Uniform tiling symmetry mutations

From Wikipedia, the free encyclopedia
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...∞)
Truncated heptagonal tiling.svg
*732 		H2-8-3-trunc-dual.svg
*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[]

Family of regular hosohedra · *n22 symmetry mutations of regular hosohedral tilings: nn
Space Spherical Euclidean
Tiling name (Monogonal)
Henagonal hosohedron 		Digonal hosohedron (Triangular)
Trigonal hosohedron 		(Tetragonal)
Square hosohedron 		Pentagonal hosohedron Hexagonal hosohedron Heptagonal hosohedron Octagonal hosohedron Enneagonal hosohedron Decagonal hosohedron Hendecagonal hosohedron Dodecagonal hosohedron ... Apeirogonal hosohedron
Tiling image Spherical henagonal hosohedron.png 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.svg
Schläfli symbol {2,1} {2,2} {2,3} {2,4} {2,5} {2,6} {2,7} {2,8} {2,9} {2,10} {2,11} {2,12} ... {2,∞}
Coxeter diagram CDel node 1.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 6.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 7.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 8.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 9.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 10.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 11.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 12.pngCDel node.png ... CDel node 1.pngCDel 2x.pngCDel node.pngCDel infin.pngCDel node.png
Faces and edges 1 2 3 4 5 6 7 8 9 10 11 12 ... ∞
Vertices 2 ... 2
Vertex config. 2 2.2 23 24 25 26 27 28 29 210 211 212 ... 2∞
Family of regular dihedra · *n22 symmetry mutations of regular dihedral tilings: nn
Space Spherical Euclidean
Tiling name (Hengonal)
Monogonal dihedron 		Digonal dihedron (Triangular)
Trigonal dihedron 		(Tetragonal)
Square dihedron 		Pentagonal dihedron Hexagonal dihedron ... Apeirogonal dihedron
Tiling image Monogonal dihedron.svg Digonal dihedron.svg Trigonal dihedron.svg Tetragonal dihedron.svg Pentagonal dihedron.svg Hexagonal dihedron.svg ... Apeirogonal tiling.svg
Schläfli symbol {1,2} {2,2} {3,2} {4,2} {5,2} {6,2} ... {∞,2}
Coxeter diagram CDel node 1.pngCDel 1x.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2x.pngCDel node.png ... CDel node 1.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel node.png
Faces 2 {1} 2 {2} 2 {3} 2 {4} 2 {5} 2 {6} ... 2 {∞}
Edges and vertices 1 2 3 4 5 6 ... ∞
Vertex config. 1.1 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.svg
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.svg
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}
  • v
Spherical Euclid. Compact hyper. Paraco. Noncompact hyperbolic
Trigonal dihedron.svg Uniform tiling 332-t2.png Uniform tiling 432-t2.png Uniform tiling 532-t2.png Uniform polyhedron-63-t2.png Order-7 triangular tiling.svg H2-8-3-primal.svg H2 tiling 23i-4.png H2 tiling 23j12-4.png H2 tiling 23j9-4.png H2 tiling 23j6-4.png H2 tiling 23j3-4.png
3.3 33 34 35 36 37 38 3∞ 312i 39i 36i 33i
*n32 symmetry mutation of regular tilings: {n,3}
  • v
Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic
Spherical trigonal hosohedron.png Uniform tiling 332-t0.png Uniform tiling 432-t0.png Uniform tiling 532-t0.png Uniform polyhedron-63-t0.png Heptagonal tiling.svg H2-8-3-dual.svg H2-I-3-dual.svg H2 tiling 23j12-1.png H2 tiling 23j9-1.png H2 tiling 23j6-1.png H2 tiling 23j3-1.png
