Orbifold notation

From Wikipedia, the free encyclopedia

In geometry, orbifold notation (or orbifold signature) is a system popularized by the mathematician John Horton 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 wallpaper groups and frieze groups on the Euclidean plane (E2), the groups on the sphere (S2), and their analogues on the hyperbolic plane (H2).

Contents

[edit] 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 1,2,3,\dots
  • the infinity symbol, \infty
  • the asterisk, *
  • the symbol o, which is called a wonder
  • the symbol x, which is called a miracle

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 point
  • an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a point and reflects through a line (or plane)
  • an x 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.

[edit] 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.

[edit] 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 \frac{n-1}{n}
  • n after an asterisk counts as \frac{n-1}{2 n}
  • asterisk and x 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.

[edit] 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.

[edit] Other objects

The pentagon has symmetry *55, the whole image with arrows 55.
The pentagon has symmetry *55, the whole image with arrows 55.

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 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 11, *11, \infty\infty and *\infty\infty.

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.

[edit] Correspondence tables

The 17 plane wallpaper symmetry groups:[1]

Orbifold
Signature		 International notation
Coxeter and Moses		 Speiser
Niggli		 Polya
Guggenhein		 Fejes Toth
Cadwell
*632 p6m C(I)6v D6 W16
632 p6 C(I)6 C6 W6
*442 p4m C(I)4 D*4 W14
4*2 p4g CII4v Do4 W24
442 p4 C(I)4 C4 W4
*333 p3m1 CII3v D*3 W13
3*3 p31m CI3v Do4 W23
333 p3 CI3 C3 W3
*2222 pmm CI2v D2kkkk W22
2*22 cmm CIV2v D2kgkg W12
22* pmg CIII2v D2kkgg W32
22x pgg CII2v D2gggg W42
2222 p2 C(I)2 C2 W2
** pm CIs D1kk W21
*x cm CIIIs D1kg W11
xx pg CII2 D1gg W31
o p1 C(I)1 C1 W1

The spherical symmetry groups: (N=1,2,3,..)[2]

Orbifold
Signature		 Coxeter Schönflies Weyl Pólya and Meyer International notation
*532 [3,5] Ih 53m
532 [3,5]+ I 532
*432 [3,4] Oh m3m
432 [3,4]+ O 432
*332 [3,3] Td 4-3m
3*2 [3+,4] Th m3
332 [3,3]+ T 23
*22N [2,N] DNh N/mmm or 2N-m2
2*N [2+,2N] DNd 2N-2m or N-m
22N [2,N]+ DN N2
*NN [N] CNv Nm
N* [2,N+] CNh N/m or 2N-
Nx [2+,2N+] S2N 2N- or N-
NN [N]+ CN N

[edit] External links

[edit] See also

[edit] References

  1. ^ Symmetry of Things, Appendix A, page 416
  2. ^ Symmetry of Things, Appendix A, page 416
  • 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 Symmetry of Things 2008, ISBN 978-1-56881-220-5