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Plastic ratio - Wikipedia
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<span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Van_der_Laan_sequence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Van_der_Laan_sequence"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Van der Laan sequence</span> </div> </a> <ul id="toc-Van_der_Laan_sequence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Geometry</span> </div> </a> <button aria-controls="toc-Geometry-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Geometry subsection</span> </button> <ul id="toc-Geometry-sublist" class="vector-toc-list"> <li id="toc-Rho-squared_rectangle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rho-squared_rectangle"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Rho-squared rectangle</span> </div> </a> <ul id="toc-Rho-squared_rectangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plastic_spiral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plastic_spiral"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Plastic spiral</span> </div> </a> <ul id="toc-Plastic_spiral-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History_and_names" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History_and_names"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History and names</span> </div> </a> <ul id="toc-History_and_names-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Plastic ratio</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 21 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-21" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">21 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A8%D9%84%D8%A7%D8%B3%D8%AA%D9%8A%D9%83%D9%8A" title="عدد بلاستيكي – Arabic" lang="ar" hreflang="ar" data-title="عدد بلاستيكي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B2%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%9F%E0%A4%BF%E0%A4%95_%E0%A4%A8%E0%A4%82%E0%A4%AC%E0%A4%B0" title="प्लास्टिक नंबर – Bhojpuri" lang="bh" hreflang="bh" data-title="प्लास्टिक नंबर" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Plasti%C4%8Dni_broj" title="Plastični broj – Bosnian" lang="bs" hreflang="bs" data-title="Plastični broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_pl%C3%A0stic" title="Nombre plàstic – Catalan" lang="ca" hreflang="ca" data-title="Nombre plàstic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Plastische_Zahl" title="Plastische Zahl – German" lang="de" hreflang="de" data-title="Plastische Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_pl%C3%A1stico" title="Número plástico – Spanish" lang="es" hreflang="es" data-title="Número plástico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_plastique" title="Nombre plastique – French" lang="fr" hreflang="fr" data-title="Nombre plastique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%8C%EB%9D%BC%EC%8A%A4%ED%8B%B1_%EC%88%98" title="플라스틱 수 – Korean" lang="ko" hreflang="ko" data-title="플라스틱 수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_plastico" title="Numero plastico – Italian" lang="it" hreflang="it" data-title="Numero plastico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%99%D7%97%D7%A1_%D7%94%D7%A4%D7%9C%D7%A1%D7%98%D7%99" title="היחס הפלסטי – Hebrew" lang="he" hreflang="he" data-title="היחס הפלסטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Plastisch_getal" title="Plastisch getal – Dutch" lang="nl" hreflang="nl" data-title="Plastisch getal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%97%E3%83%A9%E3%82%B9%E3%83%81%E3%83%83%E3%82%AF%E6%95%B0" title="プラスチック数 – Japanese" lang="ja" hreflang="ja" data-title="プラスチック数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Plastik_raqam" title="Plastik raqam – Uzbek" lang="uz" hreflang="uz" data-title="Plastik raqam" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczba_plastikowa" title="Liczba plastikowa – Polish" lang="pl" hreflang="pl" data-title="Liczba plastikowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_pl%C3%A1stico" title="Número plástico – Portuguese" lang="pt" hreflang="pt" data-title="Número plástico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Пластическое число – Russian" lang="ru" hreflang="ru" data-title="Пластическое число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Plasti%C4%8Dno_%C5%A1tevilo" title="Plastično število – Slovenian" lang="sl" hreflang="sl" data-title="Plastično število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9E%E0%B8%A5%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%B4%E0%B8%81" title="จำนวนพลาสติก – Thai" lang="th" hreflang="th" data-title="จำนวนพลาสติก" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Пластичне число – Ukrainian" lang="uk" hreflang="uk" data-title="Пластичне число" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_nh%E1%BB%B1a" title="Số nhựa – Vietnamese" lang="vi" hreflang="vi" data-title="Số nhựa" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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The present address (URL) is a <a href="/wiki/Help:Permanent_link" title="Help:Permanent link">permanent link</a> to this version.</b></p><div id="revision-info-current-plain" style="display: none;">Revision as of 16:13, 30 November 2024 by <a href="/w/index.php?title=User:Zilverspreeuw&action=edit&redlink=1" class="new mw-userlink" title="User:Zilverspreeuw (page does not exist)" data-mw-revid="1260407401"><bdi>Zilverspreeuw</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Zilverspreeuw" class="mw-usertoollinks-talk" title="User talk:Zilverspreeuw">talk</a> | <a href="/wiki/Special:Contributions/Zilverspreeuw" class="mw-usertoollinks-contribs" title="Special:Contributions/Zilverspreeuw">contribs</a>)</span> <span class="comment">(<span class="autocomment"><a href="#Plastic_spiral">→<bdi dir="ltr">Plastic spiral</bdi></a>: </span> ce)</span></div></div><div id="mw-revision-nav">(<a href="/w/index.php?title=Plastic_ratio&diff=prev&oldid=1260407401" title="Plastic ratio">diff</a>) <a href="/w/index.php?title=Plastic_ratio&direction=prev&oldid=1260407401" title="Plastic ratio">← Previous revision</a> | Latest revision (diff) | Newer revision → (diff)</div></div></div></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">Number, approximately 1.3247</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><caption class="infobox-title">Plastic ratio</caption><tbody><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Triangles_in_ratio_of_the_plastic_number_in_a_three_armed_counter_clockwise_spiral.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Triangles_in_ratio_of_the_plastic_number_in_a_three_armed_counter_clockwise_spiral.svg/220px-Triangles_in_ratio_of_the_plastic_number_in_a_three_armed_counter_clockwise_spiral.svg.png" decoding="async" width="220" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Triangles_in_ratio_of_the_plastic_number_in_a_three_armed_counter_clockwise_spiral.svg/330px-Triangles_in_ratio_of_the_plastic_number_in_a_three_armed_counter_clockwise_spiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Triangles_in_ratio_of_the_plastic_number_in_a_three_armed_counter_clockwise_spiral.svg/440px-Triangles_in_ratio_of_the_plastic_number_in_a_three_armed_counter_clockwise_spiral.svg.png 2x" data-file-width="1044" data-file-height="990" /></a></span><div class="infobox-caption">Triangles with sides in ratio <span class="texhtml">ρ</span> form a closed spiral</div></td></tr><tr><th scope="row" class="infobox-label">Rationality</th><td class="infobox-data">irrational algebraic</td></tr><tr><th scope="row" class="infobox-label">Symbol</th><td class="infobox-data"><span class="texhtml">ρ</span></td></tr><tr><th colspan="2" class="infobox-header">Representations</th></tr><tr><th scope="row" class="infobox-label">Decimal</th><td class="infobox-data"><span style="white-space:nowrap">1.32471<span style="margin-left:0.25em">79572</span><span style="margin-left:0.25em">44746</span><span style="margin-left:0.25em">02596</span><span style="margin-left:0.25em">09088</span><span style="margin-left:0.25em">...</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></td></tr><tr><th scope="row" class="infobox-label">Algebraic form</th><td class="infobox-data">real root of <span class="texhtml"><i>x</i><sup>3</sup> = <i>x</i> + 1</span></td></tr><tr><th scope="row" class="infobox-label">Continued fraction (linear)</th><td class="infobox-data"><span class="nowrap">[1;3,12,1,1,3,2,3,2,4,2,141,80,...] <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></span><br />not periodic<br />infinite</td></tr></tbody></table> <p>In mathematics, the <b>plastic ratio</b> is a geometrical <a href="/wiki/Aspect_ratio" title="Aspect ratio">proportion</a> close to <span class="texhtml">53/40</span>. Its true value is the real <a href="/wiki/Polynomial_root" class="mw-redirect" title="Polynomial root">solution</a> of the equation <span class="texhtml"><i>x</i><sup>3</sup> = <i>x</i> + 1.</span> </p><p>The adjective <i>plastic</i> does not refer to <a href="/wiki/Plastic" title="Plastic">the artificial material</a>, but to the formative and sculptural qualities of this ratio, as in <i>plastic arts</i>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plastic_number_square_spiral.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Plastic_number_square_spiral.svg/280px-Plastic_number_square_spiral.svg.png" decoding="async" width="280" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Plastic_number_square_spiral.svg/420px-Plastic_number_square_spiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Plastic_number_square_spiral.svg/560px-Plastic_number_square_spiral.svg.png 2x" data-file-width="1676" data-file-height="1265" /></a><figcaption>Squares with sides in ratio <span class="texhtml">ρ</span> form a closed spiral</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastic_ratio&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Three quantities <span class="texhtml">a > b > c > 0</span> are in the plastic ratio if </p> <dl><dd><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 {\frac {a}{b}}={\frac {b+c}{a}}={\frac {b}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}={\frac {b+c}{a}}={\frac {b}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfe0aca3b5e116af7109f9bf8f15c9e81a527a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.787ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{b}}={\frac {b+c}{a}}={\frac {b}{c}}}"></span>.</dd></dl> <p>The ratio <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 {\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbb66e57f89debc3cde3213de12228971148a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.066ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}}"></span> is commonly denoted <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 \rho .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae3f23f76f614ab4dc47bfc296699c2be740666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.849ex; height:2.176ex;" alt="{\displaystyle \rho .}"></span>⁠</span> </p><p>Let <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 a=\rho \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>ρ<!-- ρ --></mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=\rho \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/328489c87788812db347d1cf8e235d6240278288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.917ex; height:2.176ex;" alt="{\displaystyle a=\rho \,}"></span>⁠</span> and <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 \,b=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,b=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9d1afad98711cf1521d05b31a1e0b04a2eebda8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.646ex; height:2.176ex;" alt="{\displaystyle \,b=1}"></span>⁠</span>, then </p><p><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 \rho ^{2}=1+c\,\land \,\rho =1/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>c</mi> <mspace width="thinmathspace" /> <mo>∧<!-- ∧ --></mo> <mspace width="thinmathspace" /> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{2}=1+c\,\land \,\rho =1/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e726b51ccbe35f9d3a10dc68a72b571c9436d32b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.353ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}=1+c\,\land \,\rho =1/c}"></span> </p><p><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 \implies \rho ^{2}-1=\rho ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟹<!-- ⟹ --></mo> <mspace width="thickmathspace" /> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> <mo>=</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \implies \rho ^{2}-1=\rho ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a17120cdc85b71af940d04bc2a0b7be5b3ae876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.633ex; height:3.176ex;" alt="{\displaystyle \implies \rho ^{2}-1=\rho ^{-1}}"></span>. </p><p>It follows that the plastic ratio is found as the unique real solution of the <a href="/wiki/Cubic_equation" title="Cubic equation">cubic equation</a> <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 \rho ^{3}-\rho -1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>ρ<!-- ρ --></mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{3}-\rho -1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5f2b16e3803fa0ce86a4b9b154ef0283e75ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.209ex; height:3.176ex;" alt="{\displaystyle \rho ^{3}-\rho -1=0.}"></span> The decimal expansion of the <a href="/wiki/Zero_of_a_function" title="Zero of a function">root</a> begins as <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 1.324\,717\,957\,244\,746...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.324</mn> <mspace width="thinmathspace" /> <mn>717</mn> <mspace width="thinmathspace" /> <mn>957</mn> <mspace width="thinmathspace" /> <mn>244</mn> <mspace width="thinmathspace" /> <mn>746...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.324\,717\,957\,244\,746...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99ca2125ae2344625d19a9d925793662f674404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.735ex; height:2.176ex;" alt="{\displaystyle 1.324\,717\,957\,244\,746...