# Uniform tiling symmetry mutations

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. 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. 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

Family of regular hosohedra · *n22 symmetry mutations of regular hosohedral tilings: nn :
| Space | Spherical | Spherical | Spherical | Spherical | 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 | 2$^3$ | 2$^4$ | 2$^5$ | ... | 2$^∞$ |
Family of regular dihedra · *n22 symmetry mutations of regular dihedral tilings: nn :
| Space | Spherical | Spherical | Spherical | Spherical | 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

*n22 symmetry mutations of uniform prisms: n.4.4 :
| Space | Spherical | Spherical | Spherical | Spherical | Spherical | Spherical | Spherical | Spherical | Spherical | 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

*n22 symmetry mutations of antiprism tilings: Vn.3.3.3 :
| Space | Spherical | Spherical | Spherical | Spherical | Spherical | Spherical | 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
### Regular tilings

*n32 symmetry mutation of regular tilings: {3,n} vte :
| Spherical | Spherical | Spherical | Spherical | Euclid. | Compact hyper. | Compact hyper. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| 3.3 | 3$^3$ | 3$^4$ | 3$^5$ | 3$^6$ | 3$^7$ | 3$^8$ | 3$^∞$ | 3$^{12i}$ | 3$^{9i}$ | 3$^{6i}$ | 3$^{3i}$ |
*n32 symmetry mutation of regular tilings: {n,3} vte :
| Spherical | Spherical | Spherical | Spherical | Euclidean | Compact hyperb. | Compact hyperb. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic | 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

*n32 symmetry mutation of truncated tilings: t{n,3} vte :
| Symmetry *n32 [n,3] | Spherical | Spherical | Spherical | Spherical | Euclid. | Compact hyperb. | Compact hyperb. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| Symmetry *n32 [n,3] | *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 vte :
| Sym. *n42 [n,3] | Spherical | Spherical | Spherical | Spherical | Euclid. | Compact | Compact | Parac. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| Sym. *n42 [n,3] | *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

Quasiregular tilings: (3.n)$^2$ vte :
| Sym. *n32 [n,3] | Spherical | Spherical | Spherical | Euclid. | Compact hyperb. | Compact hyperb. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| Sym. *n32 [n,3] | *332 [3,3] T$_d$ | *432 [4,3] O$_h$ | *532 [5,3] I$_h$ | *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 | Spherical | Spherical | Euclidean | Hyperbolic | Hyperbolic | Hyperbolic |
| *n32 | *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

*n32 symmetry mutation of expanded tilings: 3.4.n.4 vte :
| Symmetry *n32 [n,3] | Spherical | Spherical | Spherical | Spherical | Euclid. | Compact hyperb. | Compact hyperb. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| Symmetry *n32 [n,3] | *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 | Spherical | Spherical | Spherical | Euclid. | Compact hyperb. | Compact hyperb. | Paraco. |
| Symmetry *n32 [n,3] | *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

*n32 symmetry mutation of omnitruncated tilings: 4.6.2n vte :
| Sym. *n32 [n,3] | Spherical | Spherical | Spherical | Spherical | Euclid. | Compact hyperb. | Compact hyperb. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| Sym. *n32 [n,3] | *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

n32 symmetry mutations of snub tilings: 3.3.3.3.n vte :
| Symmetry n32 | Spherical | Spherical | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry n32 | 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
### Regular tilings

*n42 symmetry mutation of regular tilings: {4,n} vte :
| Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact |
| {4,3} | {4,4} | {4,5} | {4,6} | {4,7} | {4,8}... | {4,∞} |
*n42 symmetry mutation of regular tilings: {n,4} vte :
| Spherical | Spherical | Euclidean | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings |
| 2$^4$ | 3$^4$ | 4$^4$ | 5$^4$ | 6$^4$ | 7$^4$ | 8$^4$ | ...∞$^4$ |

### Quasiregular tilings

*n42 symmetry mutations of quasiregular tilings: (4.n)$^2$ vte :
| Symmetry *4n2 [n,4] | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact | Noncompact |
| Symmetry *4n2 [n,4] | *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 | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact | Noncompact |
| Symmetry *4n2 [n,4] | *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

*n42 symmetry mutation of truncated tilings: 4.2n.2n vte :
| Symmetry *n42 | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry *n42 | *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 vte :
| Symmetry *n42 [n,4] | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact |
| Symmetry *n42 [n,4] | *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

*n42 symmetry mutation of expanded tilings: n.4.4.4 vte :
| Symmetry [n,4], (*n42) | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry [n,4], (*n42) | *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

*n42 symmetry mutation of omnitruncated tilings: 4.8.2n vte :
| Symmetry *n42 [n,4] | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry *n42 [n,4] | *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

4n2 symmetry mutations of snub tilings: 3.3.4.3.n vte :
| Symmetry 4n2 | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry 4n2 | 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
### Regular tilings

*n52 symmetry mutation of truncated tilings: 5$^n$ :
| Sphere | Hyperbolic plane | Hyperbolic plane | Hyperbolic plane | Hyperbolic plane | Hyperbolic plane | Hyperbolic plane |
| {5,3} | {5,4} | {5,5} | {5,6} | {5,7} | {5,8} | ...{5,∞} |

## *n62 symmetry
### Regular tilings

*n62 symmetry mutation of regular tilings: {6,n} vte :
| Spherical | Euclidean | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings |
| {6,2} | {6,3} | {6,4} | {6,5} | {6,6} | {6,7} | {6,8} | ... | {6,∞} |

## *n82 symmetry
### Regular tilings

n82 symmetry mutations of regular tilings: 8$^n$ vte :
| Space | Spherical | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact |
| Tiling | | | | | | | | |
| Config. | 8.8 | 8$^3$ | 8$^4$ | 8$^5$ | 8$^6$ | 8$^7$ | 8$^8$ | ...8$^∞$ |