Supersilver ratio - Wikipedia

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Supersilver rectangle subsection</span> </button> <ul id="toc-Supersilver_rectangle-sublist" class="vector-toc-list"> <li id="toc-Supersilver_spiral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Supersilver_spiral"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Supersilver spiral</span> </div> </a> <ul id="toc-Supersilver_spiral-sublist" class="vector-toc-list"> </ul> </li> </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">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">6</span> 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searchaux" style="display:none">Number, approximately 2.20557</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">Supersilver ratio</caption><tbody><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Supersilver_rectangle_RBGY.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Supersilver_rectangle_RBGY.png/220px-Supersilver_rectangle_RBGY.png" decoding="async" width="220" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Supersilver_rectangle_RBGY.png/330px-Supersilver_rectangle_RBGY.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Supersilver_rectangle_RBGY.png/440px-Supersilver_rectangle_RBGY.png 2x" data-file-width="569" data-file-height="258" /></a></span><div class="infobox-caption">A supersilver rectangle contains two scaled copies of itself, <span class="texhtml">ς = ((ς − 1)<sup>2</sup> + 2(ς − 1) + 1) / ς</span></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">ς</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">2.20556<span style="margin-left:0.25em">94304</span><span style="margin-left:0.25em">00590</span><span style="margin-left:0.25em">31170</span><span style="margin-left:0.25em">20286</span><span style="margin-left:0.25em">...</span></span></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> = 2<i>x</i><sup>2</sup> + 1</span></td></tr><tr><th scope="row" class="infobox-label">Continued fraction (linear)</th><td class="infobox-data"><span class="nowrap">[2;4,1,6,2,1,1,1,1,1,1,2,2,1,2,1,...]</span><br />not periodic<br />infinite</td></tr></tbody></table> <p>In mathematics, the <b>supersilver ratio</b> is a geometrical <a href="/wiki/Aspect_ratio" title="Aspect ratio">proportion</a> close to <span class="texhtml">75/34</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> = 2<i>x</i><sup>2</sup> + 1.</span> </p><p>The name <i>supersilver ratio</i> results from analogy with the <a href="/wiki/Silver_ratio" title="Silver ratio">silver ratio</a>, the positive solution of the equation <span class="texhtml"><i>x</i><sup>2</sup> = 2<i>x</i> + 1</span>, and the <a href="/wiki/Supergolden_ratio" title="Supergolden ratio">supergolden ratio</a>. </p> <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=Supersilver_ratio&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two quantities <span class="texhtml">a > b > 0</span> are in the supersilver ratio-squared 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 \left({\frac {2a+b}{a}}\right)^{2}={\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {2a+b}{a}}\right)^{2}={\frac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4230f77a3c840982be92e8567bb9224a5e15a5d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.706ex; height:6.509ex;" alt="{\displaystyle \left({\frac {2a+b}{a}}\right)^{2}={\frac {a}{b}}}"></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 {2a+b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2a+b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb57e8f63b3d31f463e1be020a0e52603f29de4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.066ex; height:5.343ex;" alt="{\displaystyle {\frac {2a+b}{a}}}"></span> is here 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 \varsigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/293f7951ef0a9e5377281830c2fe29d31891e857" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.589ex; height:1.843ex;" alt="{\displaystyle \varsigma .}"></span>⁠</span> </p><p>Based on this definition, 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}1&=\left({\frac {2a+b}{a}}\right)^{2}{\frac {b}{a}}\\&=\left({\frac {2a+b}{a}}\right)^{2}\left({\frac {2a+b}{a}}-2\right)\\&\implies \varsigma ^{2}\left(\varsigma -2\right)=1\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> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>ς</mi> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}1&=\left({\frac {2a+b}{a}}\right)^{2}{\frac {b}{a}}\\&=\left({\frac {2a+b}{a}}\right)^{2}\left({\frac {2a+b}{a}}-2\right)\\&\implies \varsigma ^{2}\left(\varsigma -2\right)=1\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d235d210dea583b8b2db4434e5525aa87974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; margin-top: -0.204ex; width:31.432ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}1&=\left({\frac {2a+b}{a}}\right)^{2}{\frac {b}{a}}\\&=\left({\frac {2a+b}{a}}\right)^{2}\left({\frac {2a+b}{a}}-2\right)\\&\implies \varsigma ^{2}\left(\varsigma -2\right)=1\end{aligned}}}"></span></dd></dl> <p>It follows that the supersilver 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 \varsigma ^{3}-2\varsigma ^{2}-1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma ^{3}-2\varsigma ^{2}-1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cacf573147caf448509aba2e2715018da5fdd00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.964ex; height:2.843ex;" alt="{\displaystyle \varsigma ^{3}-2\varsigma ^{2}-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 2.205\,569\,430\,400\,590...