}"></span> (sequence <span class="nowrap external"><a href="//oeis.org/A060006" class="extiw" title="oeis:A060006">A060006</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p><p>Solving the equation with <a href="/wiki/Cubic_equation#Cardano's_formula" title="Cubic equation">Cardano's formula</a>, </p> <dl><dd><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 w_{1,2}={\frac {1}{2}}\left(1\pm {\frac {1}{3}}{\sqrt {\frac {23}{3}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>23</mn> <mn>3</mn> </mfrac> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1,2}={\frac {1}{2}}\left(1\pm {\frac {1}{3}}{\sqrt {\frac {23}{3}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27004741fafd7b2af0607b3838c34490f3cc1366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.389ex; height:6.343ex;" alt="{\displaystyle w_{1,2}={\frac {1}{2}}\left(1\pm {\frac {1}{3}}{\sqrt {\frac {23}{3}}}\right)}"></span></dd> <dd><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 \rho ={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5af123af3027af4433f6998829477325d9d45d1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.449ex; height:3.009ex;" alt="{\displaystyle \rho ={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}}"></span></dd></dl> <p>or, using the <a href="/wiki/Cubic_equation#Trigonometric_and_hyperbolic_solutions" title="Cubic equation">hyperbolic cosine</a>,<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> </p> <dl><dd><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 \rho ={\frac {2}{\sqrt {3}}}\cosh \left({\frac {1}{3}}\operatorname {arcosh} \left({\frac {3{\sqrt {3}}}{2}}\right)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mi>cosh</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>arcosh</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {2}{\sqrt {3}}}\cosh \left({\frac {1}{3}}\operatorname {arcosh} \left({\frac {3{\sqrt {3}}}{2}}\right)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029c56c7606a97158c689c76fc4210a1363e9014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.475ex; height:6.843ex;" alt="{\displaystyle \rho ={\frac {2}{\sqrt {3}}}\cosh \left({\frac {1}{3}}\operatorname {arcosh} \left({\frac {3{\sqrt {3}}}{2}}\right)\right).}"></span></dd></dl> <p><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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>⁠</span> is the superstable <a href="/wiki/Fixed-point_iteration" title="Fixed-point iteration">fixed point</a> of the iteration <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 x\gets (2x^{3}+1)/(3x^{2}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">←<!-- ← --></mo> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\gets (2x^{3}+1)/(3x^{2}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7710db37b731c02be9122fda1427e18a6f0d963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.823ex; height:3.176ex;" alt="{\displaystyle x\gets (2x^{3}+1)/(3x^{2}-1)}"></span>. </p><p>The iteration <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 x\gets {\sqrt {1+{\tfrac {1}{x}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">←<!-- ← --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\gets {\sqrt {1+{\tfrac {1}{x}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddb63e0c28b6b9de8829b2aa8744cff7b4473a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.047ex; height:4.676ex;" alt="{\displaystyle x\gets {\sqrt {1+{\tfrac {1}{x}}}}}"></span> results in the continued reciprocal square root </p> <dl><dd><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 \rho ={\sqrt {1+{\cfrac {1}{\sqrt {1+{\cfrac {1}{\sqrt {1+{\cfrac {1}{\ddots }}}}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋱<!-- ⋱ --></mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\sqrt {1+{\cfrac {1}{\sqrt {1+{\cfrac {1}{\sqrt {1+{\cfrac {1}{\ddots }}}}}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/396a5e2973d2a6e74e4319d61d9dde504005c0ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.838ex; width:29.158ex; height:17.676ex;" alt="{\displaystyle \rho ={\sqrt {1+{\cfrac {1}{\sqrt {1+{\cfrac {1}{\sqrt {1+{\cfrac {1}{\ddots }}}}}}}}}}}"></span></dd></dl> <p>Dividing the defining trinomial <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 x^{3}-x-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>x</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-x-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa65123eda4efe4577f20f38f36efbb61d3f17f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.557ex; height:2.843ex;" alt="{\displaystyle x^{3}-x-1}"></span> by <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 x-\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−<!-- − --></mo> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6b79f45d30dea7dd634348aef2bd78eecae0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.372ex; height:2.509ex;" alt="{\displaystyle x-\rho }"></span>⁠</span> one obtains <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 x^{2}+\rho x+1/\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>ρ<!-- ρ --></mi> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+\rho x+1/\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c604d3f122fe78ca62744ba84a7f44d5a662729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.123ex; height:3.176ex;" alt="{\displaystyle x^{2}+\rho x+1/\rho }"></span>, and the <a href="/wiki/Conjugate_element_(field_theory)" title="Conjugate element (field theory)">conjugate elements</a> of <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>⁠</span> are </p> <dl><dd><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 x_{1,2}={\frac {1}{2}}\left(-\rho \pm i{\sqrt {3\rho ^{2}-4}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−<!-- − --></mo> <mi>ρ<!-- ρ --></mi> <mo>±<!-- ± --></mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>4</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1,2}={\frac {1}{2}}\left(-\rho \pm i{\sqrt {3\rho ^{2}-4}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9c44f14dc981e5ce33efbdea66906b0647c1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.001ex; height:6.176ex;" alt="{\displaystyle x_{1,2}={\frac {1}{2}}\left(-\rho \pm i{\sqrt {3\rho ^{2}-4}}\right),}"></span></dd></dl> <p>with <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 x_{1}+x_{2}=-\rho \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−<!-- − --></mo> <mi>ρ<!-- ρ --></mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}+x_{2}=-\rho \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3edce7ea7175949fb468ae9316a95dd6e84b68b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.362ex; height:2.509ex;" alt="{\displaystyle x_{1}+x_{2}=-\rho \;}"></span> and <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 \;x_{1}x_{2}=1/\rho .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ<!-- ρ --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;x_{1}x_{2}=1/\rho .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c552edeb3222b462f40f3890dd1df5f6dbfa5195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.685ex; height:2.843ex;" alt="{\displaystyle \;x_{1}x_{2}=1/\rho .}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastic_ratio&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PlasticSquare_6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/PlasticSquare_6.png/280px-PlasticSquare_6.png" decoding="async" width="280" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/PlasticSquare_6.png/420px-PlasticSquare_6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/PlasticSquare_6.png/560px-PlasticSquare_6.png 2x" data-file-width="816" data-file-height="816" /></a><figcaption>Rectangles in aspect ratios <span class="texhtml">ρ, ρ<sup>2</sup>, ρ<sup>3</sup> </span> (top) and <span class="texhtml">ρ<sup>2</sup>, ρ, ρ<sup>3</sup> </span> (bottom row) tile the square.</figcaption></figure> <p>The plastic ratio <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>⁠</span> and <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>⁠</span> are the only morphic numbers: real numbers <span class="texhtml"><i>x</i> > 1</span> for which there exist natural numbers m and n such that </p> <dl><dd><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 x+1=x^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+1=x^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4d3c65c7eb797ae6d07d584b462b16b5db692c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.436ex; height:2.509ex;" alt="{\displaystyle x+1=x^{m}}"></span> and <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 x-1=x^{-n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-1=x^{-n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fceec88c750382b7545f5abfc3f39f840650dcb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.258ex; height:2.676ex;" alt="{\displaystyle x-1=x^{-n}}"></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></dd></dl> <p>Morphic numbers can serve as basis for a system of measure. </p><p>Properties of <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>⁠</span> (m=3 and n=4) are related to those of <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>⁠</span> (m=2 and n=1). For example, The plastic ratio satisfies the <a href="/wiki/Nested_radical#Infinitely_nested_radicals" title="Nested radical">continued radical</a> </p> <dl><dd><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 \rho ={\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+\cdots }}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>1</mn> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+\cdots }}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69a23f5678673aa580603778df1fdf96241ac64d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.616ex; height:6.176ex;" alt="{\displaystyle \rho ={\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+\cdots }}}}}}}"></span>,</dd></dl> <p>while the golden ratio satisfies the analogous </p> <dl><dd><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 \varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6906034505de22882eba7abdad5006ee1eeeb81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.934ex; height:6.176ex;" alt="{\displaystyle \varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}"></span></dd></dl> <p>The plastic ratio can be expressed in terms of itself as the infinite <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a> </p> <dl><dd><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 \rho =\sum _{n=0}^{\infty }\rho ^{-5n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>5</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\sum _{n=0}^{\infty }\rho ^{-5n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8271602a66e5a5b3a01c7c4d6f23ab837773272d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.563ex; height:6.843ex;" alt="{\displaystyle \rho =\sum _{n=0}^{\infty }\rho ^{-5n}}"></span> and <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 \,\rho ^{2}=\sum _{n=0}^{\infty }\rho ^{-3n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>3</mn> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\rho ^{2}=\sum _{n=0}^{\infty }\rho ^{-3n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b14adc05199e7a66fb7d657a06c96e792ed1436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.652ex; height:6.843ex;" alt="{\displaystyle \,\rho ^{2}=\sum _{n=0}^{\infty }\rho ^{-3n},}"></span></dd></dl> <p>in comparison to the golden ratio identity </p> <dl><dd><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 \varphi =\sum _{n=0}^{\infty }\varphi ^{-2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\sum _{n=0}^{\infty }\varphi ^{-2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe429c16565621a66127d2056e3e31fa4ceaf2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.2ex; height:6.843ex;" alt="{\displaystyle \varphi =\sum _{n=0}^{\infty }\varphi ^{-2n}}"></span> and <i>vice versa</i>.</dd></dl> <p>Additionally, <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 1+\varphi ^{-1}+\varphi ^{-2}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\varphi ^{-1}+\varphi ^{-2}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0038ed6d78ce6839c011f72f528b4a40446431f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.81ex; height:3.176ex;" alt="{\displaystyle 1+\varphi ^{-1}+\varphi ^{-2}=2}"></span>, while <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 \sum _{n=0}^{13}\rho ^{-n}=4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </munderover> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{13}\rho ^{-n}=4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04e0aa56fdd9c1caa6958bd64243c5b6ede5f1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.349ex; height:7.343ex;" alt="{\displaystyle \sum _{n=0}^{13}\rho ^{-n}=4.}"></span> </p><p>For every integer <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> one has </p> <dl><dd><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 {\begin{aligned}\rho ^{n}&=\rho ^{n-2}+\rho ^{n-3}\\&=\rho ^{n-1}+\rho ^{n-5}\\&=\rho ^{n-3}+\rho ^{n-4}+\rho ^{n-5}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>3</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>5</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>5</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\rho ^{n}&=\rho ^{n-2}+\rho ^{n-3}\\&=\rho ^{n-1}+\rho ^{n-5}\\&=\rho ^{n-3}+\rho ^{n-4}+\rho ^{n-5}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c00819400adc03507defa99cd178cc79d92430e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:26.161ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}\rho ^{n}&=\rho ^{n-2}+\rho ^{n-3}\\&=\rho ^{n-1}+\rho ^{n-5}\\&=\rho ^{n-3}+\rho ^{n-4}+\rho ^{n-5}.\end{aligned}}}"></span></dd></dl> <p>The algebraic solution of a reduced quintic equation can be written in terms of square roots, cube roots and the <a href="/wiki/Bring_radical" title="Bring radical">Bring radical</a>. If <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 y=x^{5}+x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x^{5}+x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af4768ff67bfc4c9edbda8940b130b544571e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.808ex; height:3.009ex;" alt="{\displaystyle y=x^{5}+x}"></span> then <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 x=BR(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>B</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=BR(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ba05f4ce70616d5ada0c8324f33a85131a4098e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.921ex; height:2.843ex;" alt="{\displaystyle x=BR(y)}"></span>. Since <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 \rho ^{-5}+\rho ^{-1}=1,\quad \rho =1/BR(1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{-5}+\rho ^{-1}=1,\quad \rho =1/BR(1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7fb2a32a2ba0fedb9c43118e650e22a74c3a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.299ex; height:3.176ex;" alt="{\displaystyle \rho ^{-5}+\rho ^{-1}=1,\quad \rho =1/BR(1).}"></span> </p> <table class="floatright"> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plastic_Rauzy_cub.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Plastic_Rauzy_cub.png/290px-Plastic_Rauzy_cub.png" decoding="async" width="290" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Plastic_Rauzy_cub.png/435px-Plastic_Rauzy_cub.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Plastic_Rauzy_cub.png/580px-Plastic_Rauzy_cub.png 2x" data-file-width="2000" data-file-height="1750" /></a><figcaption>A <a href="#matrix">Rauzy fractal</a> associated with the plastic ratio-cubed. The central tile and its three subtiles have areas in the ratios <span class="texhtml">ρ<sup>5</sup> : ρ<sup>2</sup> : ρ : 1.</span></figcaption></figure> </td></tr> <tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plastic_Rauzy_sqr.