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 E³. 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 so described. The notation can be used to describe the so-called wallpaper groups, frieze groups, and three dimensional point groups. In topology, an orbifold is a generalization of manifold. ... See John B. Conway for the functional analyst. ... In topology, an orbifold is a generalization of manifold. ... Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical concept to classify repetitive designs on two-dimensional surfaces, such as walls, based on the symmetries in the pattern. ... A frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). ...

Definition of the notation

We first give an overview of the types of Euclidean transformation which can occur in a group described by orbifold notation.

• reflections through a line(or plane) l,
• translations by a vector v
• rotations of finite order around a point p
• infinite rotations around a line l in three space,
• glide-reflections: that is a reflection followed by a translation.

All translations which occur are assumed to form a discrete subgroup of the group symmetries being described.

The orbifold notation works in the following way: each group is denoted by a finite string made up from the follow symbols:

• positive integers 1,2,3,...
• the symbol
• an astrix *
• the symbol x, which is called a wonder,
• the symbol o, which is called a miracle.

A string may be written in either normal type face or bold type face. A group written in bold face is a group of symmetries of Euclidean three space. A group not written in bold face is a group of symmetries of the Eudlidean plane, which is assumed to contain two independent translations.

The strings are interpreted in the following way: each symbol corresponds to a distinct transformation T:

• an integer n to the left a * indicates a rotation T around a point p of order n,
• an integer to the right of a * indicates a transformation of order 2n which rotates around a point,

p and reflects through a line(or plane) l,

• an x indicates a glide reflection,
• the symbol 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. A frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). ...

• the exceptional symbol o indicates that are precisely are two linearly independent translations.

Chirality and achirality

An object with symmetry group G is called chiral if it contains no refelctions. Otherwise it is called achiral. The corresponding orbifold is orientable if in the chiral case and non-orientable otherwise. It has been suggested that this article or section be merged with orientable manifold. ...

The Euler characteristic and the order

The Euler characteristic of the orbifold corresponding to a group written inthe orbifold notation is give the following rule: subtract from 2 the sum of the feature values. The feature values are assigned as follows: In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... In topology, an orbifold is a generalization of manifold. ...

• n without or before * counts as (n−1)/n
• n after * counts as (n−1)/(2n)
• * and x count as 1
• o counts as 2

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.

The order of the group(in the case that the group is finite) is then 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 pentagon has symmetry *55, the whole image with arrows 55.

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. Image File history File links Download high resolution version (840x840, 113 KB) This math image (or all images in this article or category) should be recreated using vector graphics as an SVG file. ... Image File history File links Download high resolution version (840x840, 113 KB) This math image (or all images in this article or category) should be recreated using vector graphics as an SVG file. ... In geometry, a pentagon is any five-sided polygon. ...

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, and *. A pattern in 1D can be represented as a function f(x) for, say, the color at position x. ...

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. In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes...

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