Orbifold notation

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In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration.

Groups representable in this notation include the point groups on the sphere (S^2), the frieze groups and wallpaper groups of the Euclidean plane (E^2), and their analogues on the hyperbolic plane (H^2).

Definition of the notation[edit]

The following types of Euclidean transformation can occur in a group described by orbifold notation:

  • reflection through a line (or plane)
  • translation by a vector
  • rotation of finite order around a point
  • infinite rotation around a line in 3-space
  • glide-reflection, i.e. reflection followed by translation.

All translations which occur are assumed to form a discrete subgroup of the group symmetries being described.

Each group is denoted in orbifold notation by a finite string made up from the following symbols:

  • positive integers  1,2,3,\dots
  • the infinity symbol,  \infty
  • the asterisk, *
  • the symbol o (a solid circle in older documents), which is called a wonder and also a handle because it topologically represents a torus (1-handle) closed surface. Patterns repeat by two translation.
  • the symbol \times (an open circle in older documents), which is called a miracle and represents a topological crosscap where a pattern repeats as a mirror image without crossing a mirror line.

A string written in boldface represents a group of symmetries of Euclidean 3-space. A string not written in boldface represents a group of symmetries of the Euclidean plane, which is assumed to contain two independent translations.

Each symbol corresponds to a distinct transformation:

  • an integer n to the left of an asterisk indicates a rotation of order n around a gyration point
  • an integer n to the right of an asterisk indicates a transformation of order 2n wjhich rotates around a kaleidoscopic point and reflects through a line (or plane)
  • an \times indicates a glide reflection
  • the symbol  \infty indicates infinite rotational symmetry around a line; it can only occur for bold face groups. By abuse of language, we might say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way.
  • the exceptional symbol o indicates that there are precisely two linearly independent translations.

Good orbifolds[edit]

An orbifold symbol is called good if it is not one of the following: p, pq, *p, *pq, for p,q>=2, and p≠q.

Chirality and achirality[edit]

An object is chiral if its symmetry group contains no reflections; otherwise it is called achiral. The corresponding orbifold is orientable in the chiral case and non-orientable otherwise.

The Euler characteristic and the order[edit]

The Euler characteristic of an orbifold can be read from its Conway symbol, as follows. Each feature has a value:

  • n without or before an asterisk counts as  \frac{n-1}{n}
  • n after an asterisk counts as  \frac{n-1}{2 n}
  • asterisk and \times count as 1
  • o counts as 2.

Subtracting the sum of these values from 2 gives the Euler characteristic.

If the sum of the feature values is 2, the order is infinite, i.e., the notation represents a wallpaper group or a frieze group. Indeed, Conway's "Magic Theorem" indicates that the 17 wallpaper groups are exactly those with the sum of the feature values equal to 2. Otherwise, the order is 2 divided by the Euler characteristic.

Equal groups[edit]

The following groups are isomorphic:

  • 1* and *11
  • 22 and 221
  • *22 and *221
  • 2* and 2*1.

This is because 1-fold rotation is the "empty" rotation.

Two-dimensional groups[edit]

Bentley Snowflake13.jpg
A perfect snowflake would have *6• symmetry,
Pentagon symmetry as mirrors 2005-07-08.png
The pentagon has symmetry *5•, the whole image with arrows 5•.
Flag of Hong Kong.svg
The Flag of Hong Kong has 5 fold rotation symmetry, 5•.

The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a third dimension to the object which does not add or spoil symmetry. For example, for a 2D image we can consider a piece of carton with that image displayed on one side; the shape of the carton should be such that it does not spoil the symmetry, or it can be imagined to be infinite. Thus we have n• and *n•. The bullet (•) is added on one- and two-dimensional groups to imply the existence of a fixed point. (In three-dimensions these groups exist in an n-fold digonal orbifold and are represented as nn and *nn.)

Similarly, a 1D image can be drawn horizontally on a piece of carton, with a provision to avoid additional symmetry with respect to the line of the image, e.g. by drawing a horizontal bar under the image. Thus the discrete symmetry groups in one dimension are *•, *1•, ∞• and *∞•.

Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the Cartesian product of the object and an asymmetric 2D or 1D object, respectively.

