Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Orbifold notation
open-in-new
<h2 id="definition-of-the-notation">Definition of the notation</h2> <p>The following types of Euclidean transformation can occur in a group described by orbifold notation: </p> <ul><li>reflection through a line (or plane)</li> <li>translation by a vector</li> <li>rotation of finite order around a point</li> <li>infinite rotation around a line in 3-space</li> <li>glide-reflection, i.e. reflection followed by translation.</li></ul> <p>All translations which occur are assumed to form a discrete subgroup of the group symmetries being described. </p><p>Each group is denoted in orbifold notation by a finite string made up from the following symbols: </p> <ul><li>positive <i><a href="/facts/Integer/PUwwrawx">integers</a></i> 1 , 2 , 3 , … {\displaystyle 1,2,3,\dots } </li> <li>the <i><a href="/facts/Infinity/cF506uD7">infinity</a></i> symbol, ∞ {\displaystyle \infty } </li> <li>the <i><a href="/facts/Asterisk/qwmtPW5V">asterisk</a></i>, *</li> <li>the symbol <i>o</i> (a solid circle in older documents), which is called a <i>wonder</i> and also a <i>handle</i> because it topologically represents a torus (1-handle) closed surface. Patterns repeat by two translation.</li> <li>the symbol × {\displaystyle \times } (an open circle in older documents), which is called a <i>miracle</i> and represents a topological <a href="/facts/Crosscap/tkUjh2ER">crosscap</a> where a pattern repeats as a mirror image without crossing a mirror line.</li></ul> <p>A string written in <a href="/facts/Boldface/RXGfXwcG">boldface</a> 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. </p><p>Each symbol corresponds to a distinct transformation: </p> <ul><li>an integer <i>n</i> to the left of an asterisk indicates a <a href="/facts/Rotation/yChkz49x">rotation</a> of order <i>n</i> around a <a href="/facts/Gyration_point/61DXTvoA">gyration point</a></li> <li>the <i><a href="/facts/Asterisk/qwmtPW5V">asterisk</a></i>, * indicates a reflection</li> <li>an integer <i>n</i> to the right of an asterisk indicates a transformation of order 2<i>n</i> which rotates around a kaleidoscopic point and reflects through a line (or plane)</li> <li>an × {\displaystyle \times } indicates a glide reflection</li> <li>the symbol ∞ {\displaystyle \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 <a href="/facts/Frieze_group/LVknTDnt">frieze groups</a> occur in this way.</li> <li>the exceptional symbol <i>o</i> indicates that there are precisely two linearly independent translations.</li></ul> <h3>Good orbifolds</h3> <p>An orbifold symbol is called <i>good</i> if it is not one of the following: <i>p</i>, <i>pq</i>, *<i>p</i>, *<i>pq</i>, for <i>p</i>, <i>q</i> ≥ 2, and <i>p</i> ≠ <i>q</i>. </p> <h2 id="chirality-and-achirality">Chirality and achirality</h2> <p>An object is chiral if its symmetry group contains no reflections; otherwise it is called achiral. The corresponding orbifold is <a href="/facts/Orientability/1SEQM2Pt">orientable</a> in the chiral case and non-orientable otherwise. </p> <h2 id="the-euler-characteristic-and-the-order">The Euler characteristic and the order</h2> <p>The <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a> of an <a href="/facts/Orbifold/KR6dQNzH">orbifold</a> can be read from its Conway symbol, as follows. Each feature has a value: </p> <ul><li><i>n</i> without or before an asterisk counts as n − 1 n {\displaystyle {\frac {n-1}{n}}} </li> <li><i>n</i> after an asterisk counts as n − 1 2 n {\displaystyle {\frac {n-1}{2n}}} </li> <li>asterisk and × {\displaystyle \times } count as 1</li> <li><i>o</i> counts as 2.