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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">2-dimensional integer lattice</div> <table class="wikitable" align="right"> <caption>Square lattices </caption> <tbody><tr> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Square_Lattice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Square_Lattice.svg/300px-Square_Lattice.svg.png" decoding="async" width="300" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Square_Lattice.svg/450px-Square_Lattice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Square_Lattice.svg/600px-Square_Lattice.svg.png 2x" data-file-width="650" data-file-height="320" /></a></span> </td></tr> <tr> <th>Upright square<br />Simple </th> <th>diagonal square<br />Centered </th></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Square_Lattice_Tiling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Square_Lattice_Tiling.svg/220px-Square_Lattice_Tiling.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Square_Lattice_Tiling.svg/330px-Square_Lattice_Tiling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Square_Lattice_Tiling.svg/440px-Square_Lattice_Tiling.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Upright <a href="/wiki/Square_tiling" title="Square tiling">square tiling</a>. The vertices of all squares together with their centers form an upright square lattice. For each color the centers of the squares of that color form a diagonal square lattice which is in linear scale √2 times as large as the upright square lattice.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>square lattice</b> is a type of <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> in a two-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. It is the two-dimensional version of the <a href="/wiki/Integer_lattice" title="Integer lattice">integer lattice</a>, denoted as <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a807ab4cb3de13a66771b5a303aca31e0391e6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{2}}"></span>⁠</span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is one of the five types of two-dimensional lattices as classified by their <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry groups</a>;<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> its symmetry group in <a href="/wiki/IUC_notation#Symmetry_notation" class="mw-redirect" title="IUC notation">IUC notation</a> as <span class="texhtml"><a href="/wiki/Wallpaper_group#Group_p4m" title="Wallpaper group">p4m</a></span>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter notation</a> as <span class="texhtml">[4,4]</span>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Orbifold_notation" title="Orbifold notation">orbifold notation</a> as <span class="texhtml">*442</span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as the upright square lattice and diagonal square lattice; the latter is also called the <b>centered square lattice</b>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> They differ by an angle of 45°. This is related to the fact that a square lattice can be partitioned into two square sub-lattices, as is evident in the colouring of a <a href="/wiki/Checkerboard" title="Checkerboard">checkerboard</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Symmetry">Symmetry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_lattice&action=edit&section=1" title="Edit section: Symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The square lattice's <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> category is <a href="/wiki/Wallpaper_group" title="Wallpaper group">wallpaper group</a> <span class="texhtml">p4m</span>. A pattern with this lattice of <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational symmetry</a> cannot have more, but may have less symmetry than the lattice itself. An upright square lattice can be viewed as a diagonal square lattice with a mesh size that is √2 times as large, with the centers of the squares added. Correspondingly, after adding the centers of the squares of an upright square lattice one obtains a diagonal square lattice with a mesh size that is √2 times as small as that of the original lattice. A pattern with 4-fold <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> has a square lattice of 4-fold rotocenters that is a factor √2 finer and diagonally oriented relative to the lattice of <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational symmetry</a>. </p><p>With respect to <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection axes</a> there are three possibilities: </p> <ul><li>None. This is wallpaper group <span class="texhtml">p4</span>.</li> <li>In four directions. This is wallpaper group <span class="texhtml">p4m</span>.</li> <li>In two <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> directions. This is wallpaper group <span class="texhtml">p4g</span>. The points of intersection of the reflexion axes form a square grid which is as fine as, and oriented the same as, the square lattice of 4-fold rotocenters, with these rotocenters at the centers of the squares formed by the reflection axes.</li></ul> <table class="wikitable" width="600"> <tbody><tr align="center" valign="top"> <th><span class="texhtml">p4, [4,4]<sup>+</sup>, (442)</span> </th> <th><span class="texhtml">p4g, [4,4<sup>+</sup>], (4*2)</span> </th> <th><span class="texhtml">p4m, [4,4], (*442)</span> </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p4_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group_diagram_p4_square.svg/200px-Wallpaper_group_diagram_p4_square.