Orbifold notation

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 (). 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 integers the infinity symbol, the asterisk, * the symbol o (a solid circle in older documents), which is called a wonder and also a handle because it topologically represents a torus (1-handle) closed surface. Patterns repeat by two translation. the symbol (an open circle in older documents), which is called a miracle and represents a topological crosscap where a pattern repeats as a mirror image without crossing a mirror line. 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.