# User:Tomruen/List of hyperbolic symmetry groups

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## Hyperbolic plane

Poincaré disk model of fundamental domain triangles
Example right triangles (*2pq)

*237

*238

*239

*23∞

*245

*246

*247

*248

*∞42

*255

*256

*257

*266

*2∞∞
Example general triangles (*pqr)

*334

*335

*336

*337

*33∞

*344

*366

*3∞∞

*666

*∞∞∞
Example higher polygons (*pqrs...)

*2223

*2323

*3333

*22222

*222222

A first few hyperbolic groups, ordered by their Euler characteristic are:

Hyperbolic Symmetry Groups[1]
(-1/χ) Orbifolds Coxeter
(84) *237 [7,3]
(48) *238 [8,3]
(42) 237 [7,3]+
(40) *245 [5,4]
(36 - 26.4) *239, *2.3.10 [9,3], [10,3]
(26.4) *2.3.11 [11,3]
(24) *2.3.12, *246, *334, 3*4, 238 [12,3], [6,4], [(4,3,3)], [3+,8], [8,3]+
(22.3 - 21) *2.3.13, *2.3.14 [13,3], [14,3]
(20) *2.3.15, *255, 5*2, 245 [15,3], [5,5], [5+,4], [5,4]+
(19.2) *2.3.16 [16,3]
(18+2/3) *247 [7,4]
(18) *2.3.18, 239 [18,3], [9,3]+
(17.5-16.2) *2.3.19, *2.3.20, *2.3.21, *2.3.22, *2.3.23 [19,3], [20,3], [20,3], [21,3], [22,3], [23,3]
(16) *2.3.24, *248 [24,3], [8,4]
(15) *2.3.30, *256, *335, 3*5, 2.3.10 [30,3], [6,5], [(5,3,3)], [3+,10], [10,3]+
(14+2/5 - 13+1/3) *2.3.36 ... *2.3.70, *249, *2.4.10 [36,3] ... [60,3], [9,4], [10,4]
(13+1/5) *2.3.66, 2.3.11 [66,3], [11,3]+
(12+8/11) *2.3.105, *257 [105,3], [7,5]
(12+4/7) *2.3.132, *2.4.11 ... [132,3], [11,4], ...
(12) *23∞, *2.4.12, *266, 6*2, *336, 3*6, *344, 4*3, *2223, 2*23, 2.3.12, 246, 334 [∞,3] [12,4], [6,6], [6+,4], [(6,3,3)], [3+,12], [(4,4,3)], [4+,6], ... [12,3]+, [6,4]+ [(4,3,3)]+
...

## Hyperbolic groups from regular polygons

Every regular polyhedron/tiling {p,2q} represents a regular polygon reflective domain, orbifold *qp. Higher symmetry groups can be constructed by:

1. Adding a order-p gyration point in the center as p*q. (order ×p)
2. If p has divisor r, (p/r)*qr. (order ×p/r)
3. An order-p reflection point in the center creates right triangle domains, *(2q).p.2. (order ×2p)
4. If p is even,
1. you can make an isoceles triangle domain *2q.2q.(p/2), (order ×p)
2. as well as triangle *p.p.q (order ×p)
3. and also by an alternate set of p/2 central mirrors, as kite-shaped fundamental domains *2.q.2.(p/2). (order ×p)
Spherical (Platonic)/Euclidean/hyperbolic (Poincaré disc) reflective domains with their orbifold notation
p \ q 2 3 4 5 6 7 8 9
Triangle
*222/222
3*2/332
*432/432

*333/333
3*3/333
*632/632

*444/444
3*4/334
*832/832

*555/555
3*5/335
*10.3.2/10.3.2

*666/666
3*6/336
*12.3.2/12.3.2

*777/777
3*7/337
*14.3.2/14.3.2

*888/888
3*8/338
*16.3.2/16.3.2

*999/999
3*9/339
*18.3.2/18.3.2
Square
*2222/2222
4*2/442
*442/442
2*22/2222
22*

*3333/3333
4*3/443
*642/642
2*33/2323

*4444/4444
4*4/444
*842/842
2*44/2424
*5555/5555
4*5/445
*10.4.2/10.4.2
2*55/2525
*6666/6666
4*6/446
*12.4.2/12.4.2
2*66/2626
*7777/7777
4*7/447
*14.4.2/14.4.2
2*77/2727
*8888/8888
4*8/448
*16.4.2/16.4.2
2*88/2828
*9999/9999
4*9/449
*18.4.2/18.4.2
2*99/2929
Pentagon
*22222
5*2/552
*452/452

*35
5*3/553
*652/652
*45
5*4/554
*10.4.2/10.4.2
Hexagon
*222222
6*2/662
*462/462
3*22
2*222
32*

*36
6*3/663
*662
3*33
2*333
*46
6*4/664
*862/862
3*22
2*444
Heptagon
*2222222
7*2/772
*472/472
*37
7*3/773
*672/672
*47
7*4/774
*872/872
Octagon
*22222222
8*2/882
*482/482
4*22
2*2222
42*
*38
8*3/883
*682/682
4*33
2*3333
*48
8*4/884
*882/882
4*44
2*4444

## 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.[2] This table is not complete for possible hyperbolic orbifolds.

Orbifold Spherical Euclidean Hyperbolic
o - o -
pp 22 ... ∞∞ -
*pp *pp *∞∞ -
p* 2* ... ∞* -
2× ... ∞×
** - ** -
- -
×× - ×× -
ppp 222 333 444 ...
pp* - 22* 33* ...
pp× - 22× 33× ...
pqq p22, 233 244 255 ..., 433 ...
pqr 234, 235 236 237 ..., 245 ...
pq* - - 23* ...
pqx - - 23× ...
p*q 2*p 3*3, 4*2 5*2 ..., 4*3 ..., 3*4 ...
*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 - - *2223...
*ppppp - - *22222 ...
...

Example, comparing 22* symmetry of the plane to 23* symmetry of the hyperbolic plane:

## References

1. ^ Symmetries of Things, Chapter 18, More on Hyperbolic groups, Enumerating hyperbolic groups, p239
2. ^ Two Dimensional symmetry Mutations by Daniel Huson, [1]