View source for Orbifold notation

{{Short description|Notation for 2-dimensional spherical, euclidean and hyperbolic symmetry groups}}
In [[geometry]], '''[[orbifold]] notation''' (or '''orbifold signature''') is a system, invented by the mathematician [[William Thurston]] and promoted by [[John Horton Conway|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 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 ''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 <math>\times</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]]
* the ''[[asterisk]]'', * indicates a reflection
* 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 <math>\times</math> 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>\times</math> 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
*<nowiki>*</nowiki>22 and *221
*2* and 2*1.
This is because 1-fold rotation is the "empty" rotation.

==Two-dimensional groups==
{{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 [[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''• and *''n''•. The [[Bullet (typography)|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 [[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 *•, *1•, &infin;• and *&infin;•.

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 floatright" style="text-align:center;"
|+ Fundamental domains of reflective 3D point groups
|-
! (*11), C<sub>1v</sub> = C<sub>s</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>
|-
| [[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
|-
! (*221), D<sub>1h</sub> = C<sub>2v</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>
|-
| [[Image:Spherical digonal bipyramid2.svg|60px]]<br/>Order 4
| [[Image:Spherical square bipyramid2.svg|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>
|-
| 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> = C<sub>2v</sub>||1/mmm or 2{{overline|1}}m2||4
|-
|2*||[2<sup>+</sup>,2]||D<sub>1d</sub> = C<sub>2h</sub>||2{{overline|1}}2m or {{overline|1}}m||4
|-
|22||[1,2]<sup>+</sup>||D<sub>1</sub> = C<sub>2</sub>||12||2
|-
|*1||[ ]||C<sub>1v</sub> = C<sub>s</sub>||1m||2
|-
|1*||[2,1<sup>+</sup>]||C<sub>1h</sub> = C<sub>s</sub>||1/m or 2{{overline|1}}||2
|-
|1×||[2<sup>+</sup>,2<sup>+</sup>]||S<sub>2</sub> = C<sub>i</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 floatright" style="text-align:center;"
|+ Fundamental domains of Euclidean reflective groups
|-
!(*442), p4m
!(4*2), p4g
|-
|[[File:Uniform tiling 44-t1.png|200px]]
|[[Image:Tile V488 bicolor.svg|200px]]
|-
!(*333), p3m
!(632), p6
|-
|[[Image:Tile 3,6.svg|200px]]
|[[Image:Tile V46b.svg|200px]]
|}

{| class="wikitable" style="text-align:center;"
|+ 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
|-
|*632||[6,3]||p6m||C<sup>(I)</sup><sub>6v</sub>||D<sub>6</sub>||W<sup>1</sup><sub>6</sub>
|-
|632||[6,3]<sup>+</sup>||p6||C<sup>(I)</sup><sub>6</sub>||C<sub>6</sub>||W<sub>6</sub>
|-
|*442||[4,4]||p4m||C<sup>(I)</sup><sub>4</sub>||D<sup>*</sup><sub>4</sub>||W<sup>1</sup><sub>4</sub>
|-
|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>
|-
|442||[4,4]<sup>+</sup>||p4||C<sup>(I)</sup><sub>4</sub>||C<sub>4</sub>||W<sub>4</sub>
|-
|*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>
|-
|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>
|-
|333||[3<sup>[3]</sup>]<sup>+</sup>
||p3||C<sup>I</sup><sub>3</sub>||C<sub>3</sub>||W<sub>3</sub>
|-
|*2222||[∞,2,∞]||pmm||C<sup>I</sup><sub>2v</sub>||D<sub>2</sub>kkkk||W<sup>2</sup><sub>2</sub>
|-
|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>
|-
|22*||[(∞,2)<sup>+</sup>,∞]||pmg||C<sup>III</sup><sub>2v</sub>||D<sub>2</sub>kkgg||W<sup>3</sup><sub>2</sub>
|-
|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>
|-
|2222||[∞,2,∞]<sup>+</sup>||p2||C<sup>(I)</sup><sub>2</sub>||C<sub>2</sub>||W<sub>2</sub>
|-
|**||[∞<sup>+</sup>,2,∞]||pm||C<sup>I</sup><sub>s</sub>||D<sub>1</sub>kk||W<sup>2</sup><sub>1</sub>
|-
|*×||[∞<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>
|-
|××||[∞<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>
|-
|o||[∞<sup>+</sup>,2,∞<sup>+</sup>]||p1||C<sup>(I)</sup><sub>1</sub>||C<sub>1</sub>||W<sub>1</sub>
|}
{{clear}}

