Uniform tiling symmetry mutations

From Infogalactic: the planetary knowledge core
(Redirected from Mutation of orbifolds)
Jump to: navigation, search
Example *n32 symmetry mutations
Spherical tilings (n = 3..5)
Uniform tiling 332-t01-1-.png
*332		 Uniform tiling 432-t01.png
*432		 Uniform tiling 532-t01.png
*532
Euclidean plane tiling (n = 6)
Uniform tiling 63-t01.png
*632
Hyperbolic plane tilings (n = 7...∞)
H2 tiling 237-3.png
*732		 H2 tiling 238-3.png
*832		 H2 tiling 23i-3.png
... *∞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

*n22 symmetry mutations of hosohedral tilings: nn
Space Spherical Euclidean
Tiling Spherical digonal hosohedron.png Spherical trigonal hosohedron.png Spherical square hosohedron.png Spherical pentagonal hosohedron.png Spherical hexagonal hosohedron.png Spherical heptagonal hosohedron.png Spherical octagonal hosohedron.png Spherical enneagonal hosohedron.png Spherical decagonal hosohedron.png Spherical hendecagonal hosohedron.png Spherical dodecagonal hosohedron.png Apeirogonal hosohedron.png
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 Digonal dihedron.png Trigonal dihedron.png Tetragonal dihedron.png Pentagonal dihedron.png Hexagonal dihedron.png Apeirogonal tiling.png
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 Spherical triangular prism.png Spherical square prism.png Spherical pentagonal prism.png Spherical hexagonal prism.png Spherical heptagonal prism.png Spherical octagonal prism.png Spherical decagonal prism.png Infinite prism tiling.png
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 Spherical digonal antiprism.png Spherical trigonal antiprism.png Spherical square antiprism.png Spherical pentagonal antiprism.png Spherical hexagonal antiprism.png Spherical heptagonal antiprism.png Spherical octagonal antiprism.png Infinite antiprism.png
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.		 Spherical trigonal bipyramid.png
V3.4.2.4		 Spherical rhombic dodecahedron.png
V3.4.3.4		 Spherical deltoidal icositetrahedron.png
V3.4.4.4		 Spherical deltoidal icositetrahedron.png
V3.4.5.4		 Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V3.4.6.4		 Deltoidal triheptagonal til.png
V3.4.7.4		 Deltoidal trioctagonal til.png
V3.4.8.4		 Deltoidal triapeirogonal til.png
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
Uniform tiling 532-t0.png
{5,3}		 H2 tiling 245-1.png
{5,4}		 H2 tiling 255-1.png
{5,5}		 H2 tiling 256-1.png
{5,6}		 H2 tiling 257-1.png
{5,7}		 H2 tiling 258-1.png
{5,8}		 H2 tiling 25i-1.png
...{5,∞}

*n62 symmetry

Regular tilings

*n82 symmetry

Regular tilings

n82 symmetry mutations of regular tilings: 8n
Space Spherical Compact hyperbolic Paracompact
Tiling H2 tiling 238-1.png H2 tiling 248-1.png H2 tiling 258-1.png H2 tiling 268-1.png H2 tiling 278-1.png 50px H2 tiling 28i-4.png
Config. 8.8 83 84 85 86 87 88 ...8∞

References

  1. ↑ 1.0 1.1 Two Dimensional symmetry Mutations by Daniel Huson
Retrieved from "https://infogalactic.com/w/index.php?title=Uniform_tiling_symmetry_mutations&oldid=696520146#Mutations_of_orbifolds"