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Plastic_Rauzy_sqr.png/290px-Plastic_Rauzy_sqr.png" decoding="async" width="290" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Plastic_Rauzy_sqr.png/435px-Plastic_Rauzy_sqr.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Plastic_Rauzy_sqr.png/580px-Plastic_Rauzy_sqr.png 2x" data-file-width="2000" data-file-height="1750" /></a><figcaption>A Rauzy fractal associated with Ⴔ, the plastic ratio-squared; with areas as above.</figcaption></figure> </td></tr></tbody></table> <p><a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">Continued fraction</a> pattern of a few low powers </p> <dl><dd><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 \rho ^{-1}=[0;1,3,12,1,1,3,2,3,2,...]\approx 0.7549}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>12</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>≈<!-- ≈ --></mo> <mn>0.7549</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{-1}=[0;1,3,12,1,1,3,2,3,2,...]\approx 0.7549}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd7fc015b6af44c6b00e6af8a3f172f32ced18e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.713ex; height:3.176ex;" alt="{\displaystyle \rho ^{-1}=[0;1,3,12,1,1,3,2,3,2,...]\approx 0.7549}"></span> (<span class="texhtml">25/33</span>)</dd> <dd><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 \ \rho ^{0}=[1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \rho ^{0}=[1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34671b329242ecde495cc8c291ca34077bf9656a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.391ex; height:3.176ex;" alt="{\displaystyle \ \rho ^{0}=[1]}"></span></dd> <dd><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 \ \rho ^{1}=[1;3,12,1,1,3,2,3,2,4,...]\approx 1.3247}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>;</mo> <mn>3</mn> <mo>,</mo> <mn>12</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>≈<!-- ≈ --></mo> <mn>1.3247</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \rho ^{1}=[1;3,12,1,1,3,2,3,2,4,...]\approx 1.3247}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0783889c28fd45edb71c5d3a6ab0b78b7bcf191e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.015ex; height:3.176ex;" alt="{\displaystyle \ \rho ^{1}=[1;3,12,1,1,3,2,3,2,4,...]\approx 1.3247}"></span> (<span class="texhtml">45/34</span>)</dd> <dd><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 \ \rho ^{2}=[1;1,3,12,1,1,3,2,3,2,...]\approx 1.7549}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>12</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>≈<!-- ≈ --></mo> <mn>1.7549</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \rho ^{2}=[1;1,3,12,1,1,3,2,3,2,...]\approx 1.7549}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d0af90ad701a3e5b0b063b094b0cde27a78db0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.015ex; height:3.176ex;" alt="{\displaystyle \ \rho ^{2}=[1;1,3,12,1,1,3,2,3,2,...]\approx 1.7549}"></span> (<span class="texhtml">58/33</span>)</dd> <dd><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 \ \rho ^{3}=[2;3,12,1,1,3,2,3,2,4,...]\approx 2.3247}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>;</mo> <mn>3</mn> <mo>,</mo> <mn>12</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>≈<!-- ≈ --></mo> <mn>2.3247</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \rho ^{3}=[2;3,12,1,1,3,2,3,2,4,...]\approx 2.3247}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d10db878701ef2ff0aecf216b20e5bd48475422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.015ex; height:3.176ex;" alt="{\displaystyle \ \rho ^{3}=[2;3,12,1,1,3,2,3,2,4,...]\approx 2.3247}"></span> (<span class="texhtml">79/34</span>)</dd> <dd><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 \ \rho ^{4}=[3;12,1,1,3,2,3,2,4,2,...]\approx 3.0796}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>;</mo> <mn>12</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>≈<!-- ≈ --></mo> <mn>3.0796</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \rho ^{4}=[3;12,1,1,3,2,3,2,4,2,...]\approx 3.0796}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e28bc900a12ec947e2458bd6bf450f011ae7fea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.015ex; height:3.176ex;" alt="{\displaystyle \ \rho ^{4}=[3;12,1,1,3,2,3,2,4,2,...]\approx 3.0796}"></span> (<span class="texhtml">40/13</span>)</dd> <dd><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 \ \rho ^{5}=[4;12,1,1,3,2,3,2,4,2,...]\approx 4.0796}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mo>;</mo> <mn>12</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>≈<!-- ≈ --></mo> <mn>4.0796</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \rho ^{5}=[4;12,1,1,3,2,3,2,4,2,...]\approx 4.0796}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e08b39f7933e8b755840989e9a5ab913985ab62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.015ex; height:3.176ex;" alt="{\displaystyle \ \rho ^{5}=[4;12,1,1,3,2,3,2,4,2,...]\approx 4.0796}"></span> (<span class="texhtml">53/13</span>) ...</dd> <dd><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 \ \rho ^{7}=[7;6,3,1,1,4,1,1,2,1,1,...]\approx 7.1592}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>7</mn> <mo>;</mo> <mn>6</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>≈<!-- ≈ --></mo> <mn>7.1592</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \rho ^{7}=[7;6,3,1,1,4,1,1,2,1,1,...]\approx 7.1592}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e4549172dc5c89cce1557985bd020ece361951f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.049ex; height:3.176ex;" alt="{\displaystyle \ \rho ^{7}=[7;6,3,1,1,4,1,1,2,1,1,...]\approx 7.1592}"></span> (<span class="texhtml">93/13</span>) ...</dd> <dd><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 \ \rho ^{9}=[12;1,1,3,2,3,2,4,2,141,...]\approx 12.5635}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>12</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>141</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>≈<!-- ≈ --></mo> <mn>12.5635</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \rho ^{9}=[12;1,1,3,2,3,2,4,2,141,...]\approx 12.5635}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ed0cb86963afb20a44e29ad48ec6e018729c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.502ex; height:3.176ex;" alt="{\displaystyle \ \rho ^{9}=[12;1,1,3,2,3,2,4,2,141,...]\approx 12.5635}"></span> (<span class="texhtml">88/7</span>)</dd></dl> <p>The plastic ratio is the smallest <a href="/wiki/Pisot_number" class="mw-redirect" title="Pisot number">Pisot number</a>.<sup id="cite_ref-Panju_5-0" class="reference"><a href="#cite_note-Panju-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Because the <a href="/wiki/Absolute_value#Complex_numbers" title="Absolute value">absolute value</a> <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 1/{\sqrt {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ρ<!-- ρ --></mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a40d33f42895613654b14b5ed5b6f6e68a3a4e80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.463ex; height:3.176ex;" alt="{\displaystyle 1/{\sqrt {\rho }}}"></span> of the algebraic conjugates is smaller than 1, powers of <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>⁠</span> generate <a href="/wiki/Almost_integer" title="Almost integer">almost integers</a>. For example: <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 \rho ^{29}=3480.0002874...\approx 3480+1/3479.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>29</mn> </mrow> </msup> <mo>=</mo> <mn>3480.0002874...</mn> <mo>≈<!-- ≈ --></mo> <mn>3480</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3479.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{29}=3480.0002874...\approx 3480+1/3479.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/343d7c37feb4929c70daae62f4bc8a5386b75cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.761ex; height:3.176ex;" alt="{\displaystyle \rho ^{29}=3480.0002874...\approx 3480+1/3479.}"></span> After 29 rotation steps the <a href="/wiki/Phase_(waves)" title="Phase (waves)">phases</a> of the inward spiraling conjugate pair – initially close to <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 \pm 45\pi /58}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±<!-- ± --></mo> <mn>45</mn> <mi>π<!-- π --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>58</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 45\pi /58}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5be78030222a0a5a9f1f63c0488ecca5d1dd52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.952ex; height:2.843ex;" alt="{\displaystyle \pm 45\pi /58}"></span>⁠</span> – nearly align with the imaginary axis. </p><p>The <a href="/wiki/Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> of the plastic ratio <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 m(x)=x^{3}-x-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>x</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(x)=x^{3}-x-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b91e96eacbef4aa3cc4b7eaac27fd4872aebd6c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.835ex; height:3.176ex;" alt="{\displaystyle m(x)=x^{3}-x-1}"></span> has <a href="/wiki/Discriminant" title="Discriminant">discriminant</a> <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 \Delta =-23}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mn>23</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =-23}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c93206f5912fb1e5369ca369effa4c3a4a7e6237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.167ex; height:2.343ex;" alt="{\displaystyle \Delta =-23}"></span>. The <a href="/wiki/Hilbert_class_field" title="Hilbert class field">Hilbert class field</a> of imaginary <a href="/wiki/Quadratic_field" title="Quadratic field">quadratic field</a> <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 K=\mathbb {Q} ({\sqrt {\Delta }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Δ<!-- Δ --></mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} ({\sqrt {\Delta }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793e1d70ceb7d883a55e34586d1a2e3990fbe934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.654ex; height:3.176ex;" alt="{\displaystyle K=\mathbb {Q} ({\sqrt {\Delta }})}"></span> can be formed by adjoining <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>⁠</span>. With argument <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 \tau =(1+{\sqrt {\Delta }})/2\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ<!-- τ --></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Δ<!-- Δ --></mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =(1+{\sqrt {\Delta }})/2\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9173928a0685eba23f221e758cd93e0754888c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.696ex; height:3.176ex;" alt="{\displaystyle \tau =(1+{\sqrt {\Delta }})/2\,}"></span> a generator for the <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> of <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>⁠</span>, one has the special value of <a href="/wiki/Dedekind_eta_function" title="Dedekind eta function">Dedekind eta</a> quotient </p> <dl><dd><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 \rho ={\frac {e^{\pi i/24}\,\eta (\tau )}{{\sqrt {2}}\,\eta (2\tau )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π<!-- π --></mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>24</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>η<!-- η --></mi> <mo stretchy="false">(</mo> <mi>τ<!-- τ --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>η<!-- η --></mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>τ<!-- τ --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {e^{\pi i/24}\,\eta (\tau )}{{\sqrt {2}}\,\eta (2\tau )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08d6c74e6a3e447c2b761ecfd336a0a219810962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.995ex; height:7.176ex;" alt="{\displaystyle \rho ={\frac {e^{\pi i/24}\,\eta (\tau )}{{\sqrt {2}}\,\eta (2\tau )}}}"></span>.<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></dd></dl> <p>Expressed in terms of the <a href="/wiki/Weber_modular_function" title="Weber modular function">Weber-Ramanujan class invariant G<sub>n</sub></a> </p> <dl><dd><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 \rho ={\frac {{\mathfrak {f}}({\sqrt {\Delta }})}{\sqrt {2}}}={\frac {G_{23}}{\sqrt[{4}]{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">f</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Δ<!-- Δ --></mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {{\mathfrak {f}}({\sqrt {\Delta }})}{\sqrt {2}}}={\frac {G_{23}}{\sqrt[{4}]{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe26bd1400d2ccbc804a2d85756babd4083a8ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.213ex; height:7.009ex;" alt="{\displaystyle \rho ={\frac {{\mathfrak {f}}({\sqrt {\Delta }})}{\sqrt {2}}}={\frac {G_{23}}{\sqrt[{4}]{2}}}}"></span>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Properties of the related <a href="/wiki/J-invariant#The_q-expansion_and_moonshine" title="J-invariant">Klein j-invariant</a> <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 j(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo stretchy="false">(</mo> <mi>τ<!-- τ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j(\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e748f54b6308e2ffbcc2b3fc23d0c81b25c8d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:3.996ex; height:2.843ex;" alt="{\displaystyle j(\tau )}"></span>⁠</span> result in near identity <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 e^{\pi {\sqrt {-\Delta }}}\approx \left({\sqrt {2}}\,\rho \right)^{24}-24}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π<!-- π --></mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−<!-- − --></mo> <mi mathvariant="normal">Δ<!-- Δ --></mi> </msqrt> </mrow> </mrow> </msup> <mo>≈<!-- ≈ --></mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>ρ<!-- ρ --></mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>24</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\pi {\sqrt {-\Delta }}}\approx \left({\sqrt {2}}\,\rho \right)^{24}-24}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eea7f40329fca36d4879b140822cf313a22cc3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.231ex; height:3.843ex;" alt="{\displaystyle e^{\pi {\sqrt {-\Delta }}}\approx \left({\sqrt {2}}\,\rho \right)^{24}-24}"></span>. The difference is <span class="texhtml">< 1/12659</span>. </p><p>The <a href="/wiki/Modular_lambda_function#Lambda-star" title="Modular lambda function">elliptic integral singular value</a><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <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 k_{r}=\lambda ^{*}(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{r}=\lambda ^{*}(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/929b984c842823cf455c765a9a1e0945669aa5e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.551ex; height:2.843ex;" alt="{\displaystyle k_{r}=\lambda ^{*}(r)}"></span> for <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 r=23}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>23</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=23}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e419ebc249eb6319056c8119e6155cc350688423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.472ex; height:2.176ex;" alt="{\displaystyle r=23}"></span>⁠</span> has closed form expression </p> <dl><dd><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 \lambda ^{*}(23)=\sin(\arcsin \left(({\sqrt[{4}]{2}}\,\rho )^{-12}\right)/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>23</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mi>ρ<!