Orbifold notation

In geometry, orbifold notation (or orbifold signature) is a system, invented 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 follows William Thurston in describing 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 ( ), the frieze groups and wallpaper groups of the Euclidean plane ( ), and their analogues on the hyperbolic plane ( ).

Definition of the notation

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
  • the infinity symbol,
  • 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 (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 indicates a glide reflection
  • the symbol 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

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

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

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
  • n after an asterisk counts as
  • asterisk and 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

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


A perfect snowflake would have *6 symmetry,

The pentagon has symmetry *5, the whole image with arrows 5.

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

Spherical

Fundamental domains of reflective 3D point groups
(*11), C1v=Cs (*22), C2v (*33), C3v (*44), C4v (*55), C5v (*66), C6v

Order 2

Order 4

Order 6

Order 8

Order 10

Order 12
(*221), D1h=C2v (*222), D2h (*223), D3h (*224), D4h (*225), D5h (*226), D6h

Order 4

Order 8

Order 12

Order 16

Order 20

Order 24
(*332), Td (*432), Oh (*532), Ih

Order 24

Order 48

Order 120
Spherical Symmetry Groups[1]
Orbifold
Signature
Coxeter Schönflies Hermann–Mauguin Order
Polyhedral groups
*532[3,5]Ih53m120
532[3,5]+I53260
*432[3,4]Ohm3m48
432[3,4]+O43224
*332[3,3]Td43m24
3*2[3+,4]Thm324
332[3,3]+T2312
Dihedral and cyclic groups: n=3,4,5...
*22n[2,n]Dnhn/mmm or 2nm24n
2*n[2+,2n]Dnd2n2m or nm4n
22n[2,n]+Dnn22n
*nn[n]Cnvnm2n
n*[n+,2]Cnhn/m or 2n2n
[2+,2n+]S2n2n or n2n
nn[n]+Cnnn
Special cases
*222[2,2]D2h2/mmm or 22m28
2*2[2+,4]D2d222m or 2m8
222[2,2]+D2224
*22[2]C2v2m4
2*[2+,2]C2h2/m or 224
[2+,4+]S422 or 24
22[2]+C222
*22[1,2]D1h=C2v1/mmm or 21m24
2*[2+,2]D1d=C2h212m or 1m4
22[1,2]+D1=C2122
*1[ ]C1v=Cs1m2
1*[2,1+]C1h=Cs1/m or 212
[2+,2+]S2=Ci21 or 12
1[ ]+C111

Euclidean plane

Frieze groups

Frieze groups
IUC Cox Schön*
Struct.
Diagram§
Orbifold
Examples
and Conway nickname[2]
Description
p1[∞]+
C
Z

∞∞
F F F F F F F F


hop
(T) Translations only:
This group is singly generated, by a translation by the smallest distance over which the pattern is periodic.
p11g[∞+,2+]
S
Z

∞×
F ᖶ F ᖶ F ᖶ F ᖶ


step
(TG) Glide-reflections and Translations:
This group is singly generated, by a glide reflection, with translations being obtained by combining two glide reflections.
p1m1[∞]
C∞v
Dih

*∞∞
Λ Λ Λ Λ Λ Λ Λ Λ


sidle
(TV) Vertical reflection lines and Translations:
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.
p2[∞,2]+
D
Dih

22∞
S S S S S S S S


spinning hop
(TR) Translations and 180° Rotations:
The group is generated by a translation and a 180° rotation.
p2mg[∞,2+]
D∞d
Dih

2*∞
V Λ V Λ V Λ V Λ


spinning sidle
(TRVG) Vertical reflection lines, Glide reflections, Translations and 180° 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.
p11m[∞+,2]
C∞h
Z×Dih1

∞*
B B B B B B B B


jump
(THG) Translations, Horizontal reflections, Glide reflections:
This group is generated by a translation and the reflection in the horizontal axis. The glide reflection here arises as the composition of translation and horizontal reflection
p2mm[∞,2]
D∞h
Dih×Dih1

*22∞
H H H H H H H H


spinning jump
(TRHVG) Horizontal and Vertical reflection lines, Translations 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.
*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries
§The diagram shows one fundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, and 2-fold gyration points as small green squares.

Wallpaper groups

Fundamental domains of Euclidean reflective groups
(*442), p4m (4*2), p4g
(*333), p3m (632), p6
17 wallpaper groups[3]
Orbifold
Signature
Coxeter Hermann–
Mauguin
Speiser
Niggli
Polya
Guggenhein
Fejes Toth
Cadwell
*632[6,3]p6mC(I)6vD6W16
632[6,3]+p6C(I)6C6W6
*442[4,4]p4mC(I)4D*4W14
4*2[4+,4]p4gCII4vDo4W24
442[4,4]+p4C(I)4C4W4
*333[3[3]] p3m1CII3vD*3W13
3*3[3+,6]p31mCI3vDo3W23
333[3[3]]+ p3CI3C3W3
*2222[∞,2,∞]pmmCI2vD2kkkkW22
2*22[∞,2+,∞]cmmCIV2vD2kgkgW12
22*[(∞,2)+,∞]pmgCIII2vD2kkggW32
22×[∞+,2+,∞+]pggCII2vD2ggggW42
2222[∞,2,∞]+p2C(I)2C2W2
**[∞+,2,∞]pmCIsD1kkW21
[∞+,2+,∞]cmCIIIsD1kgW11
××[∞+,(2,∞)+]pgCII2D1ggW31
o[∞+,2,∞+]p1C(I)1C1W1

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

*63

*3
Example higher polygons (*pqrs...)

*2223

*(23)2

*(24)2

*34

*44

*25

*26

*27

*28

*222

*(2)2

*4

*2

*

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

Hyperbolic Symmetry Groups[4]
-1/χ Orbifolds Coxeter
84*237[7,3]
48*238[8,3]
42237[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

References

  1. Symmetries of Things, Appendix A, page 416
  2. Frieze Patterns Mathematician John Conway created names that relate to footsteps for each of the frieze groups.
  3. Symmetries of Things, Appendix A, page 416
  4. 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
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