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The Orbifold notation is a mathematical notation invented by the mathematican John Horton Conway .
It gives a description of certain subgroups of the group of three dimensional Eudlidean transformations E^3 . The advantage of the notation is that it describes these groups in 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 so described. The notation can be used to describe the so-called Wallpaper Group s, Frieze Group s, and Point Groups In Three Dimensions .


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 follow symbols:

  • positive '' Integer s'' 1,2,3,\dots

  • the '' Infinity '' symbol, \infty

  • the '' Asterisk '', ---

  • the symbol x, which is called a ''wonder''

  • the symbol o, 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 Eudlidean 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 2''n'' 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 Group s occur in this way.

  • the exceptional symbol ''o'' indicates that are precisely are two linearly independent translations.



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 rac{n-1}{n}


  • ''n'' after an asterisk counts as rac{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. 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.



OTHER OBJECTS

  • 55, the whole image with arrows 55.]]

  • ''nn''.


  • 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.


REFERENCES

  • 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, U.K., 1990; London Math. Soc. Lecture Notes Series 165. Cambridge University Press, Cambridge. pp. 438–447