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group_diagram_p4_square.svg/300px-Wallpaper_group_diagram_p4_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Wallpaper_group_diagram_p4_square.svg/400px-Wallpaper_group_diagram_p4_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p4g_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Wallpaper_group_diagram_p4g_square.svg/200px-Wallpaper_group_diagram_p4g_square.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Wallpaper_group_diagram_p4g_square.svg/300px-Wallpaper_group_diagram_p4g_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Wallpaper_group_diagram_p4g_square.svg/400px-Wallpaper_group_diagram_p4g_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Wallpaper_group_diagram_p4m_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group_diagram_p4m_square.svg/200px-Wallpaper_group_diagram_p4m_square.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group_diagram_p4m_square.svg/300px-Wallpaper_group_diagram_p4m_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group_diagram_p4m_square.svg/400px-Wallpaper_group_diagram_p4m_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a></span> </td></tr> <tr valign="top"> <td><a href="/wiki/List_of_planar_symmetry_groups#Wallpaper_groups" title="List of planar symmetry groups">Wallpaper group</a> <span class="texhtml">p4</span>, with the arrangement within a primitive cell of the 2- and 4-fold rotocenters (also applicable for <span class="texhtml">p4g</span> and <span class="texhtml">p4m</span>). <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><div class="legend"><span class="legend-color mw-no-invert" style="background-color:yellow; color:black;"> </span> Fundamental domain</div> </td> <td>Wallpaper group <span class="texhtml">p4g</span>. There are reflection axes in two directions, <i>not</i> through the 4-fold rotocenters. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:yellow; color:black;"> </span> Fundamental domain</div> </td> <td>Wallpaper group <span class="texhtml">p4m</span>. There are reflection axes in four directions, through the 4-fold rotocenters. In two directions the reflection axes are oriented the same as, and as dense as, those for <span class="texhtml">p4g</span>, but shifted. In the other two directions they are linearly a factor √2 denser. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:yellow; color:black;"> </span> Fundamental domain</div> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Crystal_classes">Crystal classes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_lattice&action=edit&section=2" title="Edit section: Crystal classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>square lattice</i> class names, <a href="/wiki/Sch%C3%B6nflies_notation" class="mw-redirect" title="Schönflies notation">Schönflies notation</a>, <a href="/wiki/Hermann-Mauguin_notation" class="mw-redirect" title="Hermann-Mauguin notation">Hermann-Mauguin notation</a>, <a href="/wiki/Orbifold_notation" title="Orbifold notation">orbifold notation</a>, <a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter notation</a>, and <a href="/wiki/Wallpaper_groups" class="mw-redirect" title="Wallpaper groups">wallpaper groups</a> are listed in the table below. </p> <table class="wikitable"> <tbody><tr> <th colspan="4">Geometric class, <a href="/wiki/Point_group" title="Point group">point group</a> </th> <th rowspan="2" colspan="2">Wallpaper groups </th></tr> <tr align="center"> <th><a href="/wiki/Sch%C3%B6nflies_notation" class="mw-redirect" title="Schönflies notation">Schön.</a></th> <th><a href="/wiki/Hermann-Mauguin_notation" class="mw-redirect" title="Hermann-Mauguin notation">Intl</a></th> <th><a href="/wiki/Orbifold_notation" title="Orbifold notation">Orb.</a></th> <th><a href="/wiki/Coxeter_notation" title="Coxeter notation">Cox.</a> </th></tr> <tr align="center"> <td>C<sub>4</sub></td> <td>4</td> <td>(44)</td> <td>[4]<sup>+</sup> </td> <td>p4<br />(442) </td> <td>  </td></tr> <tr align="center"> <td>D<sub>4</sub></td> <td>4mm</td> <td>(*44)</td> <td>[4] </td> <td>p4m<br />(*442) </td> <td>p4g<br />(4*2) </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_lattice&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square number</a></li> <li><a href="/wiki/Euclid%27s_orchard" title="Euclid's orchard">Euclid's orchard</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integer</a></li> <li><a href="/wiki/Hexagonal_lattice" title="Hexagonal lattice">Hexagonal lattice</a></li> <li><a href="/wiki/Quincunx" title="Quincunx">Quincunx</a></li> <li><a href="/wiki/Square_tiling" title="Square tiling">Square tiling</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_lattice&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFConwaySloane1999" class="citation cs2"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, John</a>; <a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, Neil J. A.</a> (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=upYwZ6cQumoC&pg=PA106"><i>Sphere Packings, Lattices and Groups</i></a>, Springer, p. 106, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387985855" title="Special:BookSources/9780387985855"><bdi>9780387985855</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sphere+Packings%2C+Lattices+and+Groups&rft.pages=106&rft.pub=Springer&rft.date=1999&rft.isbn=9780387985855&rft.aulast=Conway&rft.aufirst=John&rft.au=Sloane%2C+Neil+J.