Orbifold notation

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 $1,2,3,\backslash dots$
• the infinity symbol, $\backslash infty$
• 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 $\backslash times$ (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.

Each symbol corresponds to a distinct transformation:

• an integer n to the left of an asterisk indicates a rotation of order n around a gyration point
• the asterisk, * indicates a reflection
• an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a kaleidoscopic point and reflects through a line (or plane)
• an $\backslash times$ indicates a glide reflection
• the symbol $\backslash infty$ 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.
• the exceptional symbol o indicates that there are precisely two linearly 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:

• n without or before an asterisk counts as $\backslash frac\{n-1\}\{n\}$
• n after an asterisk counts as $\backslash frac\{n-1\}\{2\; n\}$
• asterisk and $\backslash times$ 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.

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.

{{multiple image

| align = right

| total_width = 420

| image1 = Bentley Snowflake13.jpg

| caption1 = A perfect snowflake would have *6• symmetry,

| image2 = Pentagon symmetry as mirrors 2005-07-08.png

| caption2 = The pentagon has symmetry *5•, the whole image with arrows 5•.

| image3 = Flag of Hong Kong.svg

| caption3 = The Flag of Hong Kong has 5 fold rotation symmetry, 5•.

}}

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.

{{See also|List of spherical symmetry groups}}

{{See also|List of planar symmetry groups}}

{{Frieze group notations}}

{{clear}}

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

{{Clear}}

• Mutation of orbifolds
• Fibrifold notation - an extension of orbifold notation for 3d space groups

{{reflist}}

• 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, {{isbn|978-1-56881-220-5}}
• {{citation | last = Hughes | first = Sam | title = Cohomology of Fuchsian groups and non-Euclidean crystallographic groups | journal = Manuscripta Mathematica | year = 2022 | volume = 170 | issue = 3–4 | pages = 659–676 | doi = 10.1007/s00229-022-01369-z | arxiv = 1910.00519| bibcode = 2019arXiv191000519H | s2cid = 203610179 }}

• [http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/node39.html#SECTION000390000000000000000 A field guide to the orbifolds] (Notes from class on [http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/handouts.html "Geometry and the Imagination"] in Minneapolis, with John Conway, Peter Doyle, Jane Gilman and Bill Thurston, on June 17–28, 1991. See also [http://www.math.dartmouth.edu/~doyle/docs/gi/gi.pdf PDF, 2006])
• [https://uni-tuebingen.de/fakultaeten/mathematisch-naturwissenschaftliche-fakultaet/fachbereiche/informatik/lehrstuehle/algorithms-in-bioinformatics/software/tegula/ Tegula] Software for visualizing two-dimensional tilings of the plane, sphere and hyperbolic plane, and editing their symmetry groups in orbifold notation

Category:Group theory

Category:Generalized manifolds

Category:Mathematical notation

Category:John Horton Conway