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.205</mn> <mspace width="thinmathspace" /> <mn>569</mn> <mspace width="thinmathspace" /> <mn>430</mn> <mspace width="thinmathspace" /> <mn>400</mn> <mspace width="thinmathspace" /> <mn>590...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.205\,569\,430\,400\,590...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a332864dee5911e14608c3bcd7d40f56cec361d3" 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 2.205\,569\,430\,400\,590...}"></span> (sequence <span class="nowrap external"><a href="//oeis.org/A356035" class="extiw" title="oeis:A356035">A356035</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p><p>The <a href="/wiki/Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> for the reciprocal root is the depressed cubic <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}+2x-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> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}+2x-1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf62badc3a7f2577c14016f61f9fad0d8f223c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.366ex; height:3.009ex;" alt="{\displaystyle x^{3}+2x-1,}"></span> thus the simplest solution 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}=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {59}{3}}}\right)/2}"> <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> <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>59</mn> <mn>3</mn> </mfrac> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1,2}=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {59}{3}}}\right)/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c63a25699e2002d8c4045e0bcab3ea5ba92020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.716ex; height:6.343ex;" alt="{\displaystyle w_{1,2}=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {59}{3}}}\right)/2}"></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 1/\varsigma ={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ς</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 1/\varsigma ={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa28739bc37d917473d0a0b1e36de751b58b6b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.514ex; height:3.176ex;" alt="{\displaystyle 1/\varsigma ={\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 sine</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 1/\varsigma =-2{\sqrt {\frac {2}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left(-{\frac {3}{4}}{\sqrt {\frac {3}{2}}}\right)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ς</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </msqrt> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>arsinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\varsigma =-2{\sqrt {\frac {2}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left(-{\frac {3}{4}}{\sqrt {\frac {3}{2}}}\right)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0bbb33bed44352eeabe9c2cadcd65ad431c01e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.42ex; height:6.343ex;" alt="{\displaystyle 1/\varsigma =-2{\sqrt {\frac {2}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left(-{\frac {3}{4}}{\sqrt {\frac {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 1/\varsigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ς</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\varsigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22c8b4e513e1d6082b3de08da3285598934dcbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.267ex; height:2.843ex;" alt="{\displaystyle 1/\varsigma }"></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}+2).}"> <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>2</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\gets (2x^{3}+1)/(3x^{2}+2).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd8853ffd506102e16f145a24d2573bffc30595" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.47ex; height:3.176ex;" alt="{\displaystyle x\gets (2x^{3}+1)/(3x^{2}+2).}"></span> </p><p>Rewrite the minimal polynomial 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 (x^{2}+1)^{2}=1+x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{2}+1)^{2}=1+x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63504bb9b8bbdb8e10d32e6601e42c658cb0200c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.681ex; height:3.176ex;" alt="{\displaystyle (x^{2}+1)^{2}=1+x}"></span>, then 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+{\sqrt {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> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mi>x</mi> </msqrt> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\gets {\sqrt {-1+{\sqrt {1+x}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86a64edb1e2622b520a20052dc1af68d0b8c55f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:20.347ex; height:4.843ex;" alt="{\displaystyle x\gets {\sqrt {-1+{\sqrt {1+x}}}}}"></span> results in 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 1/\varsigma ={\sqrt {-1+{\sqrt {1+{\sqrt {-1+{\sqrt {1+\cdots }}}}}}}}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ς</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mo>⋯</mo> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> </msqrt> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\varsigma ={\sqrt {-1+{\sqrt {1+{\sqrt {-1+{\sqrt {1+\cdots }}}}}}}}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d02d36d412ccb1fedc02e280f22e861481d7c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.268ex; height:7.509ex;" alt="{\displaystyle 1/\varsigma ={\sqrt {-1+{\sqrt {1+{\sqrt {-1+{\sqrt {1+\cdots }}}}}}}}\;}"></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></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}-2x^{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> <mn>2</mn> <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}-2x^{2}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b1a9becff1a2759947eaa92ee045e214bb403d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.774ex; height:2.843ex;" alt="{\displaystyle x^{3}-2x^{2}-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-\varsigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mi>ς</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-\varsigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dfad0988d7c7dd768e3faead32cc8fbe38d95ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.112ex; height:2.176ex;" alt="{\displaystyle x-\varsigma }"></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}+x/\varsigma ^{2}+1/\varsigma }"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ς</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+x/\varsigma ^{2}+1/\varsigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20c80021c66afd0c4e0e2e69aa46ff12e4f20924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.849ex; height:3.176ex;" alt="{\displaystyle x^{2}+x/\varsigma ^{2}+1/\varsigma }"></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 \varsigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1fab64d12c6ee8fbf378b73072d0cf13c8c4f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:0.942ex; height:1.843ex;" alt="{\displaystyle \varsigma }"></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}=\left(-1\pm i{\sqrt {8\varsigma ^{2}+3}}\right)/2\varsigma ^{2},}"> <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> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>±</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>8</mn> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1,2}=\left(-1\pm i{\sqrt {8\varsigma ^{2}+3}}\right)/2\varsigma ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c93ff637de127f7a1eb04d9cac6a9182261dcbf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.695ex; height:6.176ex;" alt="{\displaystyle x_{1,2}=\left(-1\pm i{\sqrt {8\varsigma ^{2}+3}}\right)/2\varsigma ^{2},}"></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}=2-\varsigma \;}"> <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> <mn>2</mn> <mo>−</mo> <mi>ς</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}+x_{2}=2-\varsigma \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/622d579c41a6486e30a60374f5f33e3bd6c266ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.297ex; height:2.509ex;" alt="{\displaystyle x_{1}+x_{2}=2-\varsigma \;}"></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/\varsigma .}"> <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>ς</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;x_{1}x_{2}=1/\varsigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08288fbeb671269b2ce5215a4d7fa421c0712f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.425ex; height:2.843ex;" alt="{\displaystyle \;x_{1}x_{2}=1/\varsigma .}"></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=Supersilver_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:SuperSilverSquare_6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/SuperSilverSquare_6.png/280px-SuperSilverSquare_6.png" decoding="async" width="280" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/SuperSilverSquare_6.png/420px-SuperSilverSquare_6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/SuperSilverSquare_6.png/560px-SuperSilverSquare_6.png 2x" data-file-width="569" data-file-height="569" /></a><figcaption>Rectangles with aspect ratios related to powers of <span class="texhtml">ς</span> tile the square.</figcaption></figure> <p>The growth rate of the average value of the n-th term of a random Fibonacci sequence is <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 \varsigma -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90a86ed8f27fba2ffd8755da6def6ec5ad66b762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.945ex; height:2.343ex;" alt="{\displaystyle \varsigma -1}"></span>⁠</span>.