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Uniform tiling symmetry mutations

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Example *n32 symmetry mutations
Spherical tilings (n = 3..5)

*332
*432
*532
Euclidean plane tiling (n = 6)

*632
Hyperbolic plane tilings (n = 7...∞)

*732
*832
... *∞32

In geometry, a symmetry mutation is a mapping of fundamental domains between two symmetry groups.[1] They are compactly expressed in orbifold notation. These mutations can occur from spherical tilings to Euclidean tilings to hyperbolic tilings. Hyperbolic tilings can also be divided between compact, paracompact and divergent cases.

The uniform tilings are the simplest application of these mutations, although more complex patterns can be expressed within a fundamental domain.

This article expressed progressive sequences of uniform tilings within symmetry families.

Mutations of orbifolds

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Orbifolds with the same structure can be mutated between different symmetry classes, including across curvature domains from spherical, to Euclidean to hyperbolic. This table shows mutation classes.[1] This table is not complete for possible hyperbolic orbifolds.

Orbifold Spherical Euclidean Hyperbolic
o - o -
pp 22, 33 ... ∞∞ -
*pp *22, *33 ... *∞∞ -
p* 2*, 3* ... ∞* -
p× 2×, 3× ... ∞×
** - ** -
*× - *× -
×× - ×× -
ppp 222 333 444 ...
pp* - 22* 33* ...
pp× - 22× 33×, 44× ...
pqq 222, 322 ... , 233 244 255 ..., 433 ...
pqr 234, 235 236 237 ..., 245 ...
pq* - - 23*, 24* ...
pq× - - 23×, 24× ...
p*q 2*2, 2*3 ... 3*3, 4*2 5*2 5*3 ..., 4*3, 4*4 ..., 3*4, 3*5 ...
*p* - - *2* ...
*p× - - *2× ...
pppp - 2222 3333 ...
pppq - - 2223...
ppqq - - 2233
pp*p - - 22*2 ...
p*qr - 2*22 3*22 ..., 2*32 ...
*ppp *222 *333 *444 ...
*pqq *p22, *233 *244 *255 ..., *344...
*pqr *234, *235 *236 *237..., *245..., *345 ...
p*ppp - - 2*222
*pqrs - *2222 *2223...
*ppppp - - *22222 ...
...

*n22 symmetry

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Regular tilings

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Family of regular hosohedra · *n22 symmetry mutations of regular hosohedral tilings: nn
Space Spherical Euclidean
Tiling
name 		Henagonal
hosohedron 		Digonal
hosohedron 		Trigonal
hosohedron 		Square
hosohedron 		Pentagonal
hosohedron 		... Apeirogonal
hosohedron
Tiling
image 		...
Schläfli
symbol 		{2,1} {2,2} {2,3} {2,4} {2,5} ... {2,∞}
Coxeter
diagram 		...
Faces and
edges 		1 2 3 4 5 ... ∞
Vertices 2 2 2 2 2 ... 2
Vertex
config. 		2 2.2 23 24 25 ... 2∞
Family of regular dihedra · *n22 symmetry mutations of regular dihedral tilings: nn
Space Spherical Euclidean
Tiling
name 		Monogonal
dihedron 		Digonal
dihedron 		Trigonal
dihedron 		Square
dihedron 		Pentagonal
dihedron 		... Apeirogonal
dihedron
Tiling
image 		...
Schläfli
symbol 		{1,2} {2,2} {3,2} {4,2} {5,2} ... {∞,2}
Coxeter
diagram 		...
Faces 2 {1} 2 {2} 2 {3} 2 {4} 2 {5} ... 2 {∞}
Edges and
vertices 		1 2 3 4 5 ... ∞
Vertex
config. 		1.1 2.2 3.3 4.4 5.5 ... ∞.∞

Prism tilings

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*n22 symmetry mutations of uniform prisms: n.4.4
Space Spherical Euclidean
Tiling
Config. 3.4.4 4.4.4 5.4.4 6.4.4 7.4.4 8.4.4 9.4.4 10.4.4 11.4.4 12.4.4 ...∞.4.4

Antiprism tilings

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*n22 symmetry mutations of antiprism tilings: Vn.3.3.3
Space Spherical Euclidean
Tiling
Config. 2.3.3.3 3.3.3.3 4.3.3.3 5.3.3.3 6.3.3.3 7.3.3.3 8.3.3.3 ...∞.3.3.3

