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 which 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]


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]
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]
Coxeter Hermann–
Fejes Toth
*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
H2checkers 238.png
Hyperbolic domains 932.png
H2checkers 23i.png
H2checkers 245.png
H2checkers 246.png
H2checkers 247.png
H2checkers 248.png
H2checkers 24i.png
H2checkers 255.png
H2checkers 256.png
H2checkers 257.png
H2checkers 266.png
H2checkers 2ii.png
Example general triangles (*pqr)
H2checkers 334.png
H2checkers 335.png
H2checkers 336.png
H2checkers 337.png
H2checkers 33i.png
H2checkers 344.png
H2checkers 366.png
H2checkers 3ii.png
H2checkers 666.png
H2checkers iii.png
Example higher polygons (*pqrs...)
Hyperbolic domains 3222.png
H2chess 246a.png
H2chess 248a.png
H2chess 246b.png
H2chess 248b.png
Uniform tiling 552-t1.png
Uniform tiling 66-t1.png
Uniform tiling 77-t1.png
Uniform tiling 88-t1.png
Hyperbolic domains i222.png
H2chess 24ia.png
H2chess 24ib.png
H2chess 24ic.png
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)]+

See also[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
  • 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-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5

External links[edit]