Uniform tiling symmetry mutations

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.

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

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
*n22 symmetry mutations of hosohedral tilings: nn
Space Spherical Euclidean
Tiling
Config. 2.2 23 24 25 26 27 28 29 210 211 212 2∞
*n22 symmetry mutations of dihedral tilings: nn
Space Spherical Euclidean
Tiling
Config. 2.2 3.3 4.4 5.5 6.6 ...∞.∞

Prism tilings
*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
*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

Regular tilings

Truncated tilings

Quasiregular tilings

Expanded tilings
*n42 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

Snub tilings

*n42 symmetry

Regular tilings

Quasiregular tilings

Truncated tilings

Expanded tilings

Omnitruncated tilings

Snub tilings

*n52 symmetry

Regular tilings
*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

Regular tilings

*n82 symmetry

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

References
  1. Two Dimensional symmetry Mutations by Daniel Huson

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