Ingeometry,orbifold notation (ororbifold signature) is a system, invented by the mathematicianWilliam Thurston and promoted byJohn Conway, for representing types ofsymmetry 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 followsWilliam Thurston in describing theorbifold obtained by taking the quotient ofEuclidean space by the group under consideration.

Groups representable in this notation include thepoint groups on thesphere (${\displaystyle S^2}$), thefrieze groups andwallpaper groups of theEuclidean plane (${\displaystyle E^2}$), and their analogues on thehyperbolic plane (${\displaystyle H^2}$).

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:

• positiveintegers${\displaystyle 1,2,3,\dots }$
• theinfinity symbol,${\displaystyle \mathrm{\infty }}$
• theasterisk, *
• the symbolo (a solid circle in older documents), which is called awonder and also ahandle because it topologically represents a torus (1-handle) closed surface. Patterns repeat by two translation.
• the symbol${\displaystyle \times }$ (an open circle in older documents), which is called amiracle and represents a topologicalcrosscap where a pattern repeats as a mirror image without crossing a mirror line.

A string written inboldface 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 integern to the left of an asterisk indicates arotation of ordern around agyration point
• theasterisk, * indicates a reflection
• an integern 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${\displaystyle \times }$ indicates a glide reflection
• the symbol${\displaystyle \mathrm{\infty }}$ 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. Thefrieze groups occur in this way.
• the exceptional symbolo indicates that there are precisely two linearly independent translations.

An orbifold symbol is calledgood if it is not one of the following:p,pq, *p, *pq, forp,q ≥ 2, andp ≠q.

An object ischiral if its symmetry group contains no reflections; otherwise it is calledachiral. The corresponding orbifold isorientable in the chiral case and non-orientable otherwise.

TheEuler characteristic of anorbifold can be read from its Conway symbol, as follows. Each feature has a value:

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

A perfectsnowflake would have *6• symmetry,

Thepentagon has symmetry *5•, the whole image with arrows 5•.

TheFlag of Hong Kong has 5 fold rotation symmetry, 5•.

Thesymmetry of a2D 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 haven• and *n•. Thebullet (•) 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-folddigonal orbifold and are represented asnn and *nn.)

Similarly, a1D 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 discretesymmetry 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 theCartesian product of the object and an asymmetric 2D or 1D object, respectively.

Fundamental domains of reflective 3D point groups

(*11), C1v = Cs	(*22), C2v	(*33), C3v	(*44), C4v	(*55), C5v	(*66), C6v
Order 2	Order 4	Order 6	Order 8	Order 10	Order 12
(*221), D1h = C2v	(*222), D2h	(*223), D3h	(*224), D4h	(*225), D5h	(*226), D6h
Order 4	Order 8	Order 12	Order 16	Order 20	Order 24
(*332), Td		(*432), Oh		(*532), Ih
Order 24		Order 48		Order 120

Frieze groups

IUC	Cox.	Schön.*	Orbifold	Diagram§	Examples and Conway nickname[2]		Description
p1	[∞]+	C∞ Z∞	∞∞			hop	(T) Translations only: This group is singly generated, by a translation by the smallest distance over which the pattern is periodic.
p11g	[∞+,2+]	S∞ Z∞	∞×			step	(TG) Glide-reflections and Translations: This group is singly generated, by a glide reflection, with translations being obtained by combining two glide reflections.
p1m1	[∞]	C∞v Dih∞	*∞∞			sidle	(TV) Vertical reflection lines and Translations: The group is the same as the non-trivial group in the one-dimensional case; it is generated by a translation and a reflection in the vertical axis.
p2	[∞,2]+	D∞ Dih∞	22∞			spinning hop	(TR) Translations and 180° Rotations: The group is generated by a translation and a 180° rotation.
p2mg	[∞,2+]	D∞d Dih∞	2*∞			spinning sidle	(TRVG) Vertical reflection lines, Glide reflections, Translations and 180° Rotations: The translations here arise from the glide reflections, so this group is generated by a glide reflection and either a rotation or a vertical reflection.
p11m	[∞+,2]	C∞h Z∞×Dih1	∞*			jump	(THG) Translations, Horizontal reflections, Glide reflections: This group is generated by a translation and the reflection in the horizontal axis. The glide reflection here arises as the composition of translation and horizontal reflection
p2mm	[∞,2]	D∞h Dih∞×Dih1	*22∞			spinning jump	(TRHVG) Horizontal and Vertical reflection lines, Translations and 180° Rotations: This group requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis.

^*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries

^§The diagram shows onefundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, and 2-fold gyration points as small green squares.

Fundamental domains of Euclidean reflective groups

(*442), p4m	(4*2), p4g
(*333), p3m	(632), p6

Poincaré disk model of fundamental domaintriangles

Example right triangles (*2pq)
*237	*238	*239	*23∞
*245	*246	*247	*248	*∞42
*255	*256	*257	*266	*2∞∞
Example general triangles (*pqr)
*334	*335	*336	*337	*33∞
*344	*366	*3∞∞	*63	*∞3
Example higher polygons (*pqrs...)
*2223	*(23)2	*(24)2	*34	*44
*25	*26	*27	*28
*222∞	*(2∞)2	*∞4	*2∞	*∞∞

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

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

• 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 Series165. 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
• Hughes, Sam (2022), "Cohomology of Fuchsian groups and non-Euclidean crystallographic groups",Manuscripta Mathematica,170 (3–4):659–676,arXiv:1910.00519,Bibcode:2019arXiv191000519H,doi:10.1007/s00229-022-01369-z,S2CID 203610179

• 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 alsoPDF, 2006)
• Tegula Software for visualizing two-dimensional tilings of the plane, sphere and hyperbolic plane, and editing their symmetry groups in orbifold notation

• Group theory
• Generalized manifolds
• Mathematical notation
• John Horton Conway

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