Uniform_tiling_symmetry_mutations

Uniform tiling symmetry mutations

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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.

More information Spherical tilings (n = 3..5), Euclidean plane tiling (n = 6) ...

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.

*n32 symmetry

*n42 symmetry

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[Huson-1] [1]
Two Dimensional symmetry Mutations by Daniel Huson

[1]

Spherical tilings (n = 3..5)
*332	*432	*532
Euclidean plane tiling (n = 6)
*632
Hyperbolic plane tilings (n = 7...∞)
*732	*832	... *∞32

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	22, 23 ...	33, 42	52 53 ..., 43, 44 ..., 34, 35 ...
p	-	-	2 ...
*p×	-	-	*2× ...
pppp	-	2222	3333 ...
pppq	-	-	2223...
ppqq	-	-	2233
pp*p	-	-	22*2 ...
p*qr	-	2*22	322 ..., 232 ...
*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 ...
...

Space	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	2³	2⁴	2⁵	...	2^∞

Space	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	...	∞.∞

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

Uniform_tiling_symmetry_mutations

Uniform tiling symmetry mutations

Mutations of orbifolds

*n22 symmetry

Regular tilings

Prism tilings

Antiprism tilings

*n32 symmetry

Regular tilings

Truncated tilings

Quasiregular tilings

Expanded tilings

Omnitruncated tilings

Snub tilings

*n42 symmetry

Regular tilings

Quasiregular tilings

Truncated tilings

Expanded tilings

Omnitruncated tilings

Snub tilings

*n52 symmetry

Regular tilings

*n62 symmetry

Regular tilings

*n82 symmetry

Regular tilings

References

Sources

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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 mutation of regular tilings: {3,n} v t e
Spherical				Euclid.	Compact hyper.		Paraco.	Noncompact hyperbolic

3.3	3³	3⁴	3⁵	3⁶	3⁷	3⁸	3^∞	3¹²ⁱ	3⁹ⁱ	3⁶ⁱ	3³ⁱ

*n32 symmetry mutation of regular tilings: {n,3} v t e
Spherical				Euclidean	Compact hyperb.		Paraco.	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}

*n32 symmetry mutation of truncated tilings: t{n,3} v t e
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	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 v t e
Sym. *n42 [n,3]	Spherical				Euclid.	Compact		Parac.	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: (3.n)² v t e
Sym. *n32 [n,3]	Spherical			Euclid.	Compact hyperb.		Paraco.	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
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			Euclidean	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

n32 symmetry mutations of snub tilings: 3.3.3.3.n v t e
Symmetry n32	Spherical				Euclidean	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 mutation of regular tilings: {n,4} v t e
Spherical		Euclidean	Hyperbolic tilings

2⁴	3⁴	4⁴	5⁴	6⁴	7⁴	8⁴	...∞⁴

*n42 symmetry mutations of quasiregular tilings: (4.n)² v t e
Symmetry *4n2 [n,4]	Spherical	Euclidean	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

4n2 symmetry mutations of snub tilings: 3.3.4.3.n v t e
Symmetry 4n2	Spherical		Euclidean	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.∞

*n42 symmetry mutation of truncated tilings: 4.2n.2n v t e
Symmetry *n42 [n,4]	Spherical		Euclidean	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]
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 expanded tilings: n.4.4.4 v t e
Symmetry [n,4], (*n42)	Spherical	Euclidean	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