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In [[geometry]], '''[[orbifold]] notation''' (or '''orbifold signature''') is a system, invented by [[William Thurston]] and popularized by the mathematician [[John Horton Conway|John Conway]], for representing types of [[symmetry groups]] in two-dimensional spaces of constant curvature.
The person who wrote the article is known as Jayson Hirano and he totally digs that name. Alaska is where I've usually been residing. Office supervising is my occupation. To play lacross is some thing I really enjoy performing.<br><br>Feel free to visit my page free psychic ([http://medialab.zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- medialab.zendesk.com])
The advantage of the notation is that it describes these groups in a 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 under consideration.
 
Groups representable in this notation include the [[point groups in three dimensions|point groups]] on the [[sphere]] (<math>S^2</math>), the [[frieze group]]s and [[wallpaper group]]s of the [[Euclidean plane]] (<math>E^2</math>), and their analogues on the [[hyperbolic geometry|hyperbolic plane]] (<math>H^2</math>).
 
== Definition of the 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 ''[[integer]]s'' <math> 1,2,3,\dots </math>
* the ''[[infinity]]'' symbol, <math> \infty </math>
* the ''[[asterisk]]'', *
* the symbol <math>o</math> (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 <math>x</math> (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]]
* an integer ''n'' to the right of an asterisk indicates a transformation of order 2''n'' which rotates around a kaleidoscopic point and reflects through a line (or plane)
* an ''×'' indicates a glide reflection
* the symbol <math> \infty </math> 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 group]]s occur in this way.
* the exceptional symbol ''o'' indicates that there are precisely two linearly independent translations.
 
=== Good orbifolds ===
 
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.
 
== Chirality and achirality ==
An object is '''chiral''' if its symmetry group contains no reflections; otherwise it is called '''achiral'''. The corresponding orbifold is [[orientability|orientable]] in the chiral case and non-orientable otherwise.
 
==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 <math> \frac{n-1}{n} </math>
 
* ''n'' after an asterisk counts as <math> \frac{n-1}{2 n} </math>
 
* asterisk and ''<math>x</math>'' count as 1
 
* ''<math>o</math>'' 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
*<nowiki>*</nowiki>22 and *221
*2* and 2*1.
This is because 1-fold rotation is the "empty" rotation.
 
==Two dimensional groups==
{| class=wikitable align=right width=420
|- valign=top
|[[File:Bentley_Snowflake13.jpg|120px]]<BR>A perfect [[snowflake]] would have *6&bull; symmetry,
|[[File:Pentagon symmetry as mirrors 2005-07-08.png|120px]]<BR>The [[pentagon]] has symmetry *5&bull;, the whole image with arrows 5&bull;.
|[[File:Flag of Hong Kong.svg|180px]]<BR>The [[Flag of Hong Kong]] has 5 fold rotation symmetry, 5&bull;.
|}
The [[symmetry]] of a [[Two dimension|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''&bull; and *''n''&bull;. The [[Bullet (typography)|bullet]] (&bull;) 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 [[digon]]al orbifold and are represented as ''nn'' and *''nn''.)
 
Similarly, a [[One dimension|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 *&bull;, *1&bull;, &infin;&bull; and *&infin;&bull;.
 
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.
 
