User:Tomruen/List of hyperbolic symmetry groups
Hyperbolic plane
[edit]| 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:
| (-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:
- Adding a order-p gyration point in the center as p*q. (order ×p)
- If p has divisor r, (p/r)*qr. (order ×p/r)
- An order-p reflection point in the center creates right triangle domains, *(2q).p.2. (order ×2p)
- If p is even,
- you can make an isoceles triangle domain *2q.2q.(p/2), (order ×p)
- as well as triangle *p.p.q (order ×p)
- and also by an alternate set of p/2 central mirrors, as kite-shaped fundamental domains *2.q.2.(p/2). (order ×p)
| 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]- Orbifold notation
- Coxeter notation
- List of regular polytopes
- List of spherical symmetry groups
- List of planar symmetry groups
References
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