</li></ul> <p>Subtracting the sum of these values from 2 gives the Euler characteristic. </p><p>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. </p> <h2 id="equal-groups">Equal groups</h2> <p>The following groups are isomorphic: </p> <ul><li>1* and *11</li> <li>22 and 221</li> <li>*22 and *221</li> <li>2* and 2*1.</li></ul> <p>This is because 1-fold rotation is the "empty" rotation. </p> <h2 id="two-dimensional-groups">Two-dimensional groups</h2> <p>The <a href="/facts/Symmetry/hpu7DK8p">symmetry</a> of a <a href="/facts/Two_dimension/f4b397W8">2D</a> 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 <i>n</i>• and *<i>n</i>•. The <a href="/facts/Bullet_(typography)/DyPRSa50">bullet</a> (•) 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 <a href="/facts/Digon/vur5IQ5d">digonal</a> orbifold and are represented as <i>nn</i> and *<i>nn</i>.) </p><p>Similarly, a <a href="/facts/One_dimension/0ktmUyac">1D</a> 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 <a href="/facts/Symmetry_groups_in_one_dimension/n2652Vyu">symmetry groups in one dimension</a> are *•, *1•, ∞• and *∞•. </p><p>Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> of the object and an asymmetric 2D or 1D object, respectively. </p> <h2 id="correspondence-tables">Correspondence tables</h2> <h3>Spherical</h3> Fundamental domains of reflective 3D point groups<table><tbody><tr><th>(*11), C1v = Cs</th><th>(*22), C2v</th><th>(*33), C3v</th><th>(*44), C4v</th><th>(*55), C5v</th><th>(*66), C6v</th></tr><tr><td>Order 2</td><td>Order 4</td><td>Order 6</td><td>Order 8</td><td>Order 10</td><td>Order 12</td></tr><tr><th>(*221), D1h = C2v</th><th>(*222), D2h</th><th>(*223), D3h</th><th>(*224), D4h</th><th>(*225), D5h</th><th>(*226), D6h</th></tr><tr><td>Order 4</td><td>Order 8</td><td>Order 12</td><td>Order 16</td><td>Order 20</td><td>Order 24</td></tr><tr><th colspan="2">(*332), Td</th><th colspan="2">(*432), Oh</th><th colspan="2">(*532), Ih</th></tr><tr><td colspan="2">Order 24</td><td colspan="2">Order 48</td><td colspan="2">Order 120</td></tr></tbody></table> <p class="note">See also: <a href="/facts/List_of_spherical_symmetry_groups/3X3YcHoQ">List of spherical symmetry groups</a></p> Spherical symmetry groups<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><table><tbody><tr><th>Orbifold signature</th><th><a href="/facts/Coxeter_notation/ArUw6goe">Coxeter</a></th><th><a href="/facts/Arthur_Moritz_Sch%25C3%25B6nflies/pRP7SGPY">Schönflies</a></th><th><a href="/facts/Hermann%25E2%2580%2593Mauguin_notation/NGHDFXKC">Hermann–Mauguin</a></th><th>Order</th></tr><tr><th colspan="5">Polyhedral groups</th></tr><tr><td>*532</td><td>[3,5]</td><td>Ih</td><td>53m</td><td>120</td></tr><tr><td>532</td><td>[3,5]+</td><td>I</td><td>532</td><td>60</td></tr><tr><td>*432</td><td>[3,4]</td><td>Oh</td><td>m3m</td><td>48</td></tr><tr><td>432</td><td>[3,4]+</td><td>O</td><td>432</td><td>24</td></tr><tr><td>*332</td><td>[3,3]</td><td>Td</td><td>43m</td><td>24</td></tr><tr><td>3*2</td><td>[3+,4]</td><td>Th</td><td>m3</td><td>24</td></tr><tr><td>332</td><td>[3,3]+</td><td>T</td><td>23</td><td>12</td></tr><tr><th colspan="5">Dihedral and cyclic groups: <i>n</i> = 3, 4, 5 ...