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

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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 π/p, π/q, and π/r. Special cases are right triangles (p q 2). Uniform solutions are constructed by a single generator point with 7 positions within the fundamental triangle, the 3 corners, along the 3 edges, and the triangle interior. All vertices exist at the generator, or a reflected copy of it. Edges exist between a generator point and its image across a mirror. Up to 3 face types exist centered on the fundamental triangle corners. Right triangle domains can have as few as 1 face type, making regular forms, while general triangles have at least 2 triangle types, leading at best to a quasiregular tiling.

There are different notations for expressing these uniform solutions, Wythoff symbol, Coxeter diagram, and Coxeter's t-notation.

Simple tiles are generated by Möbius triangles with whole numbers p,q,r, while Schwarz triangles allow rational numbers p,q,r and allow star polygon faces, and have overlapping elements.

7 generator points

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The seven generator points with each set of (and a few special forms):

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

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

There are three special cases:

  • – This is a mixture of and , containing only the faces shared by both.
  • – Snub forms (alternated) are given by this otherwise unused symbol.
  • – A unique snub form for U75 that isn't Wythoff-constructible.

Symmetry triangles

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There are 4 symmetry classes of reflection on the sphere, and three 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

(2 2 2)

(3 2 2)

(4 2 2)

(5 2 2)

(6 2 2)

(3 3 2)

(4 3 2)

(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 polyhedra.

Euclidean plane
p4m p3m p6m
*442 *333 *632

(4 4 2)

(3 3 3)

(6 3 2)
Hyperbolic plane
*732 *542 *433

(7 3 2)

(5 4 2)

(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

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Selected tilings created by the Wythoff construction are given below.

Spherical tilings (r = 2)

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(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
{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
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

(3 3 2)

{3,3}

(3.6.6)

(3.3a.3.3a)

(3.6.6)

{3,3}

(3a.4.3b.4)

(4.6a.6b)

(3.3.3a.3.3b)

(4 3 2)

{4,3}

(3.8.8)

(3.4.3.4)

(4.6.6)

{3,4}

(3.4.4a.4)

(4.6.8)

(3.3.3a.3.4)

(5 3 2)

{5,3}

(3.10.10)

(3.5.3.5)

(5.6.6)

{3,5}

(3.4.5.4)

(4.6.10)

(3.3.3a.3.5)

Some overlapping spherical tilings (r = 2)

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

(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
{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
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)
 
{3,5/2}

(5/2.6.6)

(3.5/2)2

[3.10/2.10/2]

{5/2,3}

[3.4.5/2.4]

[4.10/2.6]

(3.3.3.3.5/2)
Icosahedral
(5 5/2 2)
 
{5,5/2}

(5/2.10.10)

(5/2.5)2

[5.10/2.10/2]

{5/2,5}

(5/2.4.5.4)

[4.10/2.10]

(3.3.5/2.3.5)

Dihedral symmetry (q = r = 2)

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Spherical tilings with dihedral symmetry exist for all many with digon faces which become degenerate polyhedra. Two of the eight forms (Rectified and cantillated) 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
{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
Vertex figure (2.2p.2p) (4.4.p) 2p (4.2p.4) (3.3.p. 3)

(2 2 2)
V2.2.2

{2,2}

2.4.4
4.4.2
{2,2}

4.4.4

3.3.3.2

(3 2 2)
V3.2.2

{3,2}

2.6.6

4.4.3

{2,3}

4.4.6

3.3.3.3

(4 2 2)
V4.2.2

{4,2}
2.8.8
4.4.4

{2,4}

4.4.8

3.3.3.4

(5 2 2)
V5.2.2

{5,2}
2.10.10
4.4.5

{2,5}

4.4.10

3.3.3.5

(6 2 2)
V6.2.2

{6,2}

2.12.12

4.4.6

{2,6}

4.4.12

3.3.3.6
...

