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Uniform tiling symmetry mutations
From Wikipedia, the free encyclopedia
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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.
More information Spherical tilings (n = 3..5), Euclidean plane tiling (n = 6) ...
Spherical tilings (n = 3..5) | ||
---|---|---|
![]() *332 |
![]() *432 |
![]() *532 |
Euclidean plane tiling (n = 6) | ||
![]() *632 | ||
Hyperbolic plane tilings (n = 7...∞) | ||
![]() *732 |
![]() *832 |
![]() ... *∞32 |
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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.
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Mutations of orbifolds
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.
More information Orbifold, Spherical ...
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 ... |
... |
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*n22 symmetry
Regular tilings
More information Space, Spherical ...
Space | Spherical | Euclidean | |||||
---|---|---|---|---|---|---|---|
Tiling name |
Henagonal hosohedron |
Digonal hosohedron |
Trigonal hosohedron |
Square hosohedron |
Pentagonal hosohedron |
... | Apeirogonal hosohedron |
Tiling image |
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Schläfli symbol |
{2,1} | {2,2} | {2,3} | {2,4} | {2,5} | ... | {2,∞} |
Coxeter diagram |
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... | ![]() ![]() ![]() ![]() ![]() |
Faces and edges |
1 | 2 | 3 | 4 | 5 | ... | ∞ |
Vertices | 2 | 2 | 2 | 2 | 2 | ... | 2 |
Vertex config. |
2 | 2.2 | 23 | 24 | 25 | ... | 2∞ |
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More information Space, Spherical ...
Space | Spherical | Euclidean | |||||
---|---|---|---|---|---|---|---|
Tiling name |
Monogonal dihedron |
Digonal dihedron |
Trigonal dihedron |
Square dihedron |
Pentagonal dihedron |
... | Apeirogonal dihedron |
Tiling image |
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... | ![]() |
Schläfli symbol |
{1,2} | {2,2} | {3,2} | {4,2} | {5,2} | ... | {∞,2} |
Coxeter diagram |
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... | ![]() ![]() ![]() ![]() ![]() |
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 | ... | ∞.∞ |
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Prism tilings
Antiprism tilings
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*n32 symmetry
Regular tilings
Truncated tilings
More information Symmetry*n32 [n,3], Spherical ...
*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 |
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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 |
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Config. | V3.4.4 | V3.6.6 | V3.8.8 | V3.10.10 | V3.12.12 | V3.14.14 | V3.16.16 | V3.∞.∞ |
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More information Sym.*n42 [n,3], Spherical ...
*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 |
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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 |
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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 |
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Quasiregular tilings
More information Sym.*n32 [n,3], Spherical ...
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] | |||
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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![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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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 |
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Expanded tilings
More information Symmetry*n32 [n,3], Spherical ...
*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 | ![]() |
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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 |
Close
More information Symmetry*n32 [n,3], Spherical ...
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 |
Close
Omnitruncated tilings
More information Sym.*n32 [n,3], Spherical ...
*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 | ![]() |
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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 | ![]() |
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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 |
Close
Snub tilings
More information Symmetry n32, Spherical ...
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 |
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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 |
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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.∞ |
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*n42 symmetry
Regular tilings
Quasiregular tilings
More information Symmetry*4n2 [n,4], Spherical ...
*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 | ![]() |
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Config. | (4.3)2 | (4.4)2 | (4.5)2 | (4.6)2 | (4.7)2 | (4.8)2 | (4.∞)2 | (4.ni)2 |
Close
More information *n42 symmetry mutations of quasiregular dual tilings: V(4.n)2, Symmetry*4n2 [n,4] ...
*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.∞ |
Close
Truncated tilings
More information Symmetry*n42 [n,4], Spherical ...
*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 |
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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 |
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Config. | V4.4.4 | V4.6.6 | V4.8.8 | V4.10.10 | V4.12.12 | V4.14.14 | V4.16.16 | V4.∞.∞ |
Close
More information Symmetry*n42 [n,4], Spherical ...
*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 |
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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 |
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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 |
Close
Expanded tilings
More information Symmetry [n,4], (*n42), Spherical ...
*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 |
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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 |
Close
Omnitruncated tilings
More information Symmetry*n42 [n,4], Spherical ...
*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.∞ |
Close
Snub tilings
More information Symmetry 4n2, Spherical ...
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 |
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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 |
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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.∞ |
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*n52 symmetry
Regular tilings
*n62 symmetry
Regular tilings
*n82 symmetry
Regular tilings
References
Sources
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