Correspondence tables[edit]

Spherical[edit]

Fundamental domains of reflective 3D point groups
(*11), C1v (*22), C2v (*33), C3v (*44), C4v (*55), C5v (*66), C6v
Spherical digonal hosohedron2.png
Order 2
Spherical square hosohedron2.png
Order 4
Spherical hexagonal hosohedron2.png
Order 6
Spherical octagonal hosohedron2.png
Order 8
Spherical decagonal hosohedron2.png
Order 10
Spherical dodecagonal hosohedron2.png
Order 12
(*221), D1h (*222), D2h (*223), D3h (*224), D4h (*225), D5h (*226), D6h
Spherical digonal bipyramid2.png
Order 4
Spherical square bipyramid2.png
Order 8
Spherical hexagonal bipyramid2.png
Order 12
Spherical octagonal bipyramid2.png
Order 16
Spherical decagonal bipyramid2.png
Order 20
Spherical dodecagonal bipyramid2.png
Order 24
(*332), Td (*432), Oh (*532), Ih
Tetrahedral reflection domains.png
Order 24
Octahedral reflection domains.png
Order 48
Icosahedral reflection domains.png
Order 120
Spherical Symmetry Groups[1]
Orbifold
Signature
Coxeter Schönflies Hermann–Mauguin Order
Polyhedral groups
*532 [3,5] Ih 53m 120
532 [3,5]+ I 532 60
*432 [3,4] Oh m3m 48
432 [3,4]+ O 432 24
*332 [3,3] Td 43m 24
3*2 [3+,4] Th m3 24
332 [3,3]+ T 23 12
Dihedral and cyclic groups: n=3,4,5...
*22n [2,n] Dnh n/mmm or 2nm2 4n
2*n [2+,2n] Dnd 2n2m or nm 4n
22n [2,n]+ Dn n2 2n
*nn [n] Cnv nm 2n
n* [n+,2] Cnh n/m or 2n 2n
[2+,2n+] S2n 2n or n 2n
nn [n]+ Cn n n
Special cases
*222 [2,2] D2h 2/mmm or 22m2 8
2*2 [2+,4] D2d 222m or 2m 8
222 [2,2]+ D2 22 4
*22 [2] C2v 2m 4
2* [2+,2] C2h 2/m or 22 4
[2+,4+] S4 22 or 2 4
22 [2]+ C2 2 2
*22 [1,2] D1h 1/mmm or 21m2 4
2* [2+,2] D1d 212m or 1m 4
22 [1,2]+ D1 12 2
*1 [ ] C1v 1m 2
1* [2,1+] C1h 1/m or 21 2
[2+,2+] S2 21 or 1 2
1 [ ]+ C1 1 1

Euclidean plane[edit]

Frieze groups[edit]

Frieze groups
Notations Description Examples
IUC Orbifold Coxeter Schönflies*
p1 ∞∞ [∞]+ C (hop): Translations only. This group is singly generated, with a generator being a translation by the smallest distance over which the pattern is periodic. Abstract group: Z, the group of integers under addition. Frieze example p1.png
p1m1 *∞∞ [∞] C∞v (sidle): Translations and reflections across certain vertical lines. The group is the same as the non-trivial group in the one-dimensional case; it is generated by a translation and a reflection in the vertical axis. The elements in this group correspond to isometries (or equivalently, bijective affine transformations) of the set of integers, and so it is isomorphic to a semidirect product of the integers with Z2. Abstract group: Dih, the infinite dihedral group. Frieze example p1m1.png
p11m ∞* [∞+,2] C∞h (jump): Translations, the reflection in the horizontal axis and glide reflections. This group is generated by a translation and the reflection in the horizontal axis. Abstract group: Z × Z2 Frieze example p11m.png
p11g ∞× [∞+,2+] S (step): Glide-reflections and translations. This group is generated by a glide reflection, with translations being obtained by combining two glide reflections. Abstract group: Z Frieze example p11g.png
p2 22∞ [2,∞]+ D (spinning hop): Translations and 180° rotations. The group is generated by a translation and a 180° rotation. Abstract group: Dih Frieze example p2.png
p2mg 2*∞ [2+,∞] D∞d (spinning sidle): Reflections across certain vertical lines, glide reflections, translations and rotations. The translations here arise from the glide reflections, so this group is generated by a glide reflection and either a rotation or a vertical reflection. Abstract group: Dih Frieze example p2mg.png
p2mm *22∞ [2,∞] D∞h (spinning jump): Translations, glide reflections, reflections in both axes and 180° rotations. This group requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis. Abstract group: Dih × Z2 Frieze example p2mm.png
*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries

Wallpaper groups[edit]