<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> </p><p>The supersilver 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 \varsigma =2\sum _{k=0}^{\infty }\varsigma ^{-3k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma =2\sum _{k=0}^{\infty }\varsigma ^{-3k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16799b536dcb5ee894d523e7c9f7d528c1ba24c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.492ex; height:7.009ex;" alt="{\displaystyle \varsigma =2\sum _{k=0}^{\infty }\varsigma ^{-3k}}"></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 \,\varsigma ^{2}=-1+\sum _{k=0}^{\infty }(\varsigma -1)^{-k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>ς</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\varsigma ^{2}=-1+\sum _{k=0}^{\infty }(\varsigma -1)^{-k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de274ef1bfce61e3ade6b03e1669cd44f405794b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.445ex; height:7.009ex;" alt="{\displaystyle \,\varsigma ^{2}=-1+\sum _{k=0}^{\infty }(\varsigma -1)^{-k},}"></span></dd></dl> <p>in comparison to the <a href="/wiki/Silver_ratio" title="Silver ratio">silver ratio</a> identities </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 \sigma =2\sum _{k=0}^{\infty }\sigma ^{-2k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =2\sum _{k=0}^{\infty }\sigma ^{-2k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2788ee31782febef0cc30ba82814132c8bf2dda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.24ex; height:7.009ex;" alt="{\displaystyle \sigma =2\sum _{k=0}^{\infty }\sigma ^{-2k}}"></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 \,\sigma ^{2}=-1+2\sum _{k=0}^{\infty }(\sigma -1)^{-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\sigma ^{2}=-1+2\sum _{k=0}^{\infty }(\sigma -1)^{-k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9617b2d92afa980e190d5175435733e153a1553b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.742ex; height:7.009ex;" alt="{\displaystyle \,\sigma ^{2}=-1+2\sum _{k=0}^{\infty }(\sigma -1)^{-k}.}"></span></dd></dl> <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}\varsigma ^{n}&=2\varsigma ^{n-1}+\varsigma ^{n-3}\\&=4\varsigma ^{n-2}+\varsigma ^{n-3}+2\varsigma ^{n-4}\\&=\varsigma ^{n-1}+2\varsigma ^{n-2}+\varsigma ^{n-3}+\varsigma ^{n-4}.\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>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ς</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> <mn>4</mn> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varsigma ^{n}&=2\varsigma ^{n-1}+\varsigma ^{n-3}\\&=4\varsigma ^{n-2}+\varsigma ^{n-3}+2\varsigma ^{n-4}\\&=\varsigma ^{n-1}+2\varsigma ^{n-2}+\varsigma ^{n-3}+\varsigma ^{n-4}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0d4a7e5dc20972c457b740bae49c4a7279332e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:33.531ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}\varsigma ^{n}&=2\varsigma ^{n-1}+\varsigma ^{n-3}\\&=4\varsigma ^{n-2}+\varsigma ^{n-3}+2\varsigma ^{n-4}\\&=\varsigma ^{n-1}+2\varsigma ^{n-2}+\varsigma ^{n-3}+\varsigma ^{n-4}.\end{aligned}}}"></span></dd></dl> <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 \varsigma ^{-2}=[0;4,1,6,2,1,1,1,1,1,1,...]\approx 0.2056}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>;</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</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>0.2056</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma ^{-2}=[0;4,1,6,2,1,1,1,1,1,1,...]\approx 0.2056}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea6063fd9c917e986eff40e95be1f667e087e58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.516ex; height:3.176ex;" alt="{\displaystyle \varsigma ^{-2}=[0;4,1,6,2,1,1,1,1,1,1,...]\approx 0.2056}"></span> (<span class="texhtml">5/24</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 \varsigma ^{-1}=[0;2,4,1,6,2,1,1,1,1,1,...]\approx 0.4534}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ς</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>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</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>0.4534</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma ^{-1}=[0;2,4,1,6,2,1,1,1,1,1,...]\approx 0.4534}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5f44e0e9486d2fabf97ecf739992bfe432316b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.516ex; height:3.176ex;" alt="{\displaystyle \varsigma ^{-1}=[0;2,4,1,6,2,1,1,1,1,1,...]\approx 0.4534}"></span> (<span class="texhtml">5/11</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 \ \varsigma ^{0}=[1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>ς</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 \ \varsigma ^{0}=[1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f53f05487590d1d73db17aae5b6171d9dcbba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.161ex; height:3.176ex;" alt="{\displaystyle \ \varsigma ^{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 \varsigma ^{1}=[2;4,1,6,2,1,1,1,1,1,1,...]\approx 2.2056}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>;</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</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>2.2056</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma ^{1}=[2;4,1,6,2,1,1,1,1,1,1,...]\approx 2.2056}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16fbcc3cc1776b6cb1af32abb7b492a0d3720d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.237ex; height:3.176ex;" alt="{\displaystyle \varsigma ^{1}=[2;4,1,6,2,1,1,1,1,1,1,...]\approx 2.2056}"></span> (<span class="texhtml">53/24</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 \varsigma ^{2}=[4;1,6,2,1,1,1,1,1,1,2,...]\approx 4.8645}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>≈</mo> <mn>4.8645</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma ^{2}=[4;1,6,2,1,1,1,1,1,1,2,...]