+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DupYwZ6cQumoC%26pg%3DPA106&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+lattice" class="Z3988"></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubitskyStewart2003" class="citation cs2"><a href="/wiki/Marty_Golubitsky" title="Marty Golubitsky">Golubitsky, Martin</a>; <a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a> (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0HpyrroR9REC&pg=PA129"><i>The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space</i></a>, Progress in Mathematics, vol. 200, Springer, p. 129, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783764321710" title="Special:BookSources/9783764321710"><bdi>9783764321710</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Symmetry+Perspective%3A+From+Equilibrium+to+Chaos+in+Phase+Space+and+Physical+Space&rft.series=Progress+in+Mathematics&rft.pages=129&rft.pub=Springer&rft.date=2003&rft.isbn=9783764321710&rft.aulast=Golubitsky&rft.aufirst=Martin&rft.au=Stewart%2C+Ian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0HpyrroR9REC%26pg%3DPA129&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+lattice" class="Z3988"></span>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFieldGolubitsky2009" class="citation cs2">Field, Michael; <a href="/wiki/Marty_Golubitsky" title="Marty Golubitsky">Golubitsky, Martin</a> (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tu2Hnnc-b3YC&pg=PA47"><i>Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature</i></a> (2nd ed.), SIAM, p. 47, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780898717709" title="Special:BookSources/9780898717709"><bdi>9780898717709</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetry+in+Chaos%3A+A+Search+for+Pattern+in+Mathematics%2C+Art%2C+and+Nature&rft.pages=47&rft.edition=2nd&rft.pub=SIAM&rft.date=2009&rft.isbn=9780898717709&rft.aulast=Field&rft.aufirst=Michael&rft.au=Golubitsky%2C+Martin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dtu2Hnnc-b3YC%26pg%3DPA47&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+lattice" class="Z3988"></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnsonWeiss1999" class="citation cs2"><a href="/wiki/Norman_Johnson_(mathematician)" title="Norman Johnson (mathematician)">Johnson, Norman W.</a>; Weiss, Asia Ivić (1999), "Quadratic integers and Coxeter groups", <i>Canadian Journal of Mathematics</i>, <b>51</b> (6): 1307–1336, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-1999-060-6">10.4153/CJM-1999-060-6</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Mathematics&rft.atitle=Quadratic+integers+and+Coxeter+groups&rft.volume=51&rft.issue=6&rft.pages=1307-1336&rft.date=1999&rft_id=info%3Adoi%2F10.4153%2FCJM-1999-060-6&rft.aulast=Johnson&rft.aufirst=Norman+W.&rft.au=Weiss%2C+Asia+Ivi%C4%87&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+lattice" class="Z3988"></span>. See in particular the top of p. 1320.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchattschneiderSenechal2004" class="citation cs2"><a href="/wiki/Doris_Schattschneider" title="Doris Schattschneider">Schattschneider, Doris</a>; <a href="/wiki/Marjorie_Senechal" title="Marjorie Senechal">Senechal, Marjorie</a> (2004), "Tilings", in <a href="/wiki/Jacob_E._Goodman" title="Jacob E. Goodman">Goodman, Jacob E.</a>; <a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">O'Rourke, Joseph</a> (eds.), <i>Handbook of Discrete and Computational Geometry</i>, Discrete Mathematics and Its Applications (2nd ed.), CRC Press, pp. 53–72, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781420035315" title="Special:BookSources/9781420035315"><bdi>9781420035315</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tilings&rft.btitle=Handbook+of+Discrete+and+Computational+Geometry&rft.series=Discrete+Mathematics+and+Its+Applications&rft.pages=53-72&rft.edition=2nd&rft.pub=CRC+Press&rft.date=2004&rft.isbn=9781420035315&rft.aulast=Schattschneider&rft.aufirst=Doris&rft.au=Senechal%2C+Marjorie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+lattice" class="Z3988"></span>. See in particular the table on <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QS6vnl8WlnQC&pg=PA62">p. 62</a> relating IUC notation to orbifold notation.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnstonRichman1997" class="citation cs2">Johnston, Bernard L.; Richman, Fred (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=koUfrlgsmUcC&pg=PA159"><i>Numbers and Symmetry: An Introduction to Algebra</i></a>, CRC Press, p. 159, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780849303012" title="Special:BookSources/9780849303012"><bdi>9780849303012</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numbers+and+Symmetry%3A+An+Introduction+to+Algebra&rft.pages=159&rft.pub=CRC+Press&rft.date=1997&rft.isbn=9780849303012&rft.aulast=Johnston&rft.aufirst=Bernard+L.&rft.au=Richman%2C+Fred&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkoUfrlgsmUcC%26pg%3DPA159&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+lattice" class="Z3988"></span>.