*n32 symmetry

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Regular tilings

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*n32 symmetry mutation of regular tilings: {3,n}
Spherical Euclid. Compact hyper. Paraco. Noncompact hyperbolic
3.3 33 34 35 36 37 38 3∞ 312i 39i 36i 33i
*n32 symmetry mutation of regular tilings: {n,3}
Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic
{2,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {12i,3} {9i,3} {6i,3} {3i,3}

Truncated tilings

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*n32 symmetry mutation of truncated tilings: t{n,3}
Symmetry
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3] 		*332
[3,3] 		*432
[4,3] 		*532
[5,3] 		*632
[6,3] 		*732
[7,3] 		*832
[8,3]... 		*∞32
[∞,3] 		[12i,3] [9i,3] [6i,3]
Truncated
figures
Symbol t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3} t{12i,3} t{9i,3} t{6i,3}
Triakis
figures
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞
*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3] 		Spherical Euclid. Compact Parac. Noncompact hyperbolic
*232
[2,3] 		*332
[3,3] 		*432
[4,3] 		*532
[5,3] 		*632
[6,3] 		*732
[7,3] 		*832
[8,3]... 		*∞32
[∞,3] 		[12i,3] [9i,3] [6i,3]
Truncated
figures
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6

Quasiregular tilings

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Quasiregular tilings: (3.n)2
Sym.
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*332
[3,3]
Td 		*432
[4,3]
Oh 		*532
[5,3]
Ih 		*632
[6,3]
p6m 		*732
[7,3]
  		*832
[8,3]...
  		*∞32
[∞,3]
  		[12i,3] [9i,3] [6i,3]
Figure
Figure
Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.∞)2 (3.12i)2 (3.9i)2 (3.6i)2
Schläfli r{3,3} r{3,4} r{3,5} r{3,6} r{3,7} r{3,8} r{3,∞} r{3,12i} r{3,9i} r{3,6i}
Coxeter
Dual uniform figures
Dual
conf.
V(3.3)2
V(3.4)2
V(3.5)2
V(3.6)2
V(3.7)2
V(3.8)2
V(3.∞)2
Symmetry mutations of dual quasiregular tilings: V(3.n)2
*n32 Spherical Euclidean Hyperbolic
*332 *432 *532 *632 *732 *832... *∞32
Tiling
Conf. V(3.3)2 V(3.4)2 V(3.5)2 V(3.6)2 V(3.7)2 V(3.8)2 V(3.∞)2

Expanded tilings

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*n32 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3] 		*332
[3,3] 		*432
[4,3] 		*532
[5,3] 		*632
[6,3] 		*732
[7,3] 		*832
[8,3]... 		*∞32
[∞,3] 		 
[12i,3] 		 
[9i,3] 		 
[6i,3]
Figure
Config. 3.4.2.4 3.4.3.4 3.4.4.4 3.4.5.4 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.4.12i.4 3.4.9i.4 3.4.6i.4
*n32 symmetry mutation of dual expanded tilings: V3.4.n.4
Symmetry
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3] 		*332
[3,3] 		*432
[4,3] 		*532
[5,3] 		*632
[6,3] 		*732
[7,3] 		*832
[8,3]... 		*∞32
[∞,3]
Figure
Config.
V3.4.2.4
V3.4.3.4
V3.4.4.4
V3.4.5.4
V3.4.6.4
V3.4.7.4
V3.4.8.4
V3.4.∞.4

Omnitruncated tilings

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*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3] 		*332
[3,3] 		*432
[4,3] 		*532
[5,3] 		*632
[6,3] 		*732
[7,3] 		*832
[8,3] 		*∞32
[∞,3] 		 
[12i,3] 		 
[9i,3] 		 
[6i,3] 		 
[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i

Snub tilings

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n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry
n32 		Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
Config. 3.3.3.3.2 3.3.3.3.3 3.3.3.3.4 3.3.3.3.5 3.3.3.3.6 3.3.3.3.7 3.3.3.3.8 3.3.3.3.∞
Gyro
figures
Config. V3.3.3.3.2 V3.3.3.3.3 V3.3.3.3.4 V3.3.3.3.5 V3.3.3.3.6 V3.3.3.3.7 V3.3.3.3.8 V3.3.3.3.∞

*n42 symmetry

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Regular tilings

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*n42 symmetry mutation of regular tilings: {4,n}
Spherical Euclidean Compact hyperbolic Paracompact