== Correspondence tables ==
 
=== Spherical ===
 
{| class="wikitable" align=right
|+ Fundamental domains of reflective 3D point groups
|- align=center
!(*11), C<sub>1v</sub>
!(*22), C<sub>2v</sub>
!(*33), C<sub>3v</sub>
!(*44), C<sub>4v</sub>
!(*55), C<sub>5v</sub>
!(*66), C<sub>6v</sub>
|- align=center
|[[Image:Spherical digonal hosohedron2.png|60px]]<BR>Order 2
|[[Image:Spherical square hosohedron2.png|60px]]<BR>Order 4
|[[Image:Spherical hexagonal hosohedron2.png|60px]]<BR>Order 6
|[[Image:Spherical octagonal hosohedron2.png|60px]]<BR>Order 8
|[[Image:Spherical decagonal hosohedron2.png|60px]]<BR>Order 10
|[[Image:Spherical dodecagonal hosohedron2.png|60px]]<BR>Order 12
|- align=center
!(*221), D<sub>1h</sub>
!(*222), D<sub>2h</sub>
!(*223), D<sub>3h</sub>
!(*224), D<sub>4h</sub>
!(*225), D<sub>5h</sub>
!(*226), D<sub>6h</sub>
|- align=center
|[[Image:Spherical digonal bipyramid2.png|60px]]<BR>Order 4
|[[Image:Spherical square bipyramid2.png|60px]]<BR>Order 8
|[[Image:Spherical hexagonal bipyramid2.png|60px]]<BR>Order 12
|[[Image:Spherical octagonal bipyramid2.png|60px]]<BR>Order 16
|[[Image:Spherical decagonal bipyramid2.png|60px]]<BR>Order 20
|[[Image:Spherical dodecagonal bipyramid2.png|60px]]<BR>Order 24
|-
!colspan=2|(*332), T<sub>d</sub>
!colspan=2|(*432), O<sub>h</sub>
!colspan=2|(*532), I<sub>h</sub>
|- align=center
|colspan=2|[[Image:Tetrahedral reflection domains.png|120px]]<BR>Order 24
|colspan=2|[[Image:Octahedral reflection domains.png|120px]]<BR>Order 48
|colspan=2|[[Image:Icosahedral reflection domains.png|120px]]<BR>Order 120
|}
 
See also: [[List of spherical symmetry groups]]
 
{| class="wikitable"
|+ Spherical Symmetry Groups<ref>Symmetries of Things, Appendix A, page 416</ref>
|-
!Orbifold<br>Signature
![[Coxeter notation|Coxeter]]
![[Arthur Moritz Schönflies|Schönflies]]
![[Hermann–Mauguin notation|Hermann–Mauguin]]
!Order
|-
!colspan=5|Polyhedral groups
|-
|*532||[3,5]||I<sub>h</sub>||53m||120
|-
|532||[3,5]<sup>+</sup>||I||532||60
|-
|*432||[3,4]||O<sub>h</sub>||m3m||48
|-
|432||[3,4]<sup>+</sup>||O||432||24
|-
|*332||[3,3]||T<sub>d</sub>||{{overline|4}}3m||24
|-
|3*2||[3<sup>+</sup>,4]||T<sub>h</sub>||m3||24
|-
|332||[3,3]<sup>+</sup>||T||23||12
|-
!colspan=5|Dihedral and cyclic groups: n=3,4,5...
|-
|*22n||[2,n]||D<sub>nh</sub>||n/mmm or 2{{overline|n}}m2||4n
|-
|2*n||[2<sup>+</sup>,2n]||D<sub>nd</sub>||2{{overline|n}}2m or {{overline|n}}m||4n
|-
|22n||[2,n]<sup>+</sup>||D<sub>n</sub>||n2||2n
|-
|*nn||[n]||C<sub>nv</sub>||nm||2n
|-
|n*||[n<sup>+</sup>,2]||C<sub>nh</sub>||n/m or 2{{overline|n}}||2n
|-
|n×||[2<sup>+</sup>,2n<sup>+</sup>]||S<sub>2n</sub>||2{{overline|n}} or {{overline|n}}||2n
|-
|nn||[n]<sup>+</sup>||C<sub>n</sub>||n||n
|-
!colspan=5|Special cases
|-
|*222||[2,2]||D<sub>2h</sub>||2/mmm or 2{{overline|2}}m2||8
|-
|2*2||[2<sup>+</sup>,4]||D<sub>2d</sub>||2{{overline|2}}2m or {{overline|2}}m||8
|-
|222||[2,2]<sup>+</sup>||D<sub>2</sub>||22||4
|-
|*22||[2]||C<sub>2v</sub>||2m||4
|-
|2*||[2<sup>+</sup>,2]||C<sub>2h</sub>||2/m or 2{{overline|2}}||4
|-
|2×||[2<sup>+</sup>,4<sup>+</sup>]||S<sub>4</sub>||2{{overline|2}} or {{overline|2}}||4
|-
|22||[2]<sup>+</sup>||C<sub>2</sub>||2||2
|-
|*22||[1,2]||D<sub>1h</sub>||1/mmm or 2{{overline|1}}m2||4
|-
|2*||[2<sup>+</sup>,2]||D<sub>1d</sub>||2{{overline|1}}2m or {{overline|1}}m||4
|-
|22||[1,2]<sup>+</sup>||D<sub>1</sub>||12||2
|-
|*1||[ ]||C<sub>1v</sub>||1m||2
|-
|1*||[2,1<sup>+</sup>]||C<sub>1h</sub>||1/m or 2{{overline|1}}||2
|-
|1×||[2<sup>+</sup>,2<sup>+</sup>]||S<sub>2</sub>||2{{overline|1}} or {{overline|1}}||2
|-
|1||[ ]<sup>+</sup>||C<sub>1</sub>||1||1
|}
 