{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}
  • v
Symmetry
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*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 		Spherical triangular prism.png Uniform tiling 332-t01-1-.png Uniform tiling 432-t01.png Uniform tiling 532-t01.png Uniform tiling 63-t01.svg Truncated heptagonal tiling.svg H2-8-3-trunc-dual.svg H2 tiling 23i-3.png H2 tiling 23j12-3.png H2 tiling 23j9-3.png H2 tiling 23j6-3.png
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 		Spherical trigonal bipyramid.png Spherical triakis tetrahedron.png Spherical triakis octahedron.png Spherical triakis icosahedron.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Order-7 triakis triangular tiling.svg H2-8-3-kis-primal.svg Ord-infin triakis triang til.png
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
  • v
Sym.
*n42
[n,3] 		Spherical Euclid. Compact Parac. Noncompact hyperbolic
*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 		Hexagonal dihedron.svg Uniform tiling 332-t12.png Uniform tiling 432-t12.png Uniform tiling 532-t12.png Uniform tiling 63-t12.svg Truncated order-7 triangular tiling.svg H2-8-3-trunc-primal.svg H2 tiling 23i-6.png H2 tiling 23j12-6.png H2 tiling 23j9-6.png H2 tiling 23j-6.png
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 		Hexagonal Hosohedron.svg Spherical triakis tetrahedron.png Spherical tetrakis hexahedron.png Spherical pentakis dodecahedron.png Uniform tiling 63-t2.svg Heptakis heptagonal tiling.svg H2-8-3-kis-dual.svg H2checkers 33i.png
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
  • v
  • t
Sym.
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*332
[3,3]
Td 		*432
[4,3]
Oh 		*532
[5,3]
Ih 		*632
[6,3]
p6m 		*732
[7,3]
  		*832
[8,3]...
  		*∞32
[∞,3]
  		[12i,3] [9i,3] [6i,3]
Figure
Quasiregular fundamental domain.png 		Uniform tiling 332-t1-1-.png Uniform tiling 432-t1.png Uniform tiling 532-t1.png Uniform tiling 63-t1.svg Triheptagonal tiling.svg H2-8-3-rectified.svg H2 tiling 23i-2.png H2 tiling 23j12-2.png H2 tiling 23j9-2.png H2 tiling 23j6-2.png
Figure
Half quasiregular fundamental domain.png 		Uniform tiling 332-t02.png Uniform tiling 333-t12.png H2 tiling 334-3.png H2 tiling 33i-3.png
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
CDel node.pngCDel n.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel labelp.pngCDel branch 11.pngCDel split2.pngCDel node.png 		CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node.png CDel branch 11.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.png CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node.png
Dual uniform figures
Dual
conf. 		Uniform tiling 432-t0.png
V(3.3)2 		Spherical rhombic dodecahedron.png
V(3.4)2 		Spherical rhombic triacontahedron.png
V(3.5)2 		Rhombic star tiling.png
V(3.6)2 		7-3 rhombille tiling.svg
V(3.7)2 		H2-8-3-rhombic.svg
V(3.8)2 		Ord3infin qreg rhombic til.png
V(3.∞)2
Symmetry mutations of dual quasiregular tilings: V(3.n)2
*n32 Spherical Euclidean Hyperbolic
*332 *432 *532 *632 *732 *832... *∞32
Tiling Uniform tiling 432-t0.png Spherical rhombic dodecahedron.png Spherical rhombic triacontahedron.png Rhombic star tiling.png 7-3 rhombille tiling.svg H2-8-3-rhombic.svg Ord3infin qreg rhombic til.png
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[]