Euclidean and hyperbolic tilings (r = 2)

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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
{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
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)

V4.6.12

{6,3}

3.12.12

3.6.3.6

6.6.6

{3,6}

3.4.6.4

4.6.12

3.3.3.3.6
(Hyperbolic plane)
(7 3 2)

V4.6.14

{7,3}

3.14.14

3.7.3.7

7.6.6

{3,7}

3.4.7.4

4.6.14

3.3.3.3.7
(Hyperbolic plane)
(8 3 2)

V4.6.16

{8,3}

3.16.16

3.8.3.8

8.6.6

{3,8}

3.4.8.4

4.6.16

3.3.3.3.8
Square tiling
(4 4 2)

V4.8.8

{4,4}

4.8.8

4.4a.4.4a

4.8.8

{4,4}

4.4a.4b.4a

4.8.8

3.3.4a.3.4b
(Hyperbolic plane)
(5 4 2)

V4.8.10

{5,4}

4.10.10

4.5.4.5

5.8.8

{4,5}

4.4.5.4

4.8.10

3.3.4.3.5
(Hyperbolic plane)
(6 4 2)

V4.8.12

{6,4}

4.12.12

4.6.4.6

6.8.8

{4,6}

4.4.6.4

4.8.12

3.3.4.3.6
(Hyperbolic plane)
(7 4 2)

V4.8.14

{7,4}

4.14.14

4.7.4.7

7.8.8

{4,7}

4.4.7.4

4.8.14

3.3.4.3.7
(Hyperbolic plane)
(8 4 2)

V4.8.16

{8,4}

4.16.16

4.8.4.8

8.8.8

{4,8}

4.4.8.4

4.8.16

3.3.4.3.8
(Hyperbolic plane)
(5 5 2)

V4.10.10

{5,5}

5.10.10

5.5.5.5

5.10.10

{5,5}

5.4.5.4

4.10.10

3.3.5.3.5
(Hyperbolic plane)
(6 5 2)

V4.10.12

{6,5}

5.12.12

5.6.5.6

6.10.10

{5,6}

5.4.6.4

4.10.12

3.3.5.3.6
(Hyperbolic plane)
(7 5 2)

V4.10.14

{7,5}

5.14.14

5.7.5.7

7.10.10

{5,7}

5.4.7.4

4.10.14

3.3.5.3.7
(Hyperbolic plane)
(8 5 2)

V4.10.16

{8,5}

5.16.16

5.8.5.8

8.10.10

{5,8}

5.4.8.4

4.10.16
3.3.5.3.8
(Hyperbolic plane)
(6 6 2)

V4.12.12

{6,6}

6.12.12

6.6.6.6

6.12.12

{6,6}

6.4.6.4

4.12.12

3.3.6.3.6
(Hyperbolic plane)
(7 6 2)

V4.12.14

{7,6}

6.14.14

6.7.6.7

7.12.12

{6,7}

6.4.7.4

4.12.14
3.3.6.3.7
(Hyperbolic plane)
(8 6 2)

V4.12.16

{8,6}

6.16.16

6.8.6.8

8.12.12

{6,8}

6.4.8.4

4.12.16

3.3.6.3.8
(Hyperbolic plane)
(7 7 2)

V4.14.14

{7,7}

7.14.14

7.7.7.7

7.14.14

{7,7}

7.4.7.4

4.14.14

3.3.7.3.7
(Hyperbolic plane)
(8 7 2)

V4.14.16

{8,7}

7.16.16

7.8.7.8

8.14.14

{7,8}

7.4.8.4

4.14.16
3.3.7.3.8
(Hyperbolic plane)
(8 8 2)

V4.16.16

{8,8}

8.16.16

8.8.8.8

8.16.16

{8,8}

8.4.8.4

4.16.16

3.3.8.3.8
(Hyperbolic plane)
(∞ 3 2)

V4.6.∞

{∞,3}

3.∞.∞

3.∞.3.∞

∞.6.6

{3,∞}

3.4.∞.4

4.6.∞

3.3.3.3.∞
(Hyperbolic plane)
(∞ 4 2)

V4.8.∞

{∞,4}

4.∞.∞

4.∞.4.∞

∞.8.8

{4,∞}

4.4.∞.4

4.8.∞

3.3.4.3.∞
(Hyperbolic plane)
(∞ 5 2)

V4.10.∞

{∞,5}

5.∞.∞

5.∞.5.∞

∞.10.10

{5,∞}

5.4.∞.4

4.10.∞

3.3.5.3.∞
(Hyperbolic plane)
(∞ 6 2)

V4.12.∞

{∞,6}

6.∞.∞

6.∞.6.∞

∞.12.12

{6,∞}

6.4.∞.4

4.12.∞

3.3.6.3.∞
(Hyperbolic plane)
(∞ 7 2)

V4.14.∞

{∞,7}

7.∞.∞

7.∞.7.∞

∞.14.14

{7,∞}

7.4.∞.4

4.14.∞
3.3.7.3.∞
(Hyperbolic plane)
(∞ 8 2)