=== Euclidean plane ===
See also: [[List of planar symmetry groups]]
 
==== Frieze groups ====
{{Frieze_group_notations}}
 
==== Wallpaper groups====
{| class="wikitable" align=right
|+ Fundamental domains of Euclidean reflective groups
|- align=center
!(*442), p4m
!(4*2), p4g
|- align=center
|[[File:Uniform tiling 44-t1.png|200px]]
|[[Image:Tile V488 bicolor.svg|200px]]
|- align=center
!(*333), p3m
!(632), p6
|- align=center
|[[Image:Tile 3,6.svg|200px]]
|[[Image:Tile V46b.svg|200px]]
|}
 
{| class="wikitable"
|+ 17 [[wallpaper group]]s<ref>Symmetries of Things, Appendix A, page 416</ref>
|-
!Orbifold<br>Signature
![[Coxeter notation|Coxeter]]
![[Hermann–Mauguin notation|Hermann–<BR>Mauguin]]
![[Andreas Speiser|Speiser]]<br>[[Paul Niggli|Niggli]]
!Polya<br>Guggenhein
!Fejes Toth<br>Cadwell
|- align=center
|*632||[6,3]||p6m||C<sup>(I)</sup><sub>6v</sub>||D<sub>6</sub>||W<sup>1</sup><sub>6</sub>
|- align=center
|632||[6,3]<sup>+</sup>||p6||C<sup>(I)</sup><sub>6</sub>||C<sub>6</sub>||W<sub>6</sub>
|- align=center
|*442||[4,4]||p4m||C<sup>(I)</sup><sub>4</sub>||D<sup>*</sup><sub>4</sub>||W<sup>1</sup><sub>4</sub>
|- align=center
|4*2||[4<sup>+</sup>,4]||p4g||C<sup>II</sup><sub>4v</sub>||D<sup>o</sup><sub>4</sub>||W<sup>2</sup><sub>4</sub>
|- align=center
|442||[4,4]<sup>+</sup>||p4||C<sup>(I)</sup><sub>4</sub>||C<sub>4</sub>||W<sub>4</sub>
|- align=center
|*333||[3<sup>[3]</sup>]
||p3m1||C<sup>II</sup><sub>3v</sub>||D<sup>*</sup><sub>3</sub>||W<sup>1</sup><sub>3</sub>
|-  align=center
|3*3||[3<sup>+</sup>,6]||p31m||C<sup>I</sup><sub>3v</sub>||D<sup>o</sup><sub>3</sub>||W<sup>2</sup><sub>3</sub>
|- align=center
|333||[3<sup>[3]</sup>]<sup>+</sup>
||p3||C<sup>I</sup><sub>3</sub>||C<sub>3</sub>||W<sub>3</sub>
|- align=center
|*2222||[∞,2,∞]||pmm||C<sup>I</sup><sub>2v</sub>||D<sub>2</sub>kkkk||W<sup>2</sup><sub>2</sub>
|- align=center
|2*22||[∞,2<sup>+</sup>,∞]||cmm||C<sup>IV</sup><sub>2v</sub>||D<sub>2</sub>kgkg||W<sup>1</sup><sub>2</sub>
|- align=center
|22*||[(∞,2)<sup>+</sup>,∞]||pmg||C<sup>III</sup><sub>2v</sub>||D<sub>2</sub>kkgg||W<sup>3</sup><sub>2</sub>
|- align=center
|22×||[∞<sup>+</sup>,2<sup>+</sup>,∞<sup>+</sup>]||pgg||C<sup>II</sup><sub>2v</sub>||D<sub>2</sub>gggg||W<sup>4</sup><sub>2</sub>
|- align=center
|2222||[∞,2,∞]<sup>+</sup>||p2||C<sup>(I)</sup><sub>2</sub>||C<sub>2</sub>||W<sub>2</sub>
|- align=center
|**||[∞<sup>+</sup>,2,∞]||pm||C<sup>I</sup><sub>s</sub>||D<sub>1</sub>kk||W<sup>2</sup><sub>1</sub>
|- align=center
|*×||[∞<sup>+</sup>,2<sup>+</sup>,∞]||cm||C<sup>III</sup><sub>s</sub>||D<sub>1</sub>kg||W<sup>1</sup><sub>1</sub>
|- align=center
|××||[∞<sup>+</sup>,(2,∞)<sup>+</sup>]||pg||C<sup>II</sup><sub>2</sub>||D<sub>1</sub>gg||W<sup>3</sup><sub>1</sub>
|- align=center
|o||[∞<sup>+</sup>,2,∞<sup>+</sup>]||p1||C<sup>(I)</sup><sub>1</sub>||C<sub>1</sub>||W<sub>1</sub>
|}
 
