Uniform tiling 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
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
Regular tilings
Prism tilings
| Space | Spherical | Euclidean | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Tiling | 
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| 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
*n32 symmetry
Regular tilings
Truncated tilings
Quasiregular tilings
Expanded tilings
Omnitruncated tilings
Snub tilings
*n42 symmetry
Regular tilings
Quasiregular tilings
Truncated tilings
Expanded tilings
Omnitruncated tilings
Snub tilings
*n52 symmetry
Regular tilings
*n62 symmetry
Regular tilings
*n82 symmetry
Regular tilings
References
Sources
- 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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