Orbifold notation

Notation for 2-dimensional spherical, euclidean and hyperbolic symmetry groups

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.

The following types of Euclidean transformation can occur in a group described by orbifold notation:

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:

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.

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.

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 of an orbifold can be read from its Conway symbol, as follows. Each feature has a value:

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.

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.

*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries

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