=== Hyperbolic plane ===
{| class="wikitable" align=right
|+ [[Poincaré disk model]] of fundamental domain [[Triangle group|triangles]]
|-
!colspan=5|Example right triangles (*2pq)
|- align=center
|[[File:H2checkers_237.png|80px]]<BR>*237
|[[File:H2checkers_238.png|80px]]<BR>*238
|[[File:Hyperbolic domains 932.png|80px]]<BR>*239
|[[File:H2checkers_23i.png|80px]]<BR>*23&infin;
|- align=center
|[[File:H2checkers_245.png|80px]]<BR>*245
|[[File:H2checkers_246.png|80px]]<BR>*246
|[[File:H2checkers_247.png|80px]]<BR>*247
|[[File:H2checkers_248.png|80px]]<BR>*248
|[[File:H2checkers_24i.png|80px]]<BR>*&infin;42
|- align=center
|[[File:H2checkers_255.png|80px]]<BR>*255
|[[File:H2checkers_256.png|80px]]<BR>*256
|[[File:H2checkers_257.png|80px]]<BR>*257
|[[File:H2checkers_266.png|80px]]<BR>*266
|[[File:H2checkers_2ii.png|80px]]<BR>*2&infin;&infin;
|- align=center
!colspan=5|Example general triangles (*pqr)
|- align=center
|[[File:H2checkers_334.png|80px]]<BR>*334
|[[File:H2checkers_335.png|80px]]<BR>*335
|[[File:H2checkers_336.png|80px]]<BR>*336
|[[File:H2checkers_337.png|80px]]<BR>*337
|[[File:H2checkers 33i.png|80px]]<BR>*33&infin;
|- align=center
|[[File:H2checkers_344.png|80px]]<BR>*344
|[[File:H2checkers_366.png|80px]]<BR>*366
|[[File:H2checkers 3ii.png|80px]]<BR>*3&infin;&infin;
|[[File:H2checkers_666.png|80px]]<BR>*6<sup>3</sup>
|[[File:H2checkers iii.png|80px]]<BR>*&infin;<sup>3</sup>
|- align=center
!colspan=5|Example higher polygons (*pqrs...)
|- align=center
|[[File:Hyperbolic domains 3222.png|80px]]<BR>*2223
|[[File:H2chess 246a.png|80px]]<BR>*(23)<sup>2</sup>
|[[File:H2chess 248a.png|80px]]<BR>*(24)<sup>2</sup>
|[[File:H2chess 246b.png|80px]]<BR>*3<sup>4</sup>
|[[File:H2chess_248b.png|80px]]<BR>*4<sup>4</sup>
|- align=center
|[[File:Uniform_tiling_552-t1.png|80px]]<BR>*2<sup>5</sup>
|[[File:Uniform_tiling_66-t1.png|80px]]<BR>*2<sup>6</sup>
|[[File:Uniform_tiling_77-t1.png|80px]]<BR>*2<sup>7</sup>
|[[File:Uniform_tiling_88-t1.png|80px]]<BR>*2<sup>8</sup>
|- align=center
|[[File:Hyperbolic domains i222.png|80px]]<BR>*222&infin;
|[[File:24i-2uu.png|80px]]<BR>*(2&infin;)<sup>2</sup>
|[[File:24i-1uu.png|80px]]<BR>*&infin;<sup>4</sup>
|[[File:24i-0uu.png|80px]]<BR>*2<sup>&infin;</sup>
|[[File:H2chess iiic.png|80px]]<BR>*&infin;<sup>&infin;</sup>
|}
 