\approx 4.8645}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f43eeb5bc0b1911a450c2b8b91e056ddc1e8aba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.237ex; height:3.176ex;" alt="{\displaystyle \varsigma ^{2}=[4;1,6,2,1,1,1,1,1,1,2,...]\approx 4.8645}"></span> (<span class="texhtml">73/15</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 \varsigma ^{3}=[10;1,2,1,2,4,4,2,2,6,2,...]\approx 10.729}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <mn>10</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>≈</mo> <mn>10.729</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma ^{3}=[10;1,2,1,2,4,4,2,2,6,2,...]\approx 10.729}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9456843303f4a5d95793ec238dccc3c05131386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.4ex; height:3.176ex;" alt="{\displaystyle \varsigma ^{3}=[10;1,2,1,2,4,4,2,2,6,2,...]\approx 10.729}"></span> (<span class="texhtml">118/11</span>)</dd></dl> <p>The supersilver ratio is a <a href="/wiki/Pisot_number" class="mw-redirect" title="Pisot number">Pisot number</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> 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 {\varsigma }}}"> <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>ς</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {\varsigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84250ab2be3c91d760f9ceb5924a073a46b2c7dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.203ex; height:3.009ex;" alt="{\displaystyle 1/{\sqrt {\varsigma }}}"></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 \varsigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1fab64d12c6ee8fbf378b73072d0cf13c8c4f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:0.942ex; height:1.843ex;" alt="{\displaystyle \varsigma }"></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 \varsigma ^{10}=2724.00146856...\approx 2724+1/681.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>=</mo> <mn>2724.00146856...</mn> <mo>≈</mo> <mn>2724</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>681.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma ^{10}=2724.00146856...\approx 2724+1/681.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55ff1f48e36d07161db0d7e994b5e8125e9bf5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.531ex; height:3.176ex;" alt="{\displaystyle \varsigma ^{10}=2724.00146856...\approx 2724+1/681.}"></span> After ten 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 /82}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>45</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>82</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 45\pi /82}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fdbac2e1df0fcdb631d7b741ec2daad822fdf17" 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 /82}"></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 supersilver 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}-2x^{2}-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> <mn>2</mn> <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 m(x)=x^{3}-2x^{2}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/364e2cf39410ac0f7443555aae61f07cbf13d90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.051ex; height:3.176ex;" alt="{\displaystyle m(x)=x^{3}-2x^{2}-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 =-59}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mo>−</mo> <mn>59</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =-59}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0168c373c145dedf3692b346e4a73aecdebd0c73" 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 =-59}"></span> and factors into <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-21)^{2}(x-19){\pmod {59}};\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>21</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>19</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>59</mn> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-21)^{2}(x-19){\pmod {59}};\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ce97c19faa38d9a4b8152702307c9fe9849440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.964ex; height:3.176ex;" alt="{\displaystyle (x-21)^{2}(x-19){\pmod {59}};\;}"></span> the 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</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> has <a href="/wiki/Binary_quadratic_form#Reduction_and_class_numbers" title="Binary quadratic form">class number</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 h=3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feadb4b3f59b8647d840328ac06ab358d187abf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.247ex; height:2.176ex;" alt="{\displaystyle h=3.}"></span>⁠</span> Thus, the <a href="/wiki/Hilbert_class_field" title="Hilbert class field">Hilbert class field</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> 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 \varsigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/293f7951ef0a9e5377281830c2fe29d31891e857" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.589ex; height:1.843ex;" alt="{\displaystyle \varsigma .