-- ρ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>12</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{*}(23)=\sin(\arcsin \left(({\sqrt[{4}]{2}}\,\rho )^{-12}\right)/2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33be641cb1998634dcad48ab2e890ada533a32d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.375ex; height:3.343ex;" alt="{\displaystyle \lambda ^{*}(23)=\sin(\arcsin \left(({\sqrt[{4}]{2}}\,\rho )^{-12}\right)/2)}"></span></dd></dl> <p>(which is less than 1/3 the <a href="/wiki/Eccentricity_(mathematics)#Ellipses" title="Eccentricity (mathematics)">eccentricity</a> of the orbit of Venus). </p> <div class="mw-heading mw-heading2"><h2 id="Van_der_Laan_sequence">Van der Laan sequence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastic_ratio&action=edit&section=3" title="Edit section: Van der Laan sequence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plastic5_Rauzy_sqr.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Plastic5_Rauzy_sqr.png/290px-Plastic5_Rauzy_sqr.png" decoding="async" width="290" height="334" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Plastic5_Rauzy_sqr.png/435px-Plastic5_Rauzy_sqr.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Plastic5_Rauzy_sqr.png/580px-Plastic5_Rauzy_sqr.png 2x" data-file-width="1738" data-file-height="2000" /></a><figcaption>A fan of plastic Rauzy tiles with areas in ratio Ⴔ. The fractal boundary has <a href="/wiki/Minkowski%E2%80%93Bouligand_dimension" title="Minkowski–Bouligand dimension">box-counting</a> <a href="/wiki/Fractal_dimension" title="Fractal dimension">dimension</a> 1.11</figcaption></figure> <p>In his quest for perceptible clarity, the Dutch <a href="/wiki/Benedictine_monk" class="mw-redirect" title="Benedictine monk">Benedictine monk</a> and architect Dom <a href="/wiki/Hans_van_der_Laan" title="Hans van der Laan">Hans van der Laan</a> (1904-1991) asked for the minimum difference between two sizes, so that we will clearly perceive them as distinct. Also, what is the maximum ratio of two sizes, so that we can still relate them and perceive nearness. According to his observations, the answers are <span class="texhtml">1/4 and 7/1</span>, spanning a single <i>order of size</i>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Requiring proportional continuity, he constructed a <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a> of <a href="#types">eight measures</a> (<i>types of size</i>) with common ratio <span class="texhtml">2 / (3/4 + 1/7<sup>1/7</sup>) ≈ ρ.</span> Put in rational form, this architectonic system of measure is constructed from a subset of the numbers that bear his name. </p><p>The Van der Laan numbers have a close connection to the <a href="/wiki/Perrin_number" title="Perrin number">Perrin</a> and <a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan sequences</a>. In combinatorics, the number of <a href="/wiki/Composition_(combinatorics)" title="Composition (combinatorics)">compositions</a> of n into parts 2 and 3 is counted by the <i>n</i>th Van der Laan number. </p><p>The Van der Laan sequence is defined by the third-order <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a> </p> <dl><dd><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 V_{n}=V_{n-2}+V_{n-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}=V_{n-2}+V_{n-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78d54ce1dbc188a20c40f272b6657b19ac1b6961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.861ex; height:2.509ex;" alt="{\displaystyle V_{n}=V_{n-2}+V_{n-3}}"></span> for <span class="texhtml"><i>n</i> > 2</span>,</dd></dl> <p>with initial values </p> <dl><dd><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 V_{1}=0,V_{0}=V_{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{1}=0,V_{0}=V_{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69ab53d8da3f2726b43582393016a65c919f30aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.883ex; height:2.509ex;" alt="{\displaystyle V_{1}=0,V_{0}=V_{2}=1}"></span>.</dd></dl> <p>The first few terms are 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86,... (sequence <span class="nowrap external"><a href="//oeis.org/A182097" class="extiw" title="oeis:A182097">A182097</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). The limit ratio between consecutive terms is the plastic ratio. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Nombre_plastique2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Nombre_plastique2.svg/350px-Nombre_plastique2.svg.png" decoding="async" width="350" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Nombre_plastique2.svg/525px-Nombre_plastique2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Nombre_plastique2.svg/700px-Nombre_plastique2.svg.png 2x" data-file-width="328" data-file-height="134" /></a><figcaption>The 1924 <a href="#History_and_names">Cordonnier cut</a>. With <span class="texhtml">S<sub>1</sub> = 3, S<sub>2</sub> = 4, S<sub>3</sub> = 5</span>, the <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> of <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">S<sub>2</sub></span><span class="sr-only">/</span><span class="den">S<sub>1</sub> </span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">S<sub>1</sub> + S<sub>2</sub></span><span class="sr-only">/</span><span class="den">S<sub>3</sub> </span></span>⁠</span> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">S<sub>3</sub></span><span class="sr-only">/</span><span class="den">S<sub>2</sub> </span></span>⁠</span> </span> is <span class="texhtml">3 / <span style="font-size:133%;">(</span><span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">4</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span> </span><span style="font-size:133%;">)</span> ≈ ρ + 1/4922.</span></figcaption></figure> <table class="wikitable" id="types"> <caption>Table of the eight Van der Laan measures </caption> <tbody><tr> <th>k</th> <th>n - m</th> <th><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 V_{n}/V_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}/V_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e4ad6d68216dc8a92578e9e255f6ee399cb790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.766ex; height:2.843ex;" alt="{\displaystyle V_{n}/V_{m}}"></span>⁠</span></th> <th>err<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 (\rho ^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\rho ^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/747e90a5f9f9cca0041c0990c05783e8a93c8b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.1ex; height:3.176ex;" alt="{\displaystyle (\rho ^{k})}"></span>⁠</span></th> <th>interval </th></tr> <tr> <td>0</td> <td>3 - 3</td> <td>1 /1</td> <td>0</td> <td>minor element </td></tr> <tr> <td>1</td> <td>8 - 7</td> <td>4 /3</td> <td>1/116</td> <td>major element </td></tr> <tr> <td>2</td> <td>10 - 8</td> <td>7 /4</td> <td>-1/205</td> <td>minor piece </td></tr> <tr> <td>3</td> <td>10 - 7</td> <td>7 /3</td> <td>1/116</td> <td>major piece </td></tr> <tr> <td>4</td> <td>7 - 3</td> <td>3 /1</td> <td>-1/12</td> <td>minor part </td></tr> <tr> <td>5</td> <td>8 - 3</td> <td>4 /1</td> <td>-1/12</td> <td>major part </td></tr> <tr> <td>6</td> <td>13 - 7</td> <td>16 /3</td> <td>-1/14</td> <td>minor whole </td></tr> <tr> <td>7</td> <td>10 - 3</td> <td>7 /1</td> <td>-1/6</td> <td>major whole </td></tr></tbody></table> <p>The first 14 indices n for which <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 V_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc5a637019ce3415183f06995aeeca93547767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.509ex;" alt="{\displaystyle V_{n}}"></span>⁠</span> is prime are n = 5, 6, 7, 9, 10, 16, 21, 32, 39, 86, 130, 471, 668, 1264 (sequence <span class="nowrap external"><a href="//oeis.org/A112882" class="extiw" title="oeis:A112882">A112882</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> The last number has 154 decimal digits. </p><p>The sequence can be extended to negative indices using </p> <dl><dd><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 V_{n}=V_{n+3}-V_{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}=V_{n+3}-V_{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c97e63cd83e5995c0decffadb900434932494e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.861ex; height:2.509ex;" alt="{\displaystyle V_{n}=V_{n+3}-V_{n+1}}"></span>.</dd></dl> <p>The <a href="/wiki/Generating_function" title="Generating function">generating function</a> of the Van der Laan sequence is given by </p> <dl><dd><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 {\frac {1}{1-x^{2}-x^{3}}}=\sum _{n=0}^{\infty }V_{n}x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-x^{2}-x^{3}}}=\sum _{n=0}^{\infty }V_{n}x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d98a5de7e77041c99a064997968896fef65d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.41ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{1-x^{2}-x^{3}}}=\sum _{n=0}^{\infty }V_{n}x^{n}}"></span> for <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 x<1/\rho \;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ<!-- ρ --></mi> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<1/\rho \;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b843565f79086e18b864818dfef66518144a40f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.247ex; height:2.843ex;" alt="{\displaystyle x<1/\rho \;.}"></span><sup id="cite_ref-oeisGF_11-0" class="reference"><a href="#cite_note-oeisGF-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></dd></dl> <p>The sequence is related to sums of <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a> by </p> <dl><dd><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 V_{n}=\sum _{k=\lfloor (n+2)/3\rfloor }^{\lfloor n/2\rfloor }{k \choose n-2k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}=\sum _{k=\lfloor (n+2)/3\rfloor }^{\lfloor n/2\rfloor }{k \choose n-2k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d34924daf52b59cfacff62359a40e00f3c5a00cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.694ex; height:8.176ex;" alt="{\displaystyle V_{n}=\sum _{k=\lfloor (n+2)/3\rfloor }^{\lfloor n/2\rfloor }{k \choose n-2k}}"></span>.<sup id="cite_ref-oeispado_12-0" class="reference"><a href="#cite_note-oeispado-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></dd></dl> <p>The <a href="/wiki/Characteristic_equation_(calculus)" title="Characteristic equation (calculus)">characteristic equation</a> of the recurrence is <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 x^{3}-x-1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>x</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-x-1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d6252aae67d8cbef45e447935787debdf0455c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.818ex; height:2.843ex;" alt="{\displaystyle x^{3}-x-1=0}"></span>. If the three solutions are real root <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>⁠</span> and conjugate pair <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β<!-- β --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>⁠</span> and <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>⁠</span>, the Van der Laan numbers can be computed with the <a href="/wiki/Fibonacci_sequence#Binet's_formula" title="Fibonacci sequence">Binet formula</a> <sup id="cite_ref-oeispado_12-1" class="reference"><a href="#cite_note-oeispado-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 V_{n-1}=a\alpha ^{n}+b\beta ^{n}+c\gamma ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msup> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>β<!-- β --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>γ<!-- γ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n-1}=a\alpha ^{n}+b\beta ^{n}+c\gamma ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c9963f0fcd0df9a30a0eecc3d56e1da6dcb6d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.448ex; height:2.843ex;" alt="{\displaystyle V_{n-1}=a\alpha ^{n}+b\beta ^{n}+c\gamma ^{n}}"></span>, with real <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>⁠</span> and conjugates <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>⁠</span> and <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>⁠</span> the roots of <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 23x^{3}+x-1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>23</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 23x^{3}+x-1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa5aaccbb287f53d1e9a3cb30d9085d57376927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.143ex; height:2.843ex;" alt="{\displaystyle 23x^{3}+x-1=0}"></span>.</dd></dl> <p>Since <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 \left\vert b\beta ^{n}+c\gamma ^{n}\right\vert <1/{\sqrt {\alpha ^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>b</mi> <msup> <mi>β<!-- β --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>γ<!-- γ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\vert b\beta ^{n}+c\gamma ^{n}\right\vert <1/{\sqrt {\alpha ^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfffa5cf23d3dd8bff70040fd3dbbda90be4ddaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.258ex; height:3.176ex;" alt="{\displaystyle \left\vert b\beta ^{n}+c\gamma ^{n}\right\vert <1/{\sqrt {\alpha ^{n}}}}"></span> and <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 \alpha =\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abacc9d7b7038606f3bb6eb1b5dfc2a100f5f231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.788ex; height:2.176ex;" alt="{\displaystyle \alpha =\rho }"></span>, the number <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 V_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc5a637019ce3415183f06995aeeca93547767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.509ex;" alt="{\displaystyle V_{n}}"></span>⁠</span> is the nearest integer to <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 a\,\rho ^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\rho ^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca9f04484a2c0c53edb2561623716fb9134d826" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.138ex; height:3.176ex;" alt="{\displaystyle a\,\rho ^{n+1}}"></span>, with <span class="texhtml"><i>n</i> > 1</span> and <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 a=\rho /(3\rho ^{2}-1)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=\rho /(3\rho ^{2}-1)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a73819c213b67563e10f18a66b8350e0501e19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.377ex; height:3.176ex;" alt="{\displaystyle a=\rho /(3\rho ^{2}-1)=}"></span> <span style="white-space:nowrap">0.31062<span style="margin-left:0.25em">88296</span><span style="margin-left:0.25em">40467</span><span style="margin-left:0.25em">07776</span><span style="margin-left:0.25em">19027...