</th></tr><tr><td>*22n</td><td>[2,n]</td><td>Dnh</td><td>n/mmm or 2nm2</td><td>4n</td></tr><tr><td>2*n</td><td>[2+,2n]</td><td>Dnd</td><td>2n2m or nm</td><td>4n</td></tr><tr><td>22n</td><td>[2,n]+</td><td>Dn</td><td>n2</td><td>2n</td></tr><tr><td>*nn</td><td>[n]</td><td>Cnv</td><td>nm</td><td>2n</td></tr><tr><td>n*</td><td>[n+,2]</td><td>Cnh</td><td>n/m or 2n</td><td>2n</td></tr><tr><td>n×</td><td>[2+,2n+]</td><td>S2n</td><td>2n or n</td><td>2n</td></tr><tr><td>nn</td><td>[n]+</td><td>Cn</td><td>n</td><td>n</td></tr><tr><th colspan="5">Special cases</th></tr><tr><td>*222</td><td>[2,2]</td><td>D2h</td><td>2/mmm or 22m2</td><td>8</td></tr><tr><td>2*2</td><td>[2+,4]</td><td>D2d</td><td>222m or 2m</td><td>8</td></tr><tr><td>222</td><td>[2,2]+</td><td>D2</td><td>22</td><td>4</td></tr><tr><td>*22</td><td>[2]</td><td>C2v</td><td>2m</td><td>4</td></tr><tr><td>2*</td><td>[2+,2]</td><td>C2h</td><td>2/m or 22</td><td>4</td></tr><tr><td>2×</td><td>[2+,4+]</td><td>S4</td><td>22 or 2</td><td>4</td></tr><tr><td>22</td><td>[2]+</td><td>C2</td><td>2</td><td>2</td></tr><tr><td>*22</td><td>[1,2]</td><td>D1h = C2v</td><td>1/mmm or 21m2</td><td>4</td></tr><tr><td>2*</td><td>[2+,2]</td><td>D1d = C2h</td><td>212m or 1m</td><td>4</td></tr><tr><td>22</td><td>[1,2]+</td><td>D1 = C2</td><td>12</td><td>2</td></tr><tr><td>*1</td><td>[ ]</td><td>C1v = Cs</td><td>1m</td><td>2</td></tr><tr><td>1*</td><td>[2,1+]</td><td>C1h = Cs</td><td>1/m or 21</td><td>2</td></tr><tr><td>1×</td><td>[2+,2+]</td><td>S2 = Ci</td><td>21 or 1</td><td>2</td></tr><tr><td>1</td><td>[ ]+</td><td>C1</td><td>1</td><td>1</td></tr></tbody></table> <h3>Euclidean plane</h3> <p class="note">See also: <a href="/facts/List_of_planar_symmetry_groups/3buKYIft">List of planar symmetry groups</a></p> <h4>Frieze groups</h4> <a href="/facts/Frieze_group/LVknTDnt">Frieze groups</a><table><tbody><tr><th><a href="/facts/IUC_notation/jmrXjb5Q">IUC</a></th><th><a href="/facts/Coxeter_notation/ArUw6goe">Cox.</a></th><th><a href="/facts/Schoenflies_notation/c4y0GtEt">Schön.</a>*</th><th>Orbifold</th><th>Diagram§</th><th colspan="2">Examples and <a href="/facts/John_Horton_Conway/g3PA1zoK">Conway</a> nickname<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></th><th>Description</th></tr><tr><th>p1</th><th>[∞]+</th><th>C∞<a href="/facts/Infinite_cyclic_group/JL5zfFuL">Z∞</a></th><th>∞∞</th><td></td><td></td><td>hop</td><td>(T) Translations only:This group is singly generated, by a translation by the smallest distance over which the pattern is periodic.</td></tr><tr><th>p11g</th><th>[∞+,2+]</th><th>S∞Z∞</th><th>∞×</th><td></td><td></td><td>step</td><td>(TG) Glide-reflections and Translations:This group is singly generated, by a glide reflection, with translations being obtained by combining two glide reflections.</td></tr><tr><th>p1m1</th><th>[∞]</th><th>C∞v<a href="/facts/Infinite_dihedral_group/MYy83AcX">Dih∞</a></th><th>*∞∞</th><td></td><td></td><td>sidle</td><td>(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.</td></tr><tr><th>p2</th><th>[∞,2]+</th><th>D∞Dih∞</th><th>22∞</th><td></td><td></td><td>spinning hop</td><td>(TR) Translations and 180° Rotations:The group is generated by a translation and a 180° rotation.</td></tr><tr><th>p2mg</th><th>[∞,2+]</th><th>D∞dDih∞</th><th>2*∞</th><td></td><td></td><td>spinning sidle</td><td>(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.