}"></span>⁠</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> 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</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">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>, the real root <a href="/wiki/J-invariant" title="J-invariant"><span class="texhtml"> <big><i>j</i>(<i>τ</i>)</big></span></a> of the Hilbert class polynomial is given by <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 (\varsigma ^{-6}-27\varsigma ^{6}-6)^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mn>27</mn> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varsigma ^{-6}-27\varsigma ^{6}-6)^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ab8c035bbe634ed844cfe447d4a335ef0036c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.008ex; height:3.176ex;" alt="{\displaystyle (\varsigma ^{-6}-27\varsigma ^{6}-6)^{3}.}"></span><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><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> </p><p>The <a href="/wiki/Weber_modular_function" title="Weber modular function">Weber-Ramanujan class invariant</a> is approximated with error <span class="texhtml">< 3.5 ∙ 10<sup>−20</sup></span> 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 {\sqrt {2}}\,{\mathfrak {f}}({\sqrt {\Delta }})={\sqrt[{4}]{2}}\,G_{59}\approx (e^{\pi {\sqrt {-\Delta }}}+24)^{1/24},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <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</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>59</mn> </mrow> </msub> <mo>≈</mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi mathvariant="normal">Δ</mi> </msqrt> </mrow> </mrow> </msup> <mo>+</mo> <mn>24</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>24</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\,{\mathfrak {f}}({\sqrt {\Delta }})={\sqrt[{4}]{2}}\,G_{59}\approx (e^{\pi {\sqrt {-\Delta }}}+24)^{1/24},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd8132fde5fa9086fa21ae8705c991382212ace3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.726ex; height:3.676ex;" alt="{\displaystyle {\sqrt {2}}\,{\mathfrak {f}}({\sqrt {\Delta }})={\sqrt[{4}]{2}}\,G_{59}\approx (e^{\pi {\sqrt {-\Delta }}}+24)^{1/24},}"></span></dd></dl> <p>while its true value is the single real root of the polynomial </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_{59}(x)=x^{9}-4x^{8}+4x^{7}-2x^{6}+4x^{5}-8x^{4}+4x^{3}-8x^{2}+16x-8.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>59</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mn>8</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>8</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>16</mn> <mi>x</mi> <mo>−</mo> <mn>8.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{59}(x)=x^{9}-4x^{8}+4x^{7}-2x^{6}+4x^{5}-8x^{4}+4x^{3}-8x^{2}+16x-8.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb89241c536d9b1dbe176f174d0e9a3b640e8269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.543ex; height:3.176ex;" alt="{\displaystyle W_{59}(x)=x^{9}-4x^{8}+4x^{7}-2x^{6}+4x^{5}-8x^{4}+4x^{3}-8x^{2}+16x-8.}"></span></dd></dl> <p>The <a href="/wiki/Modular_lambda_function#Lambda-star" title="Modular lambda function">elliptic integral singular value</a><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> <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</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=59}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>59</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=59}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e36433d6d68a06dd5389d5fd3dc375f8581d7ea7" 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=59}"></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 ^{*}(59)=\sin(\arcsin \left(G_{59}^{-12}\right)/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>59</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>59</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>12</mn> </mrow> </msubsup> <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 ^{*}(59)=\sin(\arcsin \left(G_{59}^{-12}\right)/2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63113288362c842e592bb71ba17560d16975d93c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.705ex; height:3.343ex;" alt="{\displaystyle \lambda ^{*}(59)=\sin(\arcsin \left(G_{59}^{-12}\right)/2)}"></span></dd></dl> <p>(which is less than 1/294 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="Third-order_Pell_sequences">Third-order Pell sequences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supersilver_ratio&action=edit&section=3" title="Edit section: Third-order Pell sequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="floatright"> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Supersilver_Rauzy_baa.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Supersilver_Rauzy_baa.png/260px-Supersilver_Rauzy_baa.png" decoding="async" width="260" height="306" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Supersilver_Rauzy_baa.png/390px-Supersilver_Rauzy_baa.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Supersilver_Rauzy_baa.png/520px-Supersilver_Rauzy_baa.png 2x" data-file-width="1700" data-file-height="2000" /></a><figcaption><i>Hop o' my Thumb</i>: a supersilver <a href="#matrix">Rauzy fractal</a> of type <span class="nowrap">a ↦ baa.</span> 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> 1.22</figcaption></figure> </td></tr> <tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Supersilver_Rauzy_bca.