</span></span> </p><p>Coefficients <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 a=b=c=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b=c=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d8b350715922395f5781625084960c0c451e29b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.692ex; height:2.176ex;" alt="{\displaystyle a=b=c=1}"></span> result in the Binet formula for the related sequence <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 P_{n}=2V_{n}+V_{n-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}=2V_{n}+V_{n-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f856c6bb2ba3d6054d8cc1372af9de898340c14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.06ex; height:2.509ex;" alt="{\displaystyle P_{n}=2V_{n}+V_{n-3}}"></span>. </p><p>The first few terms are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119,... (sequence <span class="nowrap external"><a href="//oeis.org/A001608" class="extiw" title="oeis:A001608">A001608</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p><p>This <a href="/wiki/Perrin_sequence" class="mw-redirect" title="Perrin sequence">Perrin sequence</a> has the <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat property</a>: if p is prime, <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 P_{p}\equiv P_{1}{\bmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≡<!-- ≡ --></mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{p}\equiv P_{1}{\bmod {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38c1f7934988d83b5fd4db542dc2823ba646b049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.047ex; height:2.843ex;" alt="{\displaystyle P_{p}\equiv P_{1}{\bmod {p}}}"></span>. The converse does not hold, but the small number of <a href="/wiki/Pseudoprime" title="Pseudoprime">pseudoprimes</a> <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 \,n\mid P_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>n</mi> <mo>∣<!-- ∣ --></mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,n\mid P_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec684ab1625567a6d468ef5b66293f78493ba723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.43ex; height:2.843ex;" alt="{\displaystyle \,n\mid P_{n}}"></span> makes the sequence special.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The only 7 composite numbers below <span class="texhtml">10<sup>8</sup> </span> to pass the test are n = 271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plastic_Rauzy_ac.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Plastic_Rauzy_ac.png/290px-Plastic_Rauzy_ac.png" decoding="async" width="290" height="290" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Plastic_Rauzy_ac.png/435px-Plastic_Rauzy_ac.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Plastic_Rauzy_ac.png/580px-Plastic_Rauzy_ac.png 2x" data-file-width="2000" data-file-height="2000" /></a><figcaption>A plastic Rauzy fractal: the combined surface and the three separate tiles have areas in the ratios <span class="texhtml">ρ<sup>5</sup> : ρ<sup>2</sup> : ρ : 1.</span></figcaption></figure> <p>The Van der Laan numbers are obtained as integral powers <span class="texhtml"><i>n</i> > 2</span> of a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> with real <a href="/wiki/Eigenvalues_and_eigenvectors#Eigenvalues_and_eigenvectors_of_matrices" title="Eigenvalues and eigenvectors">eigenvalue</a> <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>⁠</span> <sup id="cite_ref-oeisGF_11-1" class="reference"><a href="#cite_note-oeisGF-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 Q={\begin{pmatrix}0&1&1\\1&0&0\\0&1&0\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\begin{pmatrix}0&1&1\\1&0&0\\0&1&0\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4b91e47734d8bb531b955b7afe1367c2c0d54a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.535ex; height:9.176ex;" alt="{\displaystyle Q={\begin{pmatrix}0&1&1\\1&0&0\\0&1&0\end{pmatrix}},}"></span></dd></dl> <dl><dd><span class="mwe-math-element" id="matrix"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{n}={\begin{pmatrix}V_{n}&V_{n+1}&V_{n-1}\\V_{n-1}&V_{n}&V_{n-2}\\V_{n-2}&V_{n-1}&V_{n-3}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{n}={\begin{pmatrix}V_{n}&V_{n+1}&V_{n-1}\\V_{n-1}&V_{n}&V_{n-2}\\V_{n-2}&V_{n-1}&V_{n-3}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2271e2074734befac73785ede2793ab57be7ad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" id="matrix" aria-hidden="true" style="vertical-align: -4.171ex; width:29.641ex; height:9.509ex;" alt="{\displaystyle Q^{n}={\begin{pmatrix}V_{n}&V_{n+1}&V_{n-1}\\V_{n-1}&V_{n}&V_{n-2}\\V_{n-2}&V_{n-1}&V_{n-3}\end{pmatrix}}}"></span></dd></dl> <p>The <a href="/wiki/Trace_(linear_algebra)#Relationship_to_eigenvalues" title="Trace (linear algebra)">trace</a> of <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 Q^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5da2f2952f8145669909f96e7f344e533b1e9a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle Q^{n}}"></span>⁠</span> gives the Perrin numbers. </p><p>Alternatively, <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span>⁠</span> can be interpreted as <a href="/wiki/Incidence_matrix" title="Incidence matrix">incidence matrix</a> for a <a href="/wiki/Morphic_word#D0L_system" title="Morphic word">D0L</a> <a href="/wiki/L-system" title="L-system">Lindenmayer system</a> on the alphabet <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 \{a,b,c\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a,b,c\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75e9bc621ced3f02e87b1c40be37867929142bf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.627ex; height:2.843ex;" alt="{\displaystyle \{a,b,c\}}"></span>⁠</span> with corresponding <a href="/wiki/Semi-Thue_system" title="Semi-Thue system">substitution rule</a> </p> <dl><dd><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 {\begin{cases}a\;\mapsto \;b\\b\;\mapsto \;ac\\c\;\mapsto \;a\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>a</mi> <mspace width="thickmathspace" /> <mo stretchy="false">↦<!-- ↦ --></mo> <mspace width="thickmathspace" /> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> <mspace width="thickmathspace" /> <mo stretchy="false">↦<!-- ↦ --></mo> <mspace width="thickmathspace" /> <mi>a</mi> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mspace width="thickmathspace" /> <mo stretchy="false">↦<!-- ↦ --></mo> <mspace width="thickmathspace" /> <mi>a</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}a\;\mapsto \;b\\b\;\mapsto \;ac\\c\;\mapsto \;a\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591004682f2b604c0ca1804e28bf458a5c7c5f8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:10.956ex; height:8.509ex;" alt="{\displaystyle {\begin{cases}a\;\mapsto \;b\\b\;\mapsto \;ac\\c\;\mapsto \;a\end{cases}}}"></span></dd></dl> <p>and initiator <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 w_{0}=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{0}=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e48401642ee1f1c77415b64951883ff450f1e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.824ex; height:2.009ex;" alt="{\displaystyle w_{0}=c}"></span>⁠</span>. The series of words <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 w_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5421b423b193df0b692addd2e4cf025e52bc09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.883ex; height:2.009ex;" alt="{\displaystyle w_{n}}"></span>⁠</span> produced by iterating the substitution have the property that the number of <span class="texhtml">c's, b's</span> and <span class="texhtml">a's</span> are equal to successive Van der Laan numbers. Their lengths are <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 l(w_{n})=V_{n+2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l(w_{n})=V_{n+2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6b64318c58423191a8c0c08014681b05c2c09b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.805ex; height:2.843ex;" alt="{\displaystyle l(w_{n})=V_{n+2}.}"></span> </p><p>Associated to this string rewriting process is a set composed of three overlapping <a href="/wiki/Self-similarity" title="Self-similarity">self-similar</a> tiles called the <a href="/wiki/Rauzy_fractal" title="Rauzy fractal">Rauzy fractal</a>, that visualizes the <a href="/wiki/Combinatorics_on_words" title="Combinatorics on words">combinatorial</a> information contained in a multiple-generation letter sequence.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Geometry">Geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastic_ratio&action=edit&section=4" title="Edit section: Geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plastic_square_partitions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Plastic_square_partitions.svg/330px-Plastic_square_partitions.svg.png" decoding="async" width="330" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Plastic_square_partitions.svg/495px-Plastic_square_partitions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Plastic_square_partitions.svg/660px-Plastic_square_partitions.svg.png 2x" data-file-width="414" data-file-height="126" /></a><figcaption>Three partitions of a square into similar rectangles, <span class="texhtml">1 = 3·<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> + 2·<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">ρ<sup>2</sup> </span></span>⁠</span> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">ρ<sup>4</sup> </span></span>⁠</span> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">ρ<sup>8</sup> </span></span>⁠</span></span>.</figcaption></figure> <p>There are precisely three ways of partitioning a square into three similar rectangles:<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <ol><li>The trivial solution given by three congruent rectangles with aspect ratio 3:1.</li> <li>The solution in which two of the three rectangles are congruent and the third one has twice the side lengths of the other two, where the rectangles have aspect ratio 3:2.</li> <li>The solution in which the three rectangles are all of different sizes and where they have aspect ratio <i>ρ</i><sup>2</sup>. The ratios of the linear sizes of the three rectangles are: <i>ρ</i> (large:medium); <i>ρ</i><sup>2</sup> (medium:small); and <i>ρ</i><sup>3</sup> (large:small). The internal, long edge of the largest rectangle (the square's fault line) divides two of the square's four edges into two segments each that stand to one another in the ratio <i>ρ.</i> The internal, coincident short edge of the medium rectangle and long edge of the small rectangle divides one of the square's other, two edges into two segments that stand to one another in the ratio <i>ρ</i><sup>4</sup>.</li></ol> <p>The fact that a rectangle of aspect ratio <i>ρ</i><sup>2</sup> can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number <i>ρ</i><sup>2</sup> related to the <a href="/wiki/Routh%E2%80%93Hurwitz_theorem" title="Routh–Hurwitz theorem">Routh–Hurwitz theorem</a>: all of its conjugates have positive real part.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Circumradius" class="mw-redirect" title="Circumradius">circumradius</a> of the <a href="/wiki/Snub_icosidodecadodecahedron" title="Snub icosidodecadodecahedron">snub icosidodecadodecahedron</a> for unit edge length is </p> <dl><dd><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 {\frac {1}{2}}{\sqrt {\frac {2\rho -1}{\rho -1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <mi>ρ<!-- ρ --></mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow> <mi>ρ<!-- ρ --></mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}{\sqrt {\frac {2\rho -1}{\rho -1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b36598d7bea3fefa9d8584aae24e4a1daf688fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.526ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{2}}{\sqrt {\frac {2\rho -1}{\rho -1}}}}"></span>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Rho-squared_rectangle">Rho-squared rectangle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastic_ratio&action=edit&section=5" title="Edit section: Rho-squared rectangle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plastic_ratio-squared.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Plastic_ratio-squared.svg/330px-Plastic_ratio-squared.svg.png" decoding="async" width="330" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Plastic_ratio-squared.svg/495px-Plastic_ratio-squared.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Plastic_ratio-squared.svg/660px-Plastic_ratio-squared.svg.png 2x" data-file-width="1100" data-file-height="640" /></a><figcaption>Nested rho-squared rectangles with side lengths in powers of <span class="texhtml">ρ</span>.</figcaption></figure> <p>Given a rectangle of height <span class="texhtml">1</span>, length <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 \rho ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b44a36e828a576858671546f5e0cb05806b742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}}"></span>⁠</span> and diagonal length <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 {\sqrt {\rho ^{5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\rho ^{5}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35351d50e1a1276d9a528c3029d6d55302d200f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:4.58ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\rho ^{5}}}}"></span> (according to <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 1+\rho ^{4}=\rho ^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\rho ^{4}=\rho ^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84dce0c0358f4caa2475c4744f80fe946e7d5c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.614ex; height:3.176ex;" alt="{\displaystyle 1+\rho ^{4}=\rho ^{5}}"></span>). The triangles on the diagonal have <a href="/wiki/Altitude_(triangle)" title="Altitude (triangle)">altitudes</a> <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 1/{\sqrt {\rho }}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ρ<!