Conway's orbifold notation

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 under consideration. The notation can be used to describe the so-called wallpaper groups, frieze groups, 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 integers $1,2,3,\dots$
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 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 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 $\frac{n-1}{n}$

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.

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

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.

Link

A field guide to the orbifolds (Notes from class on "Geometry and the Imagination" in Minneapolis, with John Conway Peter Doyle Jane Gilman Bill Thurston, on June 17-28, 1991. See also PDF, 2006)

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

Group theory | Generalized manifolds | Mathematical notation

Gourt Home::Gourt :: Conway's orbifold notation