*n42 symmetry mutation of expanded tilings: 3.4.n.4
  • v
Symmetry
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*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 Spherical triangular prism.png Uniform tiling 332-t02.png Uniform tiling 432-t02.png Uniform tiling 532-t02.png Uniform polyhedron-63-t02.png Rhombitriheptagonal tiling.svg H2-8-3-cantellated.svg H2 tiling 23i-5.png H2 tiling 23j12-5.png H2 tiling 23j9-5.png H2 tiling 23j6-5.png
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 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 hexecontahedron.png
V3.4.5.4 		Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V3.4.6.4 		Deltoidal triheptagonal tiling.svg
V3.4.7.4 		H2-8-3-deltoidal.svg
V3.4.8.4 		Deltoidal triapeirogonal til.png
V3.4.∞.4

Omnitruncated tilings[]

*n32 symmetry mutations of omnitruncated tilings: 4.6.2n
  • v
Sym.
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*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 Spherical truncated trigonal prism.png Uniform tiling 332-t012.png Uniform tiling 432-t012.png Uniform tiling 532-t012.png Uniform polyhedron-63-t012.png Truncated triheptagonal tiling.svg H2-8-3-omnitruncated.svg H2 tiling 23i-7.png H2 tiling 23j12-7.png H2 tiling 23j9-7.png H2 tiling 23j6-7.png H2 tiling 23j3-7.png
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 Spherical hexagonal bipyramid.png Spherical tetrakis hexahedron.png Spherical disdyakis dodecahedron.png Spherical disdyakis triacontahedron.png Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg H2checkers 237.png H2checkers 238.png H2checkers 23i.png H2 checkers 23j12.png H2 checkers 23j9.png H2 checkers 23j6.png H2 checkers 23j3.png
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
  • v
Symmetry
n32 		Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures 		Spherical trigonal antiprism.png Spherical snub tetrahedron.png Spherical snub cube.png Spherical snub dodecahedron.png Uniform tiling 63-snub.svg Snub triheptagonal tiling.svg H2-8-3-snub.svg Uniform tiling i32-snub.png
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 		Uniform tiling 432-t0.png Uniform tiling 532-t0.png Spherical pentagonal icositetrahedron.png Spherical pentagonal hexecontahedron.png Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg 7-3 floret pentagonal tiling.svg H2-8-3-floret.svg Order-3-infinite floret pentagonal tiling.png
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}
  • v
Spherical Euclidean Compact hyperbolic Paracompact
Uniform tiling 432-t0.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 		Uniform tiling 44-t0.svg
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png 		H2-5-4-primal.svg
{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png 		H2 tiling 246-4.png
{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png 		H2 tiling 247-4.png
{4,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png 		H2 tiling 248-4.png
{4,8}...
CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png 		H2 tiling 24i-4.png
{4,∞}
CDel node 1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png
*n42 symmetry mutation of regular tilings: {n,4}
Spherical Euclidean Hyperbolic tilings
Spherical square hosohedron.png Spherical square bipyramid.png Uniform tiling 44-t0.svg H2-5-4-dual.svg H2 tiling 246-1.png H2 tiling 247-1.png H2 tiling 248-1.png H2 tiling 24i-1.png
24 34 44 54 64 74 84 ...∞4

Quasiregular tilings[]

*n42 symmetry mutations of quasiregular tilings: (4.n)2
  • v
Symmetry
*4n2
[n,4] 		Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4] 		*442
[4,4] 		*542
[5,4] 		*642
[6,4] 		*742
[7,4] 		*842
[8,4]... 		*∞42
[∞,4] 		 
[ni,4]
Figures Uniform tiling 432-t1.png Uniform tiling 44-t1.png H2-5-4-rectified.svg H2 tiling 246-2.png H2 tiling 247-2.png H2 tiling 248-2.png H2 tiling 24i-2.png
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 Paracompact Noncompact
*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. 		Spherical rhombic dodecahedron.png
V4.3.4.3 		Uniform tiling 44-t0.svg
V4.4.4.4 		H2-5-4-rhombic.svg
V4.5.4.5 		Ord64 qreg rhombic til.png
V4.6.4.6 		Ord74 qreg rhombic til.png
V4.7.4.7 		Ord84 qreg rhombic til.png
V4.8.4.8 		Ord4infin qreg rhombic til.png
V4.∞.4.∞ 		V4.∞.4.∞

Truncated tilings[]

*n42 symmetry mutation of truncated tilings: 4.2n.2n
  • v
Symmetry
*n42
[n,4] 		Spherical Euclidean Compact hyperbolic Paracomp.
*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 		Spherical square prism.png Uniform tiling 432-t12.png Uniform tiling 44-t01.png H2-5-4-trunc-dual.svg H2 tiling 246-3.png H2 tiling 247-3.png H2 tiling 248-3.png H2 tiling 24i-3.png
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 		Spherical square bipyramid.png Spherical tetrakis hexahedron.png 1-uniform 2 dual.svg H2-5-4-kis-primal.svg Order-6 tetrakis square tiling.png Hyperbolic domains 772.png Order-8 tetrakis square tiling.png H2checkers 2ii.png
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
  • v
Symmetry
*n42
[n,4] 		Spherical Euclidean Compact hyperbolic Paracompact
*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 		Octagonal dihedron.svg Uniform tiling 432-t01.png Uniform tiling 44-t12.svg H2-5-4-trunc-primal.svg H2 tiling 246-6.png H2 tiling 247-6.png H2 tiling 248-6.png H2 tiling 24i-6.png
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 		Spherical octagonal hosohedron.png Spherical triakis octahedron.png 1-uniform 2 dual.svg H2-5-4-kis-dual.svg Order4 hexakis hexagonal til.png Order4 heptakis heptagonal til.png H2-8-3-primal.svg Ord4 apeirokis apeirogonal til.png
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
  • v
Symmetry
[n,4], (*n42) 		Spherical Euclidean Compact hyperbolic Paracomp.
*342
[3,4] 		*442
[4,4] 		*542
[5,4] 		*642
[6,4] 		*742
[7,4] 		*842
[8,4] 		*∞42
[∞,4]
Expanded
figures 		Uniform tiling 432-t02.png Uniform tiling 44-t02.png H2-5-4-cantellated.svg Uniform tiling 64-t02.png Uniform tiling 74-t02.png Uniform tiling 84-t02.png H2 tiling 24i-5.png
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. 		Spherical deltoidal icositetrahedron.png
V3.4.4.4 		Uniform tiling 44-t0.svg
V4.4.4.4 		H2-5-4-deltoidal.svg
V5.4.4.4 		Deltoidal tetrahexagonal til.png
V6.4.4.4 		Deltoidal tetraheptagonal til.png
V7.4.4.4 		Deltoidal tetraoctagonal til.png
V8.4.4.4 		Deltoidal tetraapeirogonal tiling.png
V∞.4.4.4