-- ρ --></mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {\rho }}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/986284a89f58573205b66a4bd099b4d8f0fd463d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.497ex; height:3.176ex;" alt="{\displaystyle 1/{\sqrt {\rho }}\,;}"></span> each perpendicular foot divides the diagonal in ratio <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 \rho ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7addc2fa24d6aca9e7b9f994c1b9efe2f43bfc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:3.176ex;" alt="{\displaystyle \rho ^{4}}"></span>⁠</span>. </p><p>On the left-hand side, cut off a square of side length <span class="texhtml">1</span> and mark the intersection with the falling diagonal. The remaining rectangle now has <a href="/wiki/Aspect_ratio" title="Aspect ratio">aspect ratio</a> <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 \rho :1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> <mo>:</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho :1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c216db0a7c549e5bb79f858918c26b6ae38d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.302ex; height:2.676ex;" alt="{\displaystyle \rho :1}"></span>⁠</span> (according to <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 \rho ^{2}-1=\rho ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> <mo>=</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{2}-1=\rho ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c73697341f34fa5fca05519765397f3242565939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.892ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}-1=\rho ^{-1}}"></span>). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The parent rho-squared rectangle and the two scaled copies along the diagonal have linear sizes in the ratios <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 \rho ^{3}:\rho :1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>:</mo> <mi>ρ<!-- ρ --></mi> <mo>:</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{3}:\rho :1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc687f30557c1d84968bcb07cecb2d6bcd8801f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.142ex; height:3.176ex;" alt="{\displaystyle \rho ^{3}:\rho :1.}"></span> The areas of the rectangles opposite the diagonal are both equal to <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 1/\rho ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\rho ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eeac846822f650636a9d55d47f372ff157c9612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.581ex; height:3.176ex;" alt="{\displaystyle 1/\rho ^{3}}"></span>⁠</span>, with aspect ratios <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 \rho ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf917b00016b1413771ea497ce45764014b862dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:3.176ex;" alt="{\displaystyle \rho ^{3}}"></span>⁠</span> (below) and <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>⁠</span> (above). </p><p>If the diagram is further subdivided by perpendicular lines through the feet of the altitudes, the lengths of the diagonal and its (thus far) seven distinct subsections are in ratios <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 \rho ^{6}:\rho ^{5}:\rho ^{4}:\rho ^{2}+1:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>:</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>:</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>:</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{6}:\rho ^{5}:\rho ^{4}:\rho ^{2}+1:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/322b9bf52cdb90f6f5998fc3986e8f68890761f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.131ex; height:3.176ex;" alt="{\displaystyle \rho ^{6}:\rho ^{5}:\rho ^{4}:\rho ^{2}+1:}"></span> <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 \,\rho ^{3}:\rho ^{2}:\rho :1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>:</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <mi>ρ<!-- ρ --></mi> <mo>:</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\rho ^{3}:\rho ^{2}:\rho :1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8da85e3c35be25a2cd5c53fa078690924c867f8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.722ex; height:3.176ex;" alt="{\displaystyle \,\rho ^{3}:\rho ^{2}:\rho :1,}"></span> where <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 \rho ^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a799afc70008cc75f74cdbecccd67272a707550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}+1}"></span>⁠</span> corresponds to the span between both feet. </p><p>Nested rho-squared rectangles with diagonal lengths in ratios <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 1/\rho ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\rho ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d58f2e9412b204447593b60e607461b528e638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.581ex; height:3.176ex;" alt="{\displaystyle 1/\rho ^{2}}"></span>⁠</span> converge at distance <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 t={\sqrt {\rho }}/(\rho ^{2}+1)=0.41779130...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ρ<!-- ρ --></mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.41779130...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t={\sqrt {\rho }}/(\rho ^{2}+1)=0.41779130...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d55c619cbe9b9de8b1a86177e106a30e8b74ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.455ex; height:3.509ex;" alt="{\displaystyle t={\sqrt {\rho }}/(\rho ^{2}+1)=0.41779130...}"></span> from the intersection point. This is equal to the unique positive node that optimizes cubic <a href="/wiki/Lagrange_polynomial" title="Lagrange polynomial">Lagrange interpolation</a> on the interval <span class="texhtml">[−1,1]</span>. With optimal node set <span class="texhtml">T = {−1,−t, t, 1</span>}, the <a href="/wiki/Lebesgue_constant" title="Lebesgue constant">Lebesgue function</a> <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 \lambda _{3}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{3}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79f321bd1b8e8160699cf38fb51cd5cb4016318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.548ex; height:2.843ex;" alt="{\displaystyle \lambda _{3}(x)}"></span>⁠</span> evaluates to the minimal cubic Lebesgue constant <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 \Lambda _{3}(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Λ<!-- Λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda _{3}(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c86e89174025c4935e80c441cf7b92b137440e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.113ex; height:2.843ex;" alt="{\displaystyle \Lambda _{3}(T)}"></span>⁠</span> at <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical point</a> <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 x_{c}=\rho ^{2}t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{c}=\rho ^{2}t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f76483616f1a40a317947096b6427a7d251c96c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.115ex; height:3.176ex;" alt="{\displaystyle x_{c}=\rho ^{2}t.}"></span><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Since <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 t+\rho ^{2}t={\sqrt {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>+</mo> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ρ<!-- ρ --></mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t+\rho ^{2}t={\sqrt {\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07869a02644c1f6810ae8d466cb554251249fa35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.012ex; height:3.509ex;" alt="{\displaystyle t+\rho ^{2}t={\sqrt {\rho }}}"></span>, this is also the distance from the point of convergence to the upper left vertex. </p> <div class="mw-heading mw-heading3"><h3 id="Plastic_spiral">Plastic spiral</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastic_ratio&action=edit&section=6" title="Edit section: Plastic spiral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:332px;max-width:332px"><div class="trow"><div class="tsingle" style="width:330px;max-width:330px"><div class="thumbimage" style="height:246px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Plastic_spiral.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Plastic_spiral.svg/328px-Plastic_spiral.svg.png" decoding="async" width="328" height="246" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Plastic_spiral.svg/492px-Plastic_spiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Plastic_spiral.svg/656px-Plastic_spiral.svg.png 2x" data-file-width="1400" data-file-height="1050" /></a></span></div><div class="thumbcaption">Two plastic spirals with different initial radii.</div></div></div><div class="trow"><div class="tsingle" style="width:330px;max-width:330px"><div class="thumbimage" style="height:246px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Chambered_nautilus_shell_and_plastic_spiral.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Chambered_nautilus_shell_and_plastic_spiral.svg/328px-Chambered_nautilus_shell_and_plastic_spiral.svg.png" decoding="async" width="328" height="246" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Chambered_nautilus_shell_and_plastic_spiral.svg/492px-Chambered_nautilus_shell_and_plastic_spiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Chambered_nautilus_shell_and_plastic_spiral.svg/656px-Chambered_nautilus_shell_and_plastic_spiral.svg.png 2x" data-file-width="2240" data-file-height="1680" /></a></span></div><div class="thumbcaption">Chambered nautilus shell and plastic spiral.</div></div></div></div></div> <p>A plastic spiral is a <a href="/wiki/Logarithmic_spiral" title="Logarithmic spiral">logarithmic spiral</a> that gets wider by a factor of <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>⁠</span> for every quarter turn. It is described by the <a href="/wiki/Polar_equation" class="mw-redirect" title="Polar equation">polar equation</a> <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 r(\theta )=a\exp(k\theta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>exp</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\theta )=a\exp(k\theta ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5fffde50a7039d758d06da2deb447a224b0a6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.974ex; height:2.843ex;" alt="{\displaystyle r(\theta )=a\exp(k\theta ),}"></span> with initial radius <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>⁠</span> and parameter <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 k={\frac {2\ln(\rho )}{\pi }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>ρ<!-- ρ --></mi> <mo stretchy="false">)</mo> </mrow> <mi>π<!-- π --></mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {2\ln(\rho )}{\pi }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d330c15270cb5aba7e4b55f0c78f68f95ef9c477" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.293ex; height:5.676ex;" alt="{\displaystyle k={\frac {2\ln(\rho )}{\pi }}.}"></span> If drawn on a rectangle with sides in ratio <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>⁠</span>, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of rectangles with aspect ratio <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 \rho ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b44a36e828a576858671546f5e0cb05806b742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}}"></span>⁠</span> which are orthogonally aligned and successively scaled by a factor <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 1/\rho .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ<!-- ρ --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\rho .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40bd6e5e9005c3914e5e9911b35db7560151ad1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.174ex; height:2.843ex;" alt="{\displaystyle 1/\rho .}"></span>⁠</span> </p><p><br /> In 1838 <a href="/wiki/Henry_Moseley_(mathematician)" title="Henry Moseley (mathematician)">Henry Moseley</a> noticed that whorls of a shell of the <a href="/wiki/Chambered_nautilus" title="Chambered nautilus">chambered nautilus</a> are in geometrical progression: "It will be found that the distance of any two of its whorls measured upon a radius vector is <i>one-third</i> that of the next two whorls measured upon the same radius vector ... The curve is therefore a logarithmic spiral."<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Moseley thus gave the expansion rate <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 {\sqrt[{4}]{3}}\approx \rho -1/116}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo>≈<!-- ≈ --></mo> <mi>ρ<!-- ρ --></mi> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>116</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{4}]{3}}\approx \rho -1/116}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27386c1df3740c8258a567f418d2cc25d305d418" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.051ex; height:3.176ex;" alt="{\displaystyle {\sqrt[{4}]{3}}\approx \rho -1/116}"></span> for a quarter turn.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> Considering the plastic ratio a three-dimensional equivalent of the ubiquitous golden ratio, it appears to be a natural candidate for measuring the shell.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="History_and_names">History and names</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastic_ratio&action=edit&section=7" title="Edit section: History and names"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="texhtml">ρ</span> was first studied by <a href="/wiki/Axel_Thue" title="Axel Thue">Axel Thue</a> in 1912 and by <a href="/wiki/G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a> in 1919.<sup id="cite_ref-Panju_5-1" class="reference"><a href="#cite_note-Panju-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> French high school student <a href="/w/index.php?title=G%C3%A9rard_Cordonnier&action=edit&redlink=1" class="new" title="Gérard Cordonnier (page does not exist)">Gérard Cordonnier</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://fr.wikipedia.org/wiki/G%C3%A9rard_Cordonnier" class="extiw" title="fr:Gérard Cordonnier">fr</a>]</span> discovered the ratio for himself in 1924. In his correspondence with <a href="/wiki/Hans_van_der_Laan" title="Hans van der Laan">Hans van der Laan</a> a few years later, he called it the radiant number (<a href="/wiki/French_language" title="French language">French</a>: <i lang="fr">le nombre radiant</i>). Van der Laan initially referred to it as the fundamental ratio (<a href="/wiki/Dutch_language" title="Dutch language">Dutch</a>: <i lang="nl">de grondverhouding</i>), using the plastic number (<a href="/wiki/Dutch_language" title="Dutch language">Dutch</a>: <i lang="nl">het plastische getal</i>) from the 1950s onward.<sup id="cite_ref-FOOTNOTEVoet2016note_12_27-0" class="reference"><a href="#cite_note-FOOTNOTEVoet2016note_12-27"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> In 1944 <a href="/wiki/Carl_Ludwig_Siegel" title="Carl Ludwig Siegel">Carl Siegel</a> showed that <span class="texhtml">ρ</span> is the smallest possible <a href="/wiki/Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot–Vijayaraghavan number</a> and suggested naming it in honour of Thue. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Interieur_bovenkerk,_zicht_op_de_middenbeuk_met_koorbanken_voor_de_monniken_-_Mamelis_-_20536587_-_RCE.