A first few hyperbolic groups, ordered by their Euler characteristic are:
 
{| class=wikitable
|+ Hyperbolic Symmetry Groups<ref>Symmetries of Things, Chapter 18, More on Hyperbolic groups, Enumerating hyperbolic groups, p239</ref>
|-
!(-1/&chi;)
!Orbifolds
![[Coxeter notation|Coxeter]]
|-
|(84)||*237||[7,3]
|-
|(48)||*238||[8,3]
|-
|(42)||237||[7,3]<sup>+</sup>
|-
|(40)||*245||[5,4]
|-
|(36 - 26.4)||*239, *2.3.10||[9,3], [10,3]
|-
|(26.4)||*2.3.11||[11,3]
|-
|(24)||*2.3.12, *246, *334, 3*4, 238||[12,3], [6,4], [(4,3,3)], [3<sup>+</sup>,8], [8,3]<sup>+</sup>
|-
|(22.3 - 21)||*2.3.13, *2.3.14||[13,3], [14,3]
|-
|(20)||*2.3.15, *255, 5*2, 245||[15,3], [5,5], [5<sup>+</sup>,4], [5,4]<sup>+</sup>
|-
|(19.2)||*2.3.16||[16,3]
|-
|(18+2/3)||*247||[7,4]
|-
|(18)||*2.3.18, 239||[18,3], [9,3]<sup>+</sup>
|-
|(17.5-16.2)||*2.3.19, *2.3.20, *2.3.21, *2.3.22, *2.3.23 ||[19,3], [20,3], [20,3], [21,3], [22,3], [23,3]
|-
|(16)||*2.3.24, *248||[24,3], [8,4]
|-
|(15)||*2.3.30, *256, *335, 3*5, 2.3.10||[30,3], [6,5], [(5,3,3)], [3<sup>+</sup>,10], [10,3]<sup>+</sup>
|-
|(14+2/5 - 13+1/3)||*2.3.36 ... *2.3.70, *249, *2.4.10||[36,3] ... [60,3], [9,4], [10,4]
|-
|(13+1/5)||*2.3.66, 2.3.11||[66,3], [11,3]<sup>+</sup>
|-
|(12+8/11)||*2.3.105, *257||[105,3], [7,5]
|-
|(12+4/7)||*2.3.132, *2.4.11 ... ||[132,3], [11,4], ...
|-
|(12)||*23&infin;, *2.4.12, *266, 6*2, *336, 3*6, *344, 4*3, *2223, 2*23, 2.3.12, 246, 334||[&infin;,3] [12,4], [6,6], [6<sup>+</sup>,4], [(6,3,3)], [3<sup>+</sup>,12], [(4,4,3)], [4<sup>+</sup>,6], [&infin;,3,&infin;], [12,3]<sup>+</sup>, [6,4]<sup>+</sup> [(4,3,3)]<sup>+</sup>
|-
|colspan=2|...
|}
 