Omnitruncated tilings[]

*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
  • v
Symmetry
*n42
[n,4] 		Spherical Euclidean Compact hyperbolic Paracomp.
*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 		Spherical octagonal prism2.png
4.8.4 		Uniform tiling 432-t012.png
4.8.6 		Uniform tiling 44-t012.png
4.8.8 		H2-5-4-omnitruncated.svg
4.8.10 		H2 tiling 246-7.png
4.8.12 		H2 tiling 247-7.png
4.8.14 		H2 tiling 248-7.png
4.8.16 		H2 tiling 24i-7.png
4.8.∞
Omnitruncated
duals 		Spherical octagonal bipyramid2.png
V4.8.4 		Spherical disdyakis dodecahedron.png
V4.8.6 		1-uniform 2 dual.svg
V4.8.8 		H2-5-4-kisrhombille.svg
V4.8.10 		Hyperbolic domains 642.png
V4.8.12 		Hyperbolic domains 742.png
V4.8.14 		Hyperbolic domains 842.png
V4.8.16 		H2checkers 24i.png
V4.8.∞

Snub tilings[]

4n2 symmetry mutations of snub tilings: 3.3.4.3.n
  • v
Symmetry
4n2 		Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Snub
figures 		Spherical square antiprism.png Spherical snub cube.png Uniform tiling 44-snub.png H2-5-4-snub.svg Uniform tiling 64-snub.png Uniform tiling 74-snub.png Uniform tiling 84-snub.png Uniform tiling i42-snub.png
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 		Spherical tetragonal trapezohedron.png Spherical pentagonal icositetrahedron.png Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg H2-5-4-floret.svg
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: 5n
Sphere Hyperbolic plane
Uniform tiling 532-t0.png
{5,3} 		H2-5-4-dual.svg
{5,4} 		H2 tiling 255-1.png
{5,5} 		H2 tiling 256-1.png
{5,6} 		H2 tiling 257-1.png
 H2 tiling 258-1.png
{5,8} 		H2 tiling 25i-1.png
...{5,∞}

*n62 symmetry[]

Regular tilings[]

*n62 symmetry mutation of regular tilings: {6,n}
  • v
Spherical Euclidean Hyperbolic tilings
Hexagonal dihedron.svg
{6,2} 		Uniform tiling 63-t0.svg
{6,3} 		H2 tiling 246-1.png
{6,4} 		H2 tiling 256-1.png
{6,5} 		H2 tiling 266-4.png
{6,6} 		H2 tiling 267-4.png
 H2 tiling 268-4.png
{6,8} 		... H2 tiling 26i-4.png
{6,∞}

*n82 symmetry[]

Regular tilings[]

n82 symmetry mutations of regular tilings: 8n
  • v
Space Spherical Compact hyperbolic Paracompact
Tiling H2-8-3-dual.svg H2 tiling 248-1.png H2 tiling 258-1.png H2 tiling 268-1.png H2 tiling 278-1.png H2 tiling 288-4.png H2 tiling 28i-4.png
Config. 8.8 83 84 86 88 ...

References[]

  1. ^ a b 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 [1]
  • From hyperbolic 2-space to Euclidean 3-space: Tilings and patterns via topology Stephen Hyde
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