</td></tr><tr><th>p11m</th><th>[∞+,2]</th><th>C∞hZ∞×Dih1</th><th>∞*</th><td></td><td></td><td>jump</td><td>(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</td></tr><tr><th>p2mm</th><th>[∞,2]</th><th>D∞hDih∞×Dih1</th><th>*22∞</th><td></td><td></td><td>spinning jump</td><td>(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.</td></tr></tbody></table> *Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries §The diagram shows one <a href="/facts/Fundamental_domain/e68KDYWu">fundamental domain</a> 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. <h4>Wallpaper groups</h4> Fundamental domains of Euclidean reflective groups<table><tbody><tr><th>(*442), p4m</th><th>(4*2), p4g</th></tr><tr><td></td><td></td></tr><tr><th>(*333), p3m</th><th>(632), p6</th></tr><tr><td></td><td></td></tr></tbody></table> 17 <a href="/facts/Wallpaper_group/egxEaC8q">wallpaper groups</a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><table><tbody><tr><th>Orbifoldsignature</th><th><a href="/facts/Coxeter_notation/ArUw6goe">Coxeter</a></th><th><a href="/facts/Hermann%25E2%2580%2593Mauguin_notation/NGHDFXKC">Hermann–Mauguin</a></th><th><a href="/facts/Andreas_Speiser/jlFbHboV">Speiser</a><a href="/facts/Paul_Niggli/bQMhtDMa">Niggli</a></th><th>PolyaGuggenhein</th><th>Fejes TothCadwell</th></tr><tr><td>*632</td><td>[6,3]</td><td>p6m</td><td>C(I)6v</td><td>D6</td><td>W16</td></tr><tr><td>632</td><td>[6,3]+</td><td>p6</td><td>C(I)6</td><td>C6</td><td>W6</td></tr><tr><td>*442</td><td>[4,4]</td><td>p4m</td><td>C(I)4</td><td>D*4</td><td>W14</td></tr><tr><td>4*2</td><td>[4+,4]</td><td>p4g</td><td>CII4v</td><td>Do4</td><td>W24</td></tr><tr><td>442</td><td>[4,4]+</td><td>p4</td><td>C(I)4</td><td>C4</td><td>W4</td></tr><tr><td>*333</td><td>[3[3]]</td><td>p3m1</td><td>CII3v</td><td>D*3</td><td>W13</td></tr><tr><td>3*3</td><td>[3+,6]</td><td>p31m</td><td>CI3v</td><td>Do3</td><td>W23</td></tr><tr><td>333</td><td>[3[3]]+</td><td>p3</td><td>CI3</td><td>C3</td><td>W3</td></tr><tr><td>*2222</td><td>[∞,2,∞]</td><td>pmm</td><td>CI2v</td><td>D2kkkk</td><td>W22</td></tr><tr><td>2*22</td><td>[∞,2+,∞]</td><td>cmm</td><td>CIV2v</td><td>D2kgkg</td><td>W12</td></tr><tr><td>22*</td><td>[(∞,2)+,∞]</td><td>pmg</td><td>CIII2v</td><td>D2kkgg</td><td>W32</td></tr><tr><td>22×</td><td>[∞+,2+,∞+]</td><td>pgg</td><td>CII2v</td><td>D2gggg</td><td>W42</td></tr><tr><td>2222</td><td>[∞,2,∞]+</td><td>p2</td><td>C(I)2</td><td>C2</td><td>W2</td></tr><tr><td>**</td><td>[∞+,2,∞]</td><td>pm</td><td>CIs</td><td>D1kk</td><td>W21</td></tr><tr><td>*×</td><td>[∞+,2+,∞]</td><td>cm</td><td>CIIIs</td><td>D1kg</td><td>W11</td></tr><tr><td>××</td><td>[∞+,(2,∞)+]</td><td>pg</td><td>CII2</td><td>D1gg</td><td>W31</td></tr><tr><td>o</td><td>[∞+,2,∞+]</td><td>p1</td><td>C(I)1</td><td>C1</td><td>W1</td></tr></tbody></table> <h3>Hyperbolic plane</h3> <a href="/facts/Poincar%25C3%25A9_disk_model/6nCWjbRL">Poincaré disk model</a> of fundamental domain <a href="/facts/Triangle_group/LrEs3xVw">triangles</a><table><tbody><tr><th colspan="5">Example <a href="https://www.icann.org/en/data-protection">https://www.icann.org/en/data-protection</a> triangles (*2pq)</th></tr><tr><td><a href="/facts/732_symmetry/dChEPASD">*237</a></td><td><a href="/facts/832_symmetry/jgtEuHdg">*238</a></td><td>*239</td><td><a href="/facts/I32_symmetry/z4hwBwxu">*23∞</a></td></tr><tr><td><a