{{-}}
 
=== Mutations of orbifolds ===
Orbifolds with the same structure can be mutated between different symmetry classes, including across curvature domains from spherical, to Euclidean to Hyperbolic. This table shows mutation classes.<ref>''Two Dimensional symmetry Mutations'' by Daniel Huson, [http://www.google.com/search?q=Two-Dimensional+Symmetry+Mutation]</ref> This table is not complete for possible hyperbolic orbifolds.
{| class=wikitable align=right
|+ Example *''n''32 symmetry mutations
|-
!colspan=3|Spherical tilings (n=3..5)
|- align=center
| [[File:Uniform tiling 332-t01-1-.png|120px]]<br>[[Truncated tetrahedron|*332]]
| [[File:Uniform_tiling_432-t01.png|120px]]<br>[[Truncated cube|*432]]
| [[File:Uniform_tiling_532-t01.png|120px]]<br>[[Truncated dodecahedron|*532]]
|-
!colspan=3|Euclidean plane tiling (n=6)
|- align=center
|colspan=3|[[File:Uniform tiling 63-t01.png|240px]]<br>[[Truncated hexagonal tiling|*632]]
|-
!colspan=3|Hyperbolic plane tilings (n=7...&infin;)
|- align=center
| [[File:H2 tiling 237-3.png|120px]]<br>[[Order-3 truncated heptagonal tiling|*732]]
| [[File:H2 tiling 238-3.png|120px]]<br>[[Order-3 truncated octagonal tiling|*832]]
| [[File:H2 tiling 23i-3.png|120px]]<BR>... *&infin;32
|}
 
{| class=wikitable
!Orbifold
!Spherical
!Euclidean
!Hyperbolic
|-
!o
| -
|o
| -
|-
!pp
|22 ...
|&infin;&infin;
| -
|-
!*pp
|*pp
|*&infin;&infin;
| -
|-
!p*
|2* ...
|&infin;*
| -
|-
!p×
|2× ...
|&infin;×
|
|-
!**
| -
|**
| -
|-
!*×
| -
|*×
| -
|-
!××
| -
| ××
| -
|-
!ppp
|222
|333
|444 ...
|-
!pp*
| -
|22*
|33* ...
|-
!pp×
| -
|22×
|33× ...
|-
!pqq
|p22, 233
|244
|255 ..., 433 ...
|-
!pqr
|234, 235
|236
|237 ..., 245 ...
|-
!pq*
| -
| -
|23* ...
|-
!pqx
| -
| -
|23× ...
|-
!p*q
|2*p
|3*3, 4*2
|5*2 ..., 4*3 ..., 3*4 ...
|-
!*p*
| -
| -
| *2* ...
|-
!*p×
| -
| -
| *2× ...
|-
!pppp
| -
| 2222
| 3333 ...
|-
!pppq
| -
| -
| 2223...
|-
!ppqq
| -
| -
|2233
|-
!pp*p
| -
| -
|22*2 ...
|-
!p*qr
| -
|2*22
|3*22 ..., 2*32 ...
|-
!*ppp
|*222
|*333
|*444 ...
|-
!*pqq
|*p22, *233
|*244
|*255 ..., *344...
|-
!*pqr
|*234, *235
|*236
|*237..., *245..., *345 ...
|-
!p*ppp
| -
| -
|2*222
|-
!*pqrs
| -
| -
|*2223...
|-
!*ppppp
| -
| -
|*22222 ...
|-
!...
|}
 
== See also ==
* [[Fibrifold notation]] - an extension of orbifold notation for 3d [[space group]]s
 
==References==
{{reflist}}
* John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, and William P. Thurston. On Three-dimensional Orbifolds and 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 Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5
 
==External links==
* [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])
* [http://www-ab.informatik.uni-tuebingen.de/software/2dtiler/welcome.html 2DTiler] Software for visualizing two-dimensional tilings of the plane and editing their symmetry groups in orbifold notation
 
[[Category:Group theory]]
[[Category:Generalized manifolds]]
[[Category:Mathematical notation]]

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