{4,3}
{4,4}
{4,5}
{4,6}
{4,7}
{4,8}...
{4,∞}
*n42 symmetry mutation of regular tilings: {n,4}
Spherical Euclidean Hyperbolic tilings
24 34 44 54 64 74 84 ...∞4

Quasiregular tilings

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*n42 symmetry mutations of quasiregular tilings: (4.n)2
Symmetry
*4n2
[n,4] 		Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4] 		*442
[4,4] 		*542
[5,4] 		*642
[6,4] 		*742
[7,4] 		*842
[8,4]... 		*∞42
[∞,4] 		 
[ni,4]
Figures
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2
*n42 symmetry mutations of quasiregular dual tilings: V(4.n)2
Symmetry
*4n2
[n,4] 		Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4] 		*442
[4,4] 		*542
[5,4] 		*642
[6,4] 		*742
[7,4] 		*842
[8,4]... 		*∞42
[∞,4] 		 
[iπ/λ,4]
Tiling
 
Conf.
V4.3.4.3
V4.4.4.4
V4.5.4.5
V4.6.4.6
V4.7.4.7
V4.8.4.8
V4.∞.4.∞ 		V4.∞.4.∞

Truncated tilings

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*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4] 		Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4] 		*342
[3,4] 		*442
[4,4] 		*542
[5,4] 		*642
[6,4] 		*742
[7,4] 		*842
[8,4]... 		*∞42
[∞,4]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4] 		Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4] 		*342
[3,4] 		*442
[4,4] 		*542
[5,4] 		*642
[6,4] 		*742
[7,4] 		*842
[8,4]... 		*∞42
[∞,4]
Truncated
figures
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8

Expanded tilings

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*n42 symmetry mutation of expanded tilings: n.4.4.4
Symmetry
[n,4], (*n42) 		Spherical Euclidean Compact hyperbolic Paracomp.
*342
[3,4] 		*442
[4,4] 		*542
[5,4] 		*642
[6,4] 		*742
[7,4] 		*842
[8,4] 		*∞42
[∞,4]
Expanded
figures
Config. 3.4.4.4 4.4.4.4 5.4.4.4 6.4.4.4 7.4.4.4 8.4.4.4 ∞.4.4.4
Rhombic
figures
config.
V3.4.4.4
V4.4.4.4
V5.4.4.4
V6.4.4.4
V7.4.4.4
V8.4.4.4
V∞.4.4.4

Omnitruncated tilings

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*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4] 		Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4] 		*342
[3,4] 		*442
[4,4] 		*542
[5,4] 		*642
[6,4] 		*742
[7,4] 		*842
[8,4]... 		*∞42
[∞,4]
Omnitruncated
figure
4.8.4
4.8.6
4.8.8
4.8.10
4.8.12
4.8.14
4.8.16
4.8.∞
Omnitruncated
duals
V4.8.4
V4.8.6
V4.8.8
V4.8.10
V4.8.12
V4.8.14
V4.8.16
V4.8.∞

Snub tilings

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4n2 symmetry mutations of snub tilings: 3.3.4.3.n
Symmetry
4n2 		Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Snub
figures
Config. 3.3.4.3.2 3.3.4.3.3 3.3.4.3.4 3.3.4.3.5 3.3.4.3.6 3.3.4.3.7 3.3.4.3.8 3.3.4.3.∞
Gyro
figures
Config. V3.3.4.3.2 V3.3.4.3.3 V3.3.4.3.4 V3.3.4.3.5 V3.3.4.3.6 V3.3.4.3.7 V3.3.4.3.8 V3.3.4.3.∞

*n52 symmetry

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Regular tilings

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*n52 symmetry mutation of truncated tilings: 5n
Sphere Hyperbolic plane

{5,3}
{5,4}
{5,5}
{5,6}
{5,7}
{5,8}
...{5,∞}

*n62 symmetry

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Regular tilings

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*n62 symmetry mutation of regular tilings: {6,n}
Spherical Euclidean Hyperbolic tilings

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

*n82 symmetry

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Regular tilings

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n82 symmetry mutations of regular tilings: 8n
Space Spherical Compact hyperbolic Paracompact
Tiling
Config. 8.8 83 84 85 86 87 88 ...8∞

References

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  1. ^ a b Two Dimensional symmetry Mutations by Daniel Huson

Sources

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  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
  • From hyperbolic 2-space to Euclidean 3-space: Tilings and patterns via topology Stephen Hyde
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