href="/facts/542_symmetry/r5gxSwXq">*245</a></td><td><a href="/facts/642_symmetry/ErN3lqr7">*246</a></td><td>*247</td><td><a href="/facts/842_symmetry/rqk2ML2M">*248</a></td><td><a href="/facts/I42_symmetry/5XK7XWYP">*∞42</a></td></tr><tr><td><a href="/facts/552_symmetry/Yskjrj4I">*255</a></td><td>*256</td><td>*257</td><td><a href="/facts/662_symmetry/gXdwQa9F">*266</a></td><td>*2∞∞</td></tr><tr><th colspan="5">Example general triangles (*pqr)</th></tr><tr><td><a href="/facts/433_symmetry/5pEnULQ3">*334</a></td><td>*335</td><td>*336</td><td>*337</td><td><a href="/facts/I33_symmetry/rw9ALo6e">*33∞</a></td></tr><tr><td><a href="/facts/443_symmetry/qUNkn9yb">*344</a></td><td><a href="/facts/663_symmetry/cM523f3g">*366</a></td><td>*3∞∞</td><td>*63</td><td><a href="/facts/Iii_symmetry/UEkWcKPh">*∞3</a></td></tr><tr><th colspan="5">Example higher polygons (*pqrs...)</th></tr><tr><td><a href="/facts/3222_symmetry/YDEs7484">*2223</a></td><td><a href="/facts/3232_symmetry/Hxr3upIW">*(23)2</a></td><td><a href="/facts/4242_symmetry/cuTIhX2V">*(24)2</a></td><td><a href="/facts/3333_symmetry/3deERRR3">*34</a></td><td><a href="/facts/4444_symmetry/31PjvkLS">*44</a></td></tr><tr><td><a href="/facts/22222_symmetry/aHah4SUB">*25</a></td><td><a href="/facts/222222_symmetry/KS6qS8IU">*26</a></td><td><a href="/facts/2222222_symmetry/mv3edRYK">*27</a></td><td><a href="/facts/22222222_symmetry/E9ec0oBY">*28</a></td></tr><tr><td><a href="/facts/I222_symmetry/IwNxLvr5">*222∞</a></td><td>*(2∞)2</td><td><a href="/facts/Iiii_symmetry/dMECKVSx">*∞4</a></td><td><a href="/facts/2%255Ei_symmetry/EjSUkNc8">*2∞</a></td><td><a href="/facts/I%255Ei_symmetry/WVaudE59">*∞∞</a></td></tr></tbody></table> <p>A first few hyperbolic groups, ordered by their Euler characteristic are: </p> Hyperbolic symmetry groups<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><table><tbody><tr><th>−1/χ</th><th>Orbifolds</th><th><a href="/facts/Coxeter_notation/ArUw6goe">Coxeter</a></th></tr><tr><td>84</td><td>*237</td><td>[7,3]</td></tr><tr><td>48</td><td>*238</td><td>[8,3]</td></tr><tr><td>42</td><td>237</td><td>[7,3]+</td></tr><tr><td>40</td><td>*245</td><td>[5,4]</td></tr><tr><td>36–26.4</td><td>*239, *2 3 10</td><td>[9,3], [10,3]</td></tr><tr><td>26.4</td><td>*2 3 11</td><td>[11,3]</td></tr><tr><td>24</td><td>*2 3 12, *246, *334, 3*4, 238</td><td>[12,3], [6,4], [(4,3,3)], [3+,8], [8,3]+</td></tr><tr><td>22.3–21</td><td>*2 3 13, *2 3 14</td><td>[13,3], [14,3]</td></tr><tr><td>20</td><td>*2 3 15, *255, 5*2, 245</td><td>[15,3], [5,5], [5+,4], [5,4]+</td></tr><tr><td>19.2</td><td>*2 3 16</td><td>[16,3]</td></tr><tr><td>18+2⁄3</td><td>*247</td><td>[7,4]</td></tr><tr><td>18</td><td>*2 3 18, 239</td><td>[18,3], [9,3]+</td></tr><tr><td>17.5–16.2</td><td>*2 3 19, *2 3 20, *2 3 21, *2 3 22, *2 3 23</td><td>[19,3], [20,3], [20,3], [21,3], [22,3], [23,3]</td></tr><tr><td>16</td><td>*2 3 24, *248</td><td>[24,3], [8,4]</td></tr><tr><td>15</td><td>*2 3 30, *256, *335, 3*5, 2 3 10</td><td>[30,3], [6,5], [(5,3,3)], [3+,10], [10,3]+</td></tr><tr><td>14+2⁄5–13+1⁄3</td><td>*2 3 36 ... *2 3 70, *249, *2 4 10</td><td>[36,3] ... [60,3], [9,4], [10,4]</td></tr><tr><td>13+1⁄5</td><td>*2 3 66, 2 3 11</td><td>[66,3], [11,3]+</td></tr><tr><td>12+8⁄11</td><td>*2 3 105, *257</td><td>[105,3], [7,5]</td></tr><tr><td>12+4⁄7</td><td>*2 3 132, *2 4 11 ...</td><td>[132,3], [11,4], ...