Екі өлшемді сфералық, евкалидтік және гиперболалық симметрия топтарының белгілеуі

Notation for 2 dimensional spherical, euclidean and hyperbolic symmetry groups
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by 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. Groups representable in this notation include the point groups on the sphere , the frieze groups and wallpaper groups of the Euclidean plane , and their analogues on the hyperbolic plane .

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 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.