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Conway's Orbifold notation is a mathematical notation populized by the mathematician John Horton Conway .
It gives a description of certain subgroups of the group of three-dimensional obtained by taking the quotient of Euclidean space by the group under consideration. 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 following symbols:

  • positive '' Integer s'' 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 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 there are precisely 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. 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.



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.


EXTERNAL LINKS



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