Fundamental domains of Euclidean reflective groups
(*442), p4m (4*2), p4g
Uniform tiling 44-t1.png Tile V488 bicolor.svg
(*333), p3m (632), p6
Tile 3,6.svg Tile V46b.svg
17 wallpaper groups[2]
Orbifold
Signature
Coxeter Hermann–
Mauguin
Speiser
Niggli
Polya
Guggenhein
Fejes Toth
Cadwell
*632 [6,3] p6m C(I)6v D6 W16
632 [6,3]+ p6 C(I)6 C6 W6
*442 [4,4] p4m C(I)4 D*4 W14
4*2 [4+,4] p4g CII4v Do4 W24
442 [4,4]+ p4 C(I)4 C4 W4
*333 [3[3]] p3m1 CII3v D*3 W13
3*3 [3+,6] p31m CI3v Do3 W23
333 [3[3]]+ p3 CI3 C3 W3
*2222 [∞,2,∞] pmm CI2v D2kkkk W22
2*22 [∞,2+,∞] cmm CIV2v D2kgkg W12
22* [(∞,2)+,∞] pmg CIII2v D2kkgg W32
22× [∞+,2+,∞+] pgg CII2v D2gggg W42
2222 [∞,2,∞]+ p2 C(I)2 C2 W2
** [∞+,2,∞] pm CIs D1kk W21
[∞+,2+,∞] cm CIIIs D1kg W11
×× [∞+,(2,∞)+] pg CII2 D1gg W31
o [∞+,2,∞+] p1 C(I)1 C1 W1

Hyperbolic plane[edit]

Poincaré disk model of fundamental domain triangles
Example right triangles (*2pq)
H2checkers 237.png
*237
H2checkers 238.png
*238
Hyperbolic domains 932.png
*239
H2checkers 23i.png
*23∞
H2checkers 245.png
*245
H2checkers 246.png
*246
H2checkers 247.png
*247
H2checkers 248.png
*248
H2checkers 24i.png
*∞42
H2checkers 255.png
*255
H2checkers 256.png
*256
H2checkers 257.png
*257
H2checkers 266.png
*266
H2checkers 2ii.png
*2∞∞
Example general triangles (*pqr)
H2checkers 334.png
*334
H2checkers 335.png
*335
H2checkers 336.png
*336
H2checkers 337.png
*337
H2checkers 33i.png
*33∞
H2checkers 344.png
*344
H2checkers 366.png
*366
H2checkers 3ii.png
*3∞∞
H2checkers 666.png
*63
H2checkers iii.png
*∞3
Example higher polygons (*pqrs...)
Hyperbolic domains 3222.png
*2223
H2chess 246a.png
*(23)2
H2chess 248a.png
*(24)2
H2chess 246b.png
*34
H2chess 248b.png
*44
Uniform tiling 552-t1.png
*25
Uniform tiling 66-t1.png
*26
Uniform tiling 77-t1.png
*27
Uniform tiling 88-t1.png
*28
Hyperbolic domains i222.png
*222∞
H2chess 24ia.png
*(2∞)2
H2chess 24ib.png
*∞4
H2chess 24ic.png
*2
H2chess iiic.png
*∞

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

Hyperbolic Symmetry Groups[3]
(-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], [∞,3,∞], [12,3]+, [6,4]+ [(4,3,3)]+
...


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

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...∞)
H2 tiling 237-3.png
*732
H2 tiling 238-3.png
*832
H2 tiling 23i-3.png
... *∞32
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 ...
...

See also[edit]

References[edit]

  1. ^ Symmetries of Things, Appendix A, page 416
  2. ^ Symmetries of Things, Appendix A, page 416
  3. ^ Symmetries of Things, Chapter 18, More on Hyperbolic groups, Enumerating hyperbolic groups, p239
  4. ^ Two Dimensional symmetry Mutations by Daniel Huson
  • John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, and William P. Thurston. On Three-dimensional Orbifolds and Space Groups. Contributions to Algebra and Geometry, 42(2):475-507, 2001.
  • J. H. Conway, D. H. Huson. The Orbifold Notation for Two-Dimensional Groups. Structural Chemistry, 13 (3-4): 247-257, August 2002.
  • J. H. Conway (1992). "The Orbifold Notation for Surface Groups". In: M. W. Liebeck and J. Saxl (eds.), Groups, Combinatorics and Geometry, Proceedings of the L.M.S. Durham Symposium, July 5–15, Durham, UK, 1990; London Math. Soc. Lecture Notes Series 165. Cambridge University Press, Cambridge. pp. 438–447
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5

External links[edit]