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User:Tomruen/List of hyperbolic symmetry groups

< User:Tomruen

Hyperbolic plane edit

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 edit

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 edit

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* ... ∞* -
p× 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:

 

See also edit

References edit

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