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.
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. This table is not complete for possible hyperbolic orbifolds.
| Orbifold | Spherical | Euclidean | Hyperbolic |
|---|---|---|---|
| o | - | o | - |
| 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 ... |
| - | - |
|
| - | - |
|
| pppp | - | 2222 | 3333 ... |
| pppq | - | - | 2223... |
| ppqq | - | - | 2233 |
| pp*p | - | - | 22*2 ... |
| p*qr | - | 2*22 | 3*22 ..., 2*32 ... |
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| p*ppp | - | - | 2*222 |
| - |
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| - | - |
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| ... |