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 ( S^2 ), the frieze groups and wallpaper groups of the Euclidean plane ( E^2 ), and their analogues on the hyperbolic plane ( H^2 ).

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 ( S^2 ), the frieze groups and wallpaper groups of the Euclidean plane ( E^2 ), and their analogues on the hyperbolic plane ( H^2 ). ==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 \times 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. ==Two-dimensional groups== 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. ==== Wallpaper groups==== ==References== John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, and William P. Thurston. On Three-dimensional Space Groups. Contributions to Algebra and Geometry, 42(2):475-507, 2001. J. H. Conway, D. H. Huson. The Orbifold Notation for Two-Dimensional Groups. Structural Chemistry, 13 (3-4): 247–257, August 2002. 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 John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ==External links== A field guide to the orbifolds (Notes from class on "Geometry and the Imagination" in Minneapolis, with John Conway, Peter Doyle, Jane Gilman and Bill Thurston, on June 17–28, 1991. See also PDF, 2006) Tegula Software for visualizing two-dimensional tilings of the plane, sphere and hyperbolic plane, and editing their symmetry groups in orbifold notation

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