[File:H2 tiling 346-3.png|65px]] [[Tritetrahexagonal tiling|3.12.4.12]]
|[[File:H2 tiling 346-2.png|65px]] [[Ditrigonal tetrahexagonal tiling|(4.6)3]]
|[[File:H2 tiling 346-6.png|65px]] [[Trihexatetragonal tiling|3.8.6.8]]
|[[File:H2 tiling 346-4.png|65px]] [[Dihexagonal tritetragonal tiling|(3.4)6]]
|[[File:H2 tiling 346-5.png|65px]] [[Tetrahexatrigonal tiling|4.6.6.6]]
|[[File:H2 tiling 346-7.png|65px]] [[Trigonally truncated tetrahexagonal tiling|6.8.12]]
|valign=bottom|[[Snub tritetrahexagonal tiling|3.6.3.4.3.3]]
|- align=center
|Hyperbolic (6 4 4) 
|[[File:Hyperbolic domains 644.png|72px]] V8.8.12
|[[File:H2 tiling 446-1.png|65px]] [[Ditetragonal tetrahexagonal tiling|(4.6)4]]
|[[File:H2 tiling 446-3.png|65px]] [[Tetradodecagonal tiling|4.12.4.12]]
|[[File:H2 tiling 446-2.png|65px]] [[Ditetragonal tetrahexagonal tiling|(4.6)4]]
|[[File:H2 tiling 446-6.png|65px]] [[Tetrahexatetragonal tiling|4.8.6.8]]
|[[File:H2 tiling 446-4.png|65px]] [[Order-12 square tiling|(4.4)6]]
|[[File:H2 tiling 446-5.png|65px]] [[Tetrahexatetragonal tiling|4.8.6.8]]
|[[File:H2 tiling 446-7.png|65px]] [[Truncated order-12 square tiling|8.8.12]]
|valign=bottom|[[Snub tetrahexatetragonal tiling|3.6.3.4.3.4]]
|- align=center
|Hyperbolic (∞ 3 3) 
|[[File:H2checkers 33i.png|72px]] V6.6.∞
|[[File:H2 tiling 33i-1.png|65px]] [[Ditrigonal triapeirogonal tiling|(3.∞)3]]
|[[File:H2 tiling 33i-3.png|65px]] [[Triapeirogonal tiling|3.∞.3.∞]]
|[[File:H2 tiling 33i-2.png|65px]] [[Ditrigonal triapeirogonal tiling|(3.∞)3]]
|[[File:H2 tiling 33i-6.png|65px]] [[Triapeirotrigonal tiling|3.6.∞.6]]
|[[File:H2 tiling 33i-4.png|65px]] [[Infinite-order triangular tiling|(3.3)∞]]
|[[File:H2 tiling 33i-5.png|65px]] [[Triapeirotrigonal tiling|3.6.∞.6]]
|[[File:H2 tiling 33i-7.png|65px]] [[Truncated infinite-order triangular tiling|6.6.∞]]
|valign=bottom|[[Snub triapeirotrigonal tiling|3.3.3.3.3.∞]]
|- align=center
|Hyperbolic (∞ 4 3) 
|[[File:H2checkers 34i.png|72px]] V6.8.∞
|[[File:H2 tiling 34i-1.png|65px]] [[Ditetragonal triapeirogonal tiling|(3.∞)4]]
|[[File:H2 tiling 34i-3.png|65px]] [[Tritetraapeirogonal tiling|3.∞.4.∞]]
|[[File:H2 tiling 34i-2.png|65px]] [[Ditrigonal tetraapeirogonal tiling|(4.∞)3]]
|[[File:H2 tiling 34i-6.png|65px]] [[Triapeirotetragonal tiling|3.8.∞.8]]
|[[File:H2 tiling 34i-4.png|65px]] [[Diapeirogonal tritetragonal tiling|(3.4)∞]]
|[[File:H2 tiling 34i-5.png|65px]] [[Tetraapeirotrigonal tiling|4.6.∞.6]]
|[[File:H2 tiling 34i-7.png|65px]] [[Trigonally truncated tetraapeirogonal tiling|6.8.∞]]
|valign=bottom|[[Snub tritetraapeirogonal tiling|3.∞.3.4.3.3]]
|- align=center
|Hyperbolic (∞ 4 4) 
|[[File:H2checkers 44i.png|72px]] V8.8.∞
|[[File:H2 tiling 44i-1.png|65px]] [[Ditetragonal tetraapeirogonal tiling|(4.∞)4]]
|[[File:H2 tiling 44i-3.png|65px]] [[Tetraapeirogonal tiling|4.∞.4.∞]]
|[[File:H2 tiling 44i-2.png|65px]] [[Ditetragonal tetraapeirogonal tiling|(4.∞)4]]
|[[File:H2 tiling 44i-6.png|65px]] [[Tetraapeirotetragonal tiling|4.8.∞.8]]
|[[File:H2 tiling 44i-4.png|65px]] [[Infinite-order square tiling|(4.4)∞]]
|[[File:H2 tiling 44i-5.png|65px]] [[Tetraapeirotetragonal tiling|4.8.∞.8]]
|[[File:H2 tiling 44i-7.png|65px]] [[Truncated infinite-order square tiling|8.8.∞]]
|valign=bottom|[[Snub tetraapeirotetragonal tiling|3.∞.3.4.3.4]]
|- align=center
|Hyperbolic (∞ ∞ 3) 
|[[File:H2checkers 3ii.png|72px]] V6.∞.∞
|[[File:H2 tiling 3ii-1.png|65px]] [[Diapeirogonal triapeirogonal tiling|(3.∞)∞]]
|[[File:H2 tiling 3ii-3.png|65px]] [[Triapeiroapeirogonal tiling|3.∞.∞.∞]]
|[[File:H2 tiling 3ii-2.png|65px]] [[Order-6 apeirogonal tiling|(∞.∞)3]]
|[[File:H2 tiling 3ii-6.png|65px]] [[Triapeiroapeirogonal tiling|3.∞.∞.∞]]
|[[File:H2 tiling 3ii-4.png|65px]] [[Diapeirogonal triapeirogonal tiling|(3.∞)∞]]
|[[File:H2 tiling 3ii-5.png|65px]] [[Hexaapeirogonal tiling|∞.6.∞.6]]
|[[File:H2 tiling 3ii-7.png|65px]] [[Truncated order-6 apeirogonal tiling|6.∞.∞]]
|valign=bottom|[[Snub apeirotriapeirogonal tiling|3.∞.3.∞.3.3]]
|- align=center
|Hyperbolic (∞ ∞ 4) 
|[[File:H2checkers 4ii.png|72px]] V8.∞.∞
|[[File:H2 tiling 4ii-1.png|65px]] [[Diapeirogonal tetraapeirogonal tiling|(4.∞)∞]]
|[[File:H2 tiling 4ii-3.png|65px]] [[Tetraapeiroapeirogonal tiling|4.∞.∞.∞]]
|[[File:H2 tiling 4ii-2.png|65px]] [[Order-8 apeirogonal tiling|(∞.∞)4]]
|[[File:H2 tiling 4ii-6.png|65px]] [[Tetraapeiroapeirogonal tiling|4.∞.∞.∞]]
|[[File:H2 tiling 4ii-4.png|65px]] [[Diapeirogonal tetraapeirogonal tiling|(4.∞)∞]]
|[[File:H2 tiling 4ii-5.png|65px]] [[Octaapeirogonal tiling|∞.8.∞.8]]
|[[File:H2 tiling 4ii-7.png|65px]] [[Truncated order-8 apeirogonal tiling|8.∞.∞]]
|valign=bottom|[[Snub apeirotetraapeirogonal tiling|3.∞.3.∞.3.4]]
|- align=center
|Hyperbolic (∞ ∞ ∞) 
|[[File:Infinite-order triangular tiling.svg|72px]] V∞.∞.∞
|[[File:H2 tiling iii-1.png|65px]] [[Infinite-order apeirogonal tiling|(∞.∞)∞]]
|[[File:H2 tiling iii-3.png|65px]] [[Order-4 apeirogonal tiling|∞.∞.∞.∞]]
|[[File:H2 tiling iii-2.png|65px]] [[Infinite-order apeirogonal tiling|(∞.∞)∞]]
|[[File:H2 tiling iii-6.png|65px]] [[Order-4 apeirogonal tiling|∞.∞.∞.∞]]
|[[File:H2 tiling iii-4.png|65px]] [[Infinite-order apeirogonal tiling|(∞.∞)∞]]
|[[File:H2 tiling iii-5.png|65px]] [[Order-4 apeirogonal tiling|∞.∞.∞.∞]]
|[[File:H2 tiling iii-7.png|65px]] [[Order-3 apeirogonal tiling|∞.∞.∞]]
|[[File:Uniform tiling iii-snub.png|65px]] [[Ditrigonal triapeirogonal tiling|3.∞.3.∞.3.∞]]
|}