=== Hyperbolic plane ===
{| class="wikitable floatright" style="text-align:center;"
|+ [[Poincaré disk model]] of fundamental domain [[triangle group|triangles]]
|-
! colspan=5 | Example right triangles (*2pq)
|-
|[[File:H2checkers 237.png|60px]]<BR>[[732 symmetry|*237]]
|[[File:H2checkers 238.png|60px]]<BR>[[832 symmetry|*238]]
|[[File:Hyperbolic domains 932 black.png|60px]]<BR>*239
|[[File:H2checkers 23i.png|60px]]<BR>[[i32 symmetry|*23&infin;]]
|-
|[[File:H2checkers 245.png|60px]]<BR>[[542 symmetry|*245]]
|[[File:H2checkers 246.png|60px]]<BR>[[642 symmetry|*246]]
|[[File:H2checkers 247.png|60px]]<BR>*247
|[[File:H2checkers 248.png|60px]]<BR>[[842 symmetry|*248]]
|[[File:H2checkers 24i.png|60px]]<BR>[[i42 symmetry|*&infin;42]]
|-
|[[File:H2checkers 255.png|60px]]<BR>[[552 symmetry|*255]]
|[[File:H2checkers 256.png|60px]]<BR>*256
|[[File:H2checkers 257.png|60px]]<BR>*257
|[[File:H2checkers 266.png|60px]]<BR>[[662 symmetry|*266]]
|[[File:H2checkers 2ii.png|60px]]<BR>*2&infin;&infin;
|-
! colspan=5 | Example general triangles (*pqr)
|-
|[[File:H2checkers 334.png|60px]]<BR>[[433 symmetry|*334]]
|[[File:H2checkers 335.png|60px]]<BR>*335
|[[File:H2checkers 336.png|60px]]<BR>*336
|[[File:H2checkers 337.png|60px]]<BR>*337
|[[File:H2checkers 33i.png|60px]]<BR>[[i33 symmetry|*33&infin;]]
|-
|[[File:H2checkers 344.png|60px]]<BR>[[443 symmetry|*344]]
|[[File:H2checkers 366.png|60px]]<BR>[[663 symmetry|*366]]
|[[File:H2checkers 3ii.png|60px]]<BR>*3&infin;&infin;
|[[File:H2checkers 666.png|60px]]<BR>*6<sup>3</sup>
|[[File:Infinite-order triangular tiling.svg|60px]]<BR>[[iii symmetry|*&infin;<sup>3</sup>]]
|-
! colspan=5 | Example higher polygons (*pqrs...)
|-
|[[File:Hyperbolic domains 3222.png|60px]]<BR>[[3222 symmetry|*2223]]
|[[File:H2chess 246a.png|60px]]<BR>[[3232 symmetry|*(23)<sup>2</sup>]]
|[[File:H2chess 248a.png|60px]]<BR>[[4242 symmetry|*(24)<sup>2</sup>]]
|[[File:H2chess 246b.png|60px]]<BR>[[3333 symmetry|*3<sup>4</sup>]]
|[[File:H2chess 248b.png|60px]]<BR>[[4444 symmetry|*4<sup>4</sup>]]
|-
|[[File:Uniform tiling 552-t1.png|60px]]<BR>[[22222 symmetry|*2<sup>5</sup>]]
|[[File:Uniform tiling 66-t1.png|60px]]<BR>[[222222 symmetry|*2<sup>6</sup>]]
|[[File:Uniform tiling 77-t1.png|60px]]<BR>[[2222222 symmetry|*2<sup>7</sup>]]
|[[File:Uniform tiling 88-t1.png|60px]]<BR>[[22222222 symmetry|*2<sup>8</sup>]]
|-
|[[File:Hyperbolic domains i222.png|60px]]<BR>[[i222 symmetry|*222&infin;]]
|[[File:H2chess 24ia.png|60px]]<BR>*(2&infin;)<sup>2</sup>
|[[File:H2chess 24ib.png|60px]]<BR>[[iiii symmetry|*&infin;<sup>4</sup>]]
|[[File:H2chess 24ic.png|60px]]<BR>[[2^i symmetry|*2<sup>&infin;</sup>]]
|[[File:H2chess iiic.png|60px]]<BR>[[i^i symmetry|*&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]
|-
|{{frac|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>
|-
|{{frac|14|2|5}}–{{frac|13|1|3}}||*2 3 36 ... *2 3 70, *249, *2 4 10||[36,3] ... [60,3], [9,4], [10,4]
|-
|{{frac|13|1|5}}||*2 3 66, 2 3 11||[66,3], [11,3]<sup>+</sup>
|-
|{{frac|12|8|11}}||*2 3 105, *257||[105,3], [7,5]
|-
|{{frac|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|...
|}

{{Clear}}

== See also ==
*[[Mutation of orbifolds]]
* [[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 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.&nbsp;438–447
* [[John Horton Conway|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 }}

==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])
* [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

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[[Category:Generalized manifolds]]
[[Category:Mathematical notation]]
[[Category:John Horton Conway]]