</td></tr><tr><td>12</td><td>*23∞, *2 4 12, *266, 6*2, *336, 3*6, *344, 4*3, *2223, 2*23, 2 3 12, 246, 334</td><td>[∞,3] [12,4], [6,6], [6+,4], [(6,3,3)], [3+,12], [(4,4,3)], [4+,6], [∞,3,∞], [12,3]+, [6,4]+ [(4,3,3)]+</td></tr><tr><td colspan="2">...</td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Mutation_of_orbifolds/32bw45AW">Mutation of orbifolds</a></li> <li><a href="/facts/Fibrifold_notation/2Eb13dNI">Fibrifold notation</a> - an extension of orbifold notation for 3d <a href="/facts/Space_group/2XQx0J7M">space groups</a></li></ul> <ul><li>John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, and William P. Thurston. On Three-dimensional Space Groups. <i>Contributions to Algebra and Geometry</i>, 42(2):475-507, 2001.</li> <li>J. H. Conway, D. H. Huson. The Orbifold Notation for Two-Dimensional Groups. Structural Chemistry, 13 (3-4): 247–257, August 2002.</li> <li>J. H. Conway (1992). "The Orbifold Notation for Surface Groups". In: M. W. Liebeck and J. Saxl (eds.), <i>Groups, Combinatorics and Geometry</i>, 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</li> <li><a href="/facts/John_Horton_Conway/g3PA1zoK">John H. Conway</a>, Heidi Burgiel, <a href="/facts/Chaim_Goodman-Strauss/4IKWbx8w">Chaim Goodman-Strauss</a>, <i>The Symmetries of Things</i> 2008, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-220-5</li> <li>Hughes, Sam (2022), "Cohomology of Fuchsian groups and non-Euclidean crystallographic groups", <i>Manuscripta Mathematica</i>, 170 (3–4): 659–676, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1910.00519">1910.00519</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2019arXiv191000519H">2019arXiv191000519H</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00229-022-01369-z">10.1007/s00229-022-01369-z</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:203610179">203610179</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/node39.html#SECTION000390000000000000000">A field guide to the orbifolds</a> (Notes from class on <a href="http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/handouts.html">"Geometry and the Imagination"</a> in Minneapolis, with John Conway, Peter Doyle, Jane Gilman and Bill Thurston, on June 17–28, 1991. See also <a href="http://www.math.dartmouth.edu/~doyle/docs/gi/gi.pdf">PDF, 2006</a>)</li> <li><a href="https://uni-tuebingen.de/fakultaeten/mathematisch-naturwissenschaftliche-fakultaet/fachbereiche/informatik/lehrstuehle/algorithms-in-bioinformatics/software/tegula/">Tegula</a> Software for visualizing two-dimensional tilings of the plane, sphere and hyperbolic plane, and editing their symmetry groups in orbifold notation</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Symmetries of Things, Appendix A, page 416 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Frieze Patterns Mathematician John Conway created names that relate to footsteps for each of the frieze groups. <a href="https://www.maa.org/sites/default/files/images/upload_library/4/vol1/architecture/Math/seven.html" target="_blank">https://www.maa.org/sites/default/files/images/upload_library/4/vol1/architecture/Math/seven.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Symmetries of Things, Appendix A, page 416 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Symmetries of Things, Chapter 18, More on Hyperbolic groups, Enumerating hyperbolic groups, p239 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>