== See also ==
*[[Regular polytope]]
*[[Regular polyhedron]]
*[[List of uniform tilings]]
*[[Uniform tilings in hyperbolic plane]]
*[[List of uniform polyhedra]]
*[[List of uniform polyhedra by Schwarz triangle]]

==References ==
* [[Harold Scott MacDonald Coxeter|Coxeter]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', Third edition, (1973), Dover edition, (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
* [[Harold Scott MacDonald Coxeter|Coxeter]] ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes)
* [[Coxeter]], Longuet-Higgins, Miller, ''Uniform polyhedra'', '''Phil. Trans.''' 1954, 246 A, 401–50.
* pp. 9–10.

==External links==
*
*[https://ift.tt/2J6i2ON The Wythoff symbol]
*[https://ift.tt/2XkwPJI Wythoff symbol]
*[https://ift.tt/2J5cMeg Greg Egan's applet to display uniform polyhedra using Wythoff's construction method]
*[https://ift.tt/2DmMQVo A Shadertoy renderization of Wythoff's construction method]
*[https://ift.tt/1kYJ7kI KaleidoTile 3] Free educational software for Windows by [[Jeffrey Weeks (mathematician)|Jeffrey Weeks]] that generated many of the images on the page.
*

[[Category:Polyhedra]]
[[Category:Polytopes]]
[[Category:Mathematical notation]]

July 03, 2019 at 07:32AM

English FAN

Tuesday, July 2, 2019

Lists of uniform tilings on the sphere, plane, and hyperbolic plane

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