</span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Crystal_systems" title="Template:Crystal systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Crystal_systems&action=edit&redlink=1" class="new" title="Template talk:Crystal systems (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Crystal_systems" title="Special:EditPage/Template:Crystal systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Crystal_systems" style="font-size:114%;margin:0 4em"><a href="/wiki/Crystal_system" title="Crystal system">Crystal systems</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattice</a></li> <li><a href="/wiki/Crystallographic_point_group" title="Crystallographic point group">Crystallographic point group</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Seven 3D systems</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triclinic_crystal_system" title="Triclinic crystal system">triclinic</a> (anorthic) <span typeof="mw:File"><a href="/wiki/File:Triclinic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Triclinic.svg/30px-Triclinic.svg.png" decoding="async" width="30" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Triclinic.svg/45px-Triclinic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Triclinic.svg/60px-Triclinic.svg.png 2x" data-file-width="129" data-file-height="149" /></a></span></li> <li><a href="/wiki/Monoclinic_crystal_system" title="Monoclinic crystal system">monoclinic</a> <span typeof="mw:File"><a href="/wiki/File:Monoclinic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Monoclinic.svg/30px-Monoclinic.svg.png" decoding="async" width="30" height="38" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Monoclinic.svg/45px-Monoclinic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Monoclinic.svg/60px-Monoclinic.svg.png 2x" data-file-width="114" data-file-height="143" /></a></span></li> <li><a href="/wiki/Orthorhombic_crystal_system" title="Orthorhombic crystal system">orthorhombic</a> <span typeof="mw:File"><a href="/wiki/File:Orthorhombic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Orthorhombic.svg/30px-Orthorhombic.svg.png" decoding="async" width="30" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Orthorhombic.svg/45px-Orthorhombic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Orthorhombic.svg/60px-Orthorhombic.svg.png 2x" data-file-width="108" data-file-height="142" /></a></span></li> <li><a href="/wiki/Tetragonal_crystal_system" title="Tetragonal crystal system">tetragonal</a> <span typeof="mw:File"><a href="/wiki/File:Tetragonal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Tetragonal.svg/30px-Tetragonal.svg.png" decoding="async" width="30" height="46" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Tetragonal.svg/45px-Tetragonal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Tetragonal.svg/60px-Tetragonal.svg.png 2x" data-file-width="108" data-file-height="165" /></a></span></li> <li><a href="/wiki/Hexagonal_crystal_family" title="Hexagonal crystal family">trigonal & hexagonal</a> <span typeof="mw:File"><a href="/wiki/File:Rhombohedral.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/30px-Rhombohedral.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/45px-Rhombohedral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Rhombohedral.svg/60px-Rhombohedral.svg.png 2x" data-file-width="139" data-file-height="141" /></a></span><span typeof="mw:File"><a href="/wiki/File:Hexagonal_lattice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/30px-Hexagonal_lattice.svg.png" decoding="async" width="30" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/45px-Hexagonal_lattice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Hexagonal_lattice.svg/60px-Hexagonal_lattice.svg.png 2x" data-file-width="160" data-file-height="186" /></a></span></li> <li><a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">cubic</a> (isometric) <span typeof="mw:File"><a href="/wiki/File:Cubic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/30px-Cubic.svg.png" decoding="async" width="30" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/45px-Cubic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/60px-Cubic.svg.png 2x" data-file-width="109" data-file-height="127" /></a></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Four 2D systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Oblique_lattice" title="Oblique lattice">oblique</a> <span typeof="mw:File"><a href="/wiki/File:2d_mp.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/2d_mp.svg/30px-2d_mp.svg.png" decoding="async" width="30" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/2d_mp.svg/45px-2d_mp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/2d_mp.svg/60px-2d_mp.svg.png 2x" data-file-width="140" data-file-height="170" /></a></span></li> <li><a href="/wiki/Rectangular_lattice" title="Rectangular lattice">rectangular</a> <span typeof="mw:File"><a href="/wiki/File:2d_op_rectangular.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2d_op_rectangular.svg/30px-2d_op_rectangular.svg.png" decoding="async" width="30" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2d_op_rectangular.svg/45px-2d_op_rectangular.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/2d_op_rectangular.svg/60px-2d_op_rectangular.svg.png 2x" data-file-width="210" data-file-height="170" /></a></span></li> <li><a class="mw-selflink selflink">square</a> <span typeof="mw:File"><a href="/wiki/File:2d_tp.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/2d_tp.svg/30px-2d_tp.svg.png" decoding="async" width="30" height="30" class="mw-file-element" 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