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Interieur_bovenkerk%2C_zicht_op_de_middenbeuk_met_koorbanken_voor_de_monniken_-_Mamelis_-_20536587_-_RCE.jpg/290px-Interieur_bovenkerk%2C_zicht_op_de_middenbeuk_met_koorbanken_voor_de_monniken_-_Mamelis_-_20536587_-_RCE.jpg" decoding="async" width="290" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Interieur_bovenkerk%2C_zicht_op_de_middenbeuk_met_koorbanken_voor_de_monniken_-_Mamelis_-_20536587_-_RCE.jpg/435px-Interieur_bovenkerk%2C_zicht_op_de_middenbeuk_met_koorbanken_voor_de_monniken_-_Mamelis_-_20536587_-_RCE.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Interieur_bovenkerk%2C_zicht_op_de_middenbeuk_met_koorbanken_voor_de_monniken_-_Mamelis_-_20536587_-_RCE.jpg/580px-Interieur_bovenkerk%2C_zicht_op_de_middenbeuk_met_koorbanken_voor_de_monniken_-_Mamelis_-_20536587_-_RCE.jpg 2x" data-file-width="4724" data-file-height="3512" /></a><figcaption>The 1967 St. Benedictusberg Abbey church designed by Hans van der Laan.</figcaption></figure> <p>Unlike the names of the <a href="/wiki/Golden_ratio" title="Golden ratio">golden</a> and <a href="/wiki/Silver_ratio" title="Silver ratio">silver ratios</a>, the word plastic was not intended by van der Laan to refer to a specific substance, but rather in its adjectival sense, meaning something that can be given a three-dimensional shape.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> This, according to <a href="/wiki/Richard_Padovan" title="Richard Padovan">Richard Padovan</a>, is because the characteristic ratios of the number, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span>, relate to the limits of human perception in relating one physical size to another. Van der Laan designed the 1967 <a href="/wiki/St._Benedictusberg_Abbey" title="St. Benedictusberg Abbey">St. Benedictusberg Abbey</a> church to these plastic number proportions.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>The plastic number is also sometimes called the silver number, a name given to it by <a href="/wiki/Midhat_J._Gazal%C3%A9" title="Midhat J. Gazalé">Midhat J. Gazalé</a><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> and subsequently used by <a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a>,<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> but that name is more commonly used for the <a href="/wiki/Silver_ratio" title="Silver ratio">silver ratio</a> <span class="texhtml">1 + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></span>, one of the ratios from the family of <a href="/wiki/Metallic_mean" title="Metallic mean">metallic means</a> first described by <a href="/wiki/Vera_W._de_Spinadel" title="Vera W. de Spinadel">Vera W. de Spinadel</a>. Gardner suggested referring to <span class="texhtml">ρ<sup>2</sup></span> as "high phi", and <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a> created a special typographic mark for this name, a variant of the Greek letter phi ("φ") with its central circle raised, resembling the Georgian letter <a href="/wiki/Pari_(letter)" title="Pari (letter)">pari</a> ("Ⴔ"). </p> <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=Plastic_ratio&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Solutions of equations similar to <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 x^{3}=x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}=x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0345ddb11d4101e841917b7a75f926704e633e5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.815ex; height:2.843ex;" alt="{\displaystyle x^{3}=x+1}"></span>: <ul><li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio</a> – the only positive solution of the equation <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 x^{2}=x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f05f1f6f113f4a1ae6c6e425c7cc3113de458980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.815ex; height:2.843ex;" alt="{\displaystyle x^{2}=x+1}"></span></li> <li><a href="/wiki/Supergolden_ratio" title="Supergolden ratio">Supergolden ratio</a> – the only real solution of the equation <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 x^{3}=x^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}=x^{2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cab1579a96db79d879b29b8c893eaea62f2a8e4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.869ex; height:2.843ex;" alt="{\displaystyle x^{3}=x^{2}+1}"></span></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastic_ratio&action=edit&section=9" title="Edit section: Notes"><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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><span class="texhtml">V<sub>n</sub> = Pa<sub> n+3</sub></span> </span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">For a typical 8" nautilus shell the difference in diameter between the apertures of perfect 3<sup>1/4</sup> and <span class="texhtml">ρ</span>−sized specimens is about 1 mm. Allowing for <a href="/wiki/Phenotypic_plasticity" title="Phenotypic plasticity">phenotypic plasticity</a>, they may well be indistinguishable.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">An alternative is the <a href="/wiki/Omega_constant" title="Omega constant">omega constant</a> <span class="texhtml">0.567143...</span> which satisfies <span class="texhtml">Ω⋅exp(Ω) = 1.</span> Resembling <span class="texhtml">φ (φ−1) = 1,</span> <i>Mathworld</i> suggests it is like a "golden ratio for exponentials".<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The interval <span class="texhtml">3<sup>1/4</sup> < ρ < Ω<sup>−1/2</sup> </span> is smaller than <span class="texhtml">0.012.</span></span> </li> </ol></div></div> <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=Plastic_ratio&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><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="CITEREFSloane_"A060006"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A060006">"Sequence A060006"</a>. <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA060006&rft_id=https%3A%2F%2Foeis.org%2FA060006&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" 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="CITEREFSloane_"A072117"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A072117">"Sequence A072117"</a>. <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA072117&rft_id=https%3A%2F%2Foeis.org%2FA072117&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" 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="CITEREFTabrizian2022" class="citation web cs1">Tabrizian, Peyam (2022). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=CpBeRv-P5RM">"What is the plastic ratio?"</a>. <i>YouTube</i><span class="reference-accessdate">. Retrieved <span class="nowrap">26 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=YouTube&rft.atitle=What+is+the+plastic+ratio%3F&rft.date=2022&rft.aulast=Tabrizian&rft.aufirst=Peyam&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DCpBeRv-P5RM&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" 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="CITEREFAartsFokkinkKruijtzer2001" class="citation journal cs1">Aarts, Jan; Fokkink, Robbert; Kruijtzer, Godfried (2001). <a rel="nofollow" class="external text" href="https://www.nieuwarchief.nl/serie5/pdf/naw5-2001-02-1-056.pdf">"Morphic numbers"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Nieuw_Archief_voor_Wiskunde" title="Nieuw Archief voor Wiskunde">Nieuw Archief voor Wiskunde</a></i>. 5. <b>2</b> (1): 56–58<span class="reference-accessdate">. Retrieved <span class="nowrap">26 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nieuw+Archief+voor+Wiskunde&rft.atitle=Morphic+numbers&rft.volume=2&rft.issue=1&rft.pages=56-58&rft.date=2001&rft.aulast=Aarts&rft.aufirst=Jan&rft.au=Fokkink%2C+Robbert&rft.au=Kruijtzer%2C+Godfried&rft_id=https%3A%2F%2Fwww.nieuwarchief.nl%2Fserie5%2Fpdf%2Fnaw5-2001-02-1-056.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-Panju-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Panju_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Panju_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPanju2011" class="citation journal cs1">Panju, Maysum (2011). <a rel="nofollow" class="external text" href="https://mathreview.uwaterloo.ca/archive/voli/2/panju.pdf">"A systematic construction of almost integers"</a> <span class="cs1-format">(PDF)</span>. <i>The Waterloo Mathematics Review</i>. <b>1</b> (2): 35–43<span class="reference-accessdate">. Retrieved <span class="nowrap">29 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Waterloo+Mathematics+Review&rft.atitle=A+systematic+construction+of+almost+integers&rft.volume=1&rft.issue=2&rft.pages=35-43&rft.date=2011&rft.aulast=Panju&rft.aufirst=Maysum&rft_id=https%3A%2F%2Fmathreview.uwaterloo.ca%2Farchive%2Fvoli%2F2%2Fpanju.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></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"><span class="citation mathworld" id="Reference-Mathworld-Plastic_constant"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PlasticConstant.html">"Plastic constant"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Plastic+constant&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPlasticConstant.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="/w/index.php?title=Ramanujan_G-function&action=edit&redlink=1" class="new" title="Ramanujan G-function (page does not exist)">Ramanujan G-function</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://de.wikipedia.org/wiki/Ramanujansche_g-Funktion_und_G-Funktion#Spezielle_Werte" class="extiw" title="de:Ramanujansche g-Funktion und G-Funktion">de</a>]</span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Elliptic_integral_singular_value"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/EllipticIntegralSingularValue.html">"Elliptic integral singular value"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Elliptic+integral+singular+value&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FEllipticIntegralSingularValue.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoet2019" class="citation web cs1 cs1-prop-interwiki-linked-name"><a href="https://nl.wikipedia.org/wiki/Caroline_Voet" class="extiw" title="nl:Caroline Voet">Voet, Caroline</a> <span class="cs1-format">[in Dutch]</span> (2019). <a rel="nofollow" class="external text" href="https://domhansvanderlaan.nl/theory-practice/theory/the-plastic-number-series-8/">"1:7 and a series of 8"</a>. <i>The digital study room of Dom Hans van der Laan</i>. Van der Laan Foundation<span class="reference-accessdate">. Retrieved <span class="nowrap">28 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+digital+study+room+of+Dom+Hans+van+der+Laan&rft.atitle=1%3A7+and+a+series+of+8&rft.date=2019&rft.aulast=Voet&rft.aufirst=Caroline&rft_id=https%3A%2F%2Fdomhansvanderlaan.nl%2Ftheory-practice%2Ftheory%2Fthe-plastic-number-series-8%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-oeisGF-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-oeisGF_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-oeisGF_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">(sequence <span class="nowrap external"><a href="//oeis.org/A182097" class="extiw" title="oeis:A182097">A182097</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</span> </li> <li id="cite_note-oeispado-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-oeispado_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-oeispado_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">(sequence <span class="nowrap external"><a href="//oeis.org/A000931" class="extiw" title="oeis:A000931">A000931</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdamsShanks1982" class="citation journal cs1">Adams, William; <a href="/wiki/Daniel_Shanks" title="Daniel Shanks">Shanks, Daniel</a> (1982). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2007637">"Strong primality tests that are not sufficient"</a>. <i>Math. Comp</i>. <b>39</b> (159). AMS: 255–300. <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/10.2307%2F2007637">10.2307/2007637</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2007637">2007637</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Comp.&rft.atitle=Strong+primality+tests+that+are+not+sufficient&rft.volume=39&rft.issue=159&rft.pages=255-300&rft.date=1982&rft_id=info%3Adoi%2F10.2307%2F2007637&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2007637%23id-name%3DJSTOR&rft.aulast=Adams&rft.aufirst=William&rft.au=Shanks%2C+Daniel&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F2007637&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">(sequence <span class="nowrap external"><a href="//oeis.org/A013998" class="extiw" title="oeis:A013998">A013998</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSiegelThuswaldner2009" class="citation journal cs1">Siegel, Anne; Thuswaldner, Jörg M. (2009). <a rel="nofollow" class="external text" href="http://numdam.org/item/MSMF_2009_2_118__1_0/">"Topological properties of Rauzy fractals"</a>. <i>Mémoires de la Société Mathématique de France</i>. 2. <b>118</b>: 1–140. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fmsmf.430">10.24033/msmf.430</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=M%C3%A9moires+de+la+Soci%C3%A9t%C3%A9+Math%C3%A9matique+de+France&rft.atitle=Topological+properties+of+Rauzy+fractals&rft.volume=118&rft.pages=1-140&rft.date=2009&rft_id=info%3Adoi%2F10.24033%2Fmsmf.430&rft.aulast=Siegel&rft.aufirst=Anne&rft.au=Thuswaldner%2C+J%C3%B6rg+M.&rft_id=http%3A%2F%2Fnumdam.org%2Fitem%2FMSMF_2009_2_118__1_0%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart1996" class="citation journal cs1">Stewart, Ian (1996). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120320051231/http://members.fortunecity.com/templarser/padovan.html">"Tales of a neglected number"</a>. <i>Scientific American</i>. <b>274</b> (6): 102–103. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996SciAm.274f.102S">1996SciAm.274f.102S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0696-102">10.1038/scientificamerican0696-102</a>. Archived from <a rel="nofollow" class="external text" href="http://members.fortunecity.com/templarser/padovan.html">the original</a> on 2012-03-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Tales+of+a+neglected+number&rft.volume=274&rft.issue=6&rft.pages=102-103&rft.date=1996&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0696-102&rft_id=info%3Abibcode%2F1996SciAm.274f.102S&rft.aulast=Stewart&rft.aufirst=Ian&rft_id=http%3A%2F%2Fmembers.fortunecity.com%2Ftemplarser%2Fpadovan.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span> Feedback in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart1996" class="citation journal cs1"><a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a> (1996). "A guide to computer dating". <i>Scientific American</i>. <b>275</b> (5): 118. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996SciAm.275e.116S">1996SciAm.275e.116S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican1196-116">10.1038/scientificamerican1196-116</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=A+guide+to+computer+dating&rft.volume=275&rft.issue=5&rft.pages=118&rft.date=1996&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican1196-116&rft_id=info%3Abibcode%2F1996SciAm.275e.116S&rft.aulast=Stewart&rft.aufirst=Ian&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpinadelRedondo_Buitrago2009" class="citation cs2"><a href="/wiki/Vera_W._de_Spinadel" title="Vera W. de Spinadel">Spinadel, Vera W. de</a>; Redondo Buitrago, Antonia (2009), <a rel="nofollow" class="external text" href="http://www.heldermann-verlag.de/jgg/jgg13/j13h2spin.pdf">"Towards van der Laan's plastic number in the plane"</a> <span class="cs1-format">(PDF)</span>, <i>Journal for Geometry and Graphics</i>, <b>13</b> (2): 163–175</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+for+Geometry+and+Graphics&rft.atitle=Towards+van+der+Laan%27s+plastic+number+in+the+plane&rft.volume=13&rft.issue=2&rft.pages=163-175&rft.date=2009&rft.aulast=Spinadel&rft.aufirst=Vera+W.+de&rft.au=Redondo+Buitrago%2C+Antonia&rft_id=http%3A%2F%2Fwww.heldermann-verlag.de%2Fjgg%2Fjgg13%2Fj13h2spin.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreilingRinne1994" class="citation cs2">Freiling, C.; Rinne, D. (1994), "Tiling a square with similar rectangles", <i>Mathematical Research Letters</i>, <b>1</b> (5): 547–558, <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/10.4310%2FMRL.1994.v1.n5.a3">10.4310/MRL.1994.v1.n5.a3</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1295549">1295549</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Research+Letters&rft.atitle=Tiling+a+square+with+similar+rectangles&rft.volume=1&rft.issue=5&rft.pages=547-558&rft.date=1994&rft_id=info%3Adoi%2F10.4310%2FMRL.1994.v1.n5.a3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1295549%23id-name%3DMR&rft.aulast=Freiling&rft.aufirst=C.&rft.au=Rinne%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaczkovichSzekeres1995" class="citation cs2">Laczkovich, M.; <a href="/wiki/George_Szekeres" title="George Szekeres">Szekeres, G.</a> (1995), "Tilings of the square with similar rectangles", <i><a href="/wiki/Discrete_%26_Computational_Geometry" title="Discrete & Computational Geometry">Discrete & Computational Geometry</a></i>, <b>13</b> (3–4): 569–572, <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/10.1007%2FBF02574063">10.1007/BF02574063</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1318796">1318796</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+%26+Computational+Geometry&rft.atitle=Tilings+of+the+square+with+similar+rectangles&rft.volume=13&rft.issue=3%E2%80%934&rft.pages=569-572&rft.date=1995&rft_id=info%3Adoi%2F10.1007%2FBF02574063&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1318796%23id-name%3DMR&rft.aulast=Laczkovich&rft.aufirst=M.&rft.au=Szekeres%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Snub_icosidodecadodecahedron"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SnubIcosidodecadodecahedron.html">"Snub icosidodecadodecahedron"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Snub+icosidodecadodecahedron&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSnubIcosidodecadodecahedron.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Analogue to the construction in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrilly1994" class="citation journal cs1">Crilly, Tony (1994). "A supergolden rectangle". <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>. <b>78</b> (483): 320–325. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3620208">10.2307/3620208</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3620208">3620208</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=A+supergolden+rectangle&rft.volume=78&rft.issue=483&rft.pages=320-325&rft.date=1994&rft_id=info%3Adoi%2F10.2307%2F3620208&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3620208%23id-name%3DJSTOR&rft.aulast=Crilly&rft.aufirst=Tony&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRack2013" class="citation book cs1">Rack, Heinz-Joachim (2013). "An example of optimal nodes for interpolation revisited". In Anastassiou, George A.; Duman, Oktay (eds.). <i>Advances in applied Mathematics and Approximation Theory 2012</i>. Springer Proceedings in Mathematics and Statistics. Vol. 41. pp. 117–120. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4614-6393-1">10.1007/978-1-4614-6393-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-6393-1" title="Special:BookSources/978-1-4614-6393-1"><bdi>978-1-4614-6393-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An+example+of+optimal+nodes+for+interpolation+revisited&rft.btitle=Advances+in+applied+Mathematics+and+Approximation+Theory+2012&rft.series=Springer+Proceedings+in+Mathematics+and+Statistics&rft.pages=117-120&rft.date=2013&rft_id=info%3Adoi%2F10.1007%2F978-1-4614-6393-1&rft.isbn=978-1-4614-6393-1&rft.aulast=Rack&rft.aufirst=Heinz-Joachim&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoseley1838" class="citation journal cs1">Moseley, Henry (1838). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/108202">"On the Geometrical Forms of Turbinated and Discoid Shells"</a>. <i>Philosophical Transactions of the Royal Society of London</i>. <b>128</b>: 351–370 [355–356]. <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/10.1098%2Frstl.1838.0018">10.1098/rstl.1838.0018</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=On+the+Geometrical+Forms+of+Turbinated+and+Discoid+Shells&rft.volume=128&rft.pages=351-370+355-356&rft.date=1838&rft_id=info%3Adoi%2F10.1098%2Frstl.1838.0018&rft.aulast=Moseley&rft.aufirst=Henry&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F108202&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Omega_constant"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/OmegaConstant.html">"Omega constant"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Omega+constant&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FOmegaConstant.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span></span> </li> <li id="cite_note-FOOTNOTEVoet2016note_12-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVoet2016note_12_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVoet2016">Voet 2016</a>, note 12.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShannonAndersonHoradam2006" class="citation journal cs1">Shannon, A. G.; Anderson, P. G.; Horadam, A. F. (2006). "Properties of Cordonnier, Perrin and Van der Laan numbers". <i>International Journal of Mathematical Education in Science and Technology</i>. <b>37</b> (7): 825–831. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00207390600712554">10.1080/00207390600712554</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119808971">119808971</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Mathematical+Education+in+Science+and+Technology&rft.atitle=Properties+of+Cordonnier%2C+Perrin+and+Van+der+Laan+numbers&rft.volume=37&rft.issue=7&rft.pages=825-831&rft.date=2006&rft_id=info%3Adoi%2F10.1080%2F00207390600712554&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119808971%23id-name%3DS2CID&rft.aulast=Shannon&rft.aufirst=A.+G.&rft.au=Anderson%2C+P.+G.&rft.au=Horadam%2C+A.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPadovan2002" class="citation cs2"><a href="/wiki/Richard_Padovan" title="Richard Padovan">Padovan, Richard</a> (2002), <a rel="nofollow" class="external text" href="https://www.nexusjournal.com/the-nexus-conferences/nexus-2002/148-n2002-padovan.html">"Dom Hans van der Laan and The plastic number"</a>, <i>Nexus IV: Architecture and Mathematics</i>, Fucecchio (Florence): Kim Williams Books: 181–193</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nexus+IV%3A+Architecture+and+Mathematics&rft.atitle=Dom+Hans+van+der+Laan+and+The+plastic+number&rft.pages=181-193&rft.date=2002&rft.aulast=Padovan&rft.aufirst=Richard&rft_id=https%3A%2F%2Fwww.nexusjournal.com%2Fthe-nexus-conferences%2Fnexus-2002%2F148-n2002-padovan.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span>.</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGazalé1999" class="citation book cs1"><a href="/wiki/Midhat_J._Gazal%C3%A9" title="Midhat J. Gazalé">Gazalé, Midhat J.</a> (1999). "Chapter VII: The silver number". <i>Gnomon: From Pharaohs to Fractals</i>. Princeton, NJ: Princeton University Press. pp. 135–150.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+VII%3A+The+silver+number&rft.btitle=Gnomon%3A+From+Pharaohs+to+Fractals&rft.place=Princeton%2C+NJ&rft.pages=135-150&rft.pub=Princeton+University+Press&rft.date=1999&rft.aulast=Gazal%C3%A9&rft.aufirst=Midhat+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner2001" class="citation book cs1"><a href="/wiki/Martin_Gardner" title="Martin Gardner">Gardner, Martin</a> (2001). <a rel="nofollow" class="external text" href="https://static.nsta.org/pdfs/QuantumV4N5.pdf">"Six challenging dissection tasks"</a> <span class="cs1-format">(PDF)</span>. <i>A Gardner's Workout</i>. Natick, MA: A K Peters. pp. 121–128.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Six+challenging+dissection+tasks&rft.btitle=A+Gardner%27s+Workout&rft.place=Natick%2C+MA&rft.pages=121-128&rft.pub=A+K+Peters&rft.date=2001&rft.aulast=Gardner&rft.aufirst=Martin&rft_id=https%3A%2F%2Fstatic.nsta.org%2Fpdfs%2FQuantumV4N5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span> (Link to the 1994 Quantum article without Gardner's Postscript.)</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastic_ratio&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaan,_van_der1960" class="citation cs2"><a href="/wiki/Hans_van_der_Laan" title="Hans van der Laan">Laan, van der, Hans</a> (1960), <i>Le nombre plastique, Quinze leçons sur l'ordonnance architectonique</i>, Leiden: Brill</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Le+nombre+plastique%2C+Quinze+le%C3%A7ons+sur+l%27ordonnance+architectonique&rft.place=Leiden&rft.pub=Brill&rft.date=1960&rft.aulast=Laan%2C+van+der&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPadovanEckScheepmaker1994" class="citation cs2"><a href="/wiki/Richard_Padovan" title="Richard Padovan">Padovan, Richard</a>; <a href="/wiki/Caroline_van_Eck" title="Caroline van Eck">Eck, Caroline van</a>; Scheepmaker, H.J. (1994), <i>Dom Hans van der Laan: Modern Primitive</i>, Amsterdam: Architectura & Natura</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dom+Hans+van+der+Laan%3A+Modern+Primitive&rft.place=Amsterdam&rft.pub=Architectura+%26+Natura&rft.date=1994&rft.aulast=Padovan&rft.aufirst=Richard&rft.au=Eck%2C+Caroline+van&rft.au=Scheepmaker%2C+H.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoet2016" class="citation cs2 cs1-prop-interwiki-linked-name"><a href="https://nl.wikipedia.org/wiki/Caroline_Voet" class="extiw" title="nl:Caroline Voet">Voet, Caroline</a> <span class="cs1-format">[in Dutch]</span> (2016), <a rel="nofollow" class="external text" href="https://journal.eahn.org/article/id/7510/">"Between Looking and Making: Unravelling Dom Hans van der Laan's Plastic Number"</a>, <i><a href="/wiki/Architectural_Histories" title="Architectural Histories">Architectural Histories</a></i>, <b>4</b> (1), London: European Architectural History Network</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Architectural+Histories&rft.atitle=Between+Looking+and+Making%3A+Unravelling+Dom+Hans+van+der+Laan%27s+Plastic+Number&rft.volume=4&rft.issue=1&rft.date=2016&rft.aulast=Voet&rft.aufirst=Caroline&rft_id=https%3A%2F%2Fjournal.eahn.org%2Farticle%2Fid%2F7510%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastic_ratio&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.maecla.it/tartapelago/museo/oro/rettangoli/en%20plasticrectangle.htm">Plastic rectangle and Padovan sequence</a> at Tartapelago by Giorgio Pietrocola.</li> <li><a rel="nofollow" class="external text" href="https://domhansvanderlaan.nl/theory-practice/">The digital study room of Dom Hans van der Laan</a> at The Van der Laan Archives.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarriss2019" class="citation cs2"><a href="/wiki/Edmund_Harriss" title="Edmund Harriss">Harriss, Edmund</a> (15 March 2019), <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=PsGUEj4w9Cc&t=488s">"The Plastic Ratio"</a> <span class="cs1-format">(video)</span>, <i>youtube</i>, <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>, <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211221/PsGUEj4w9Cc">archived</a> from the original on 2021-12-21<span class="reference-accessdate">, retrieved <span class="nowrap">15 March</span> 2019</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=youtube&rft.atitle=The+Plastic+Ratio&rft.date=2019-03-15&rft.aulast=Harriss&rft.aufirst=Edmund&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DPsGUEj4w9Cc%26t%3D488s&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlastic+ratio" class="Z3988"></span>.</li></ul> <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 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cube</a></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integer</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integer</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio (<span class="texhtml mvar" style="font-style:italic;">φ</span>)</a></li> <li><a href="/wiki/Perron_number" title="Perron number">Perron number</a></li> <li><a href="/wiki/Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot–Vijayaraghavan number</a></li> <li><a class="mw-selflink selflink">Plastic ratio (<span class="texhtml mvar" style="font-style:italic;">ρ</span>)</a></li> <li><a href="/wiki/Quadratic_irrational_number" title="Quadratic irrational number">Quadratic irrational number</a></li> <li><a href="/wiki/Rational_number" title="Rational number">Rational number</a></li> <li><a href="/wiki/Root_of_unity" title="Root of unity">Root of unity</a></li> <li><a href="/wiki/Salem_number" title="Salem number">Salem number</a></li> <li><a href="/wiki/Silver_ratio" title="Silver ratio">Silver ratio (<span class="texhtml mvar" style="font-style:italic;">δ</span><sub><span class="texhtml mvar" style="font-style:italic;">S</span></sub>)</a></li> <li><a href="/wiki/Square_root_of_2" title="Square root of 2">Square root of 2</a></li> <li><a href="/wiki/Square_root_of_3" title="Square root of 3">Square root of 3</a></li> <li><a href="/wiki/Square_root_of_5" title="Square root of 5">Square root of 5</a></li> <li><a href="/wiki/Square_root_of_6" title="Square root of 6">Square root of 6</a></li> <li><a href="/wiki/Square_root_of_7" title="Square root of 7">Square root of 7</a></li> <li><a href="/wiki/Supergolden_ratio" title="Supergolden ratio">Supergolden ratio (<span class="texhtml mvar" style="font-style:italic;">ψ</span>)</a></li> <li><a href="/wiki/Supersilver_ratio" title="Supersilver ratio">Supersilver ratio (<span class="texhtml mvar" style="font-style:italic;">ς</span>)</a></li> <li><a href="/wiki/Twelfth_root_of_2" class="mw-redirect" title="Twelfth root of 2">Twelfth root of 2</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics 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