V4.16.∞

{∞,8}

8.∞.∞

8.∞.8.∞

∞.16.16

{8,∞}

8.4.∞.4

4.16.∞
3.3.8.3.∞
(Hyperbolic plane)
(∞ ∞ 2)

V4.∞.∞

{∞,∞}

∞.∞.∞

∞.∞.∞.∞

∞.∞.∞

{∞,∞}

∞.4.∞.4

4.∞.∞

3.3.∞.3.∞

Euclidean and hyperbolic tilings (r > 2)

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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
Vertex figure (p.r)q (r.2p.q.2p) (p.q)r (r.2q.p. 2q) (q.r)p (p. 2r.q.2r) (2p.2q.2r) (3.r.3.q.3.p)
Euclidean
(3 3 3)

V6.6.6

(3.3)3

3.6.3.6

(3.3)3

3.6.3.6

(3.3)3

3.6.3.6

6.6.6

3.3.3.3.3.3
Hyperbolic
(4 3 3)

V6.6.8

(3.4)3

3.8.3.8

(3.4)3

3.6.4.6

(3.3)4

3.6.4.6

6.6.8

3.3.3.3.3.4
Hyperbolic
(4 4 3)

V6.8.8

(3.4)4

3.8.4.8

(4.4)3

3.8.4.8

(3.4)4

4.6.4.6

6.8.8

3.3.3.4.3.4
Hyperbolic
(4 4 4)

V8.8.8

(4.4)4

4.8.4.8

(4.4)4

4.8.4.8

(4.4)4

4.8.4.8

8.8.8

3.4.3.4.3.4
Hyperbolic
(5 3 3)

V6.6.10

(3.5)3

3.10.3.10

(3.5)3

3.6.5.6

(3.3)5

3.6.5.6

6.6.10
3.3.3.3.3.5
Hyperbolic
(5 4 3)

V6.8.10

(3.5)4

3.10.4.10

(4.5)3

3.8.5.8

(3.4)5

4.6.5.6

6.8.10

3.5.3.4.3.3
Hyperbolic
(5 4 4)

V8.8.10

(4.5)4

4.10.4.10

(4.5)4

4.8.5.8

(4.4)5

4.8.5.8

8.8.10
3.4.3.4.3.5
Hyperbolic
(6 3 3)

V6.6.12

(3.6)3

3.12.3.12

(3.6)3

3.6.6.6

(3.3)6

3.6.6.6

6.6.12
3.3.3.3.3.6
Hyperbolic
(6 4 3)

V6.8.12

(3.6)4

3.12.4.12

(4.6)3

3.8.6.8

(3.4)6

4.6.6.6

6.8.12
3.6.3.4.3.3
Hyperbolic
(6 4 4)

V8.8.12

(4.6)4

4.12.4.12

(4.6)4

4.8.6.8

(4.4)6

4.8.6.8

8.8.12
3.6.3.4.3.4
Hyperbolic
(∞ 3 3)

V6.6.∞

(3.∞)3

3.∞.3.∞

(3.∞)3

3.6.∞.6

(3.3)

3.6.∞.6

6.6.∞
3.3.3.3.3.∞
Hyperbolic
(∞ 4 3)

V6.8.∞

(3.∞)4

3.∞.4.∞

(4.∞)3

3.8.∞.8

(3.4)

4.6.∞.6

6.8.∞
3.∞.3.4.3.3
Hyperbolic
(∞ 4 4)

V8.8.∞

(4.∞)4

4.∞.4.∞

(4.∞)4

4.8.∞.8

(4.4)

4.8.∞.8

8.8.∞
3.∞.3.4.3.4
Hyperbolic
(∞ ∞ 3)

V6.∞.∞

(3.∞)

3.∞.∞.∞

(∞.∞)3

3.∞.∞.∞

(3.∞)

∞.6.∞.6

6.∞.∞
3.∞.3.∞.3.3
Hyperbolic
(∞ ∞ 4)

V8.∞.∞

(4.∞)

4.∞.∞.∞

(∞.∞)4

4.∞.∞.∞

(4.∞)

∞.8.∞.8

8.∞.∞
3.∞.3.∞.3.4
Hyperbolic
(∞ ∞ ∞)

V∞.∞.∞

(∞.∞)

∞.∞.∞.∞

(∞.∞)

∞.∞.∞.∞

(∞.∞)

∞.∞.∞.∞

∞.∞.∞

3.∞.3.∞.3.∞

See also

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References

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  • Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
  • Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
  • Coxeter, Longuet-Higgins, Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401–50.
  • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. pp. 9–10.
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