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Supersilver_Rauzy_bca.png/260px-Supersilver_Rauzy_bca.png" decoding="async" width="260" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Supersilver_Rauzy_bca.png/390px-Supersilver_Rauzy_bca.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Supersilver_Rauzy_bca.png/520px-Supersilver_Rauzy_bca.png 2x" data-file-width="2000" data-file-height="1750" /></a><figcaption>A supersilver Rauzy fractal of type <span class="nowrap">c ↦ bca,</span> with areas in the ratios <span class="texhtml">ς<sup>2</sup> + 1 : ς (ς − 1) : ς : 1.</span></figcaption></figure> </td></tr></tbody></table> <p>These numbers are related to the supersilver ratio as the <a href="/wiki/Pell_number" title="Pell number">Pell numbers</a> and <a href="/wiki/Pell_number#Pell–Lucas_numbers" title="Pell number">Pell-Lucas numbers</a> are to the <a href="/wiki/Silver_ratio" title="Silver ratio">silver ratio</a>. </p><p>The fundamental 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 S_{n}=2S_{n-1}+S_{n-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=2S_{n-1}+S_{n-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/507a1ee314c0dd1de158e61b5eaae0551578e029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.233ex; height:2.509ex;" alt="{\displaystyle S_{n}=2S_{n-1}+S_{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 S_{0}=1,S_{1}=2,S_{2}=4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{0}=1,S_{1}=2,S_{2}=4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cae0b2de17cb89c6928ee71676d99baced52ca05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.935ex; height:2.509ex;" alt="{\displaystyle S_{0}=1,S_{1}=2,S_{2}=4.}"></span></dd></dl> <p>The first few terms are 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064,... (sequence <span class="nowrap external"><a href="//oeis.org/A008998" class="extiw" title="oeis:A008998">A008998</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 supersilver ratio. </p><p>The first 8 indices n for which <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 S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}"></span> is prime are n = 1, 6, 21, 114, 117, 849, 2418, 6144. The last number has 2111 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 S_{n}=S_{n+3}-2S_{n+2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=S_{n+3}-2S_{n+2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d6e0c6ab8ec62affbb2fba7cea5ac2ee4d1537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.233ex; height:2.509ex;" alt="{\displaystyle S_{n}=S_{n+3}-2S_{n+2}}"></span>.</dd></dl> <p>The <a href="/wiki/Generating_function" title="Generating function">generating function</a> of the 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}{1-2x-x^{3}}}=\sum _{n=0}^{\infty }S_{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> <mn>2</mn> <mi>x</mi> <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>S</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}{1-2x-x^{3}}}=\sum _{n=0}^{\infty }S_{n}x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6659dd44397468b7ddad2b8c567fbdda91c238" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.588ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{1-2x-x^{3}}}=\sum _{n=0}^{\infty }S_{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/\varsigma \;.}"> <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>ς</mi> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<1/\varsigma \;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671df3422da7d55c9d5e0a69141790111c9ee725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.987ex; height:2.843ex;" alt="{\displaystyle x<1/\varsigma \;.}"></span><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></dd></dl> <p>The third-order Pell numbers are 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 S_{n}=\sum _{k=0}^{\lfloor n/3\rfloor }{n-2k \choose k}\cdot 2^{n-3k}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</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> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</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"> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mi>k</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=\sum _{k=0}^{\lfloor n/3\rfloor }{n-2k \choose k}\cdot 2^{n-3k}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6440e4d70da473c62ffcd1bfe25eb36e6ccb9820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.911ex; height:7.843ex;" alt="{\displaystyle S_{n}=\sum _{k=0}^{\lfloor n/3\rfloor }{n-2k \choose k}\cdot 2^{n-3k}\;}"></span>.<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></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}-2x^{2}-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> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-2x^{2}-1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656c97eb6fe1b5ba35d57c85ede234c0ade9e63b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.681ex; height:2.843ex;" alt="{\displaystyle x^{3}-2x^{2}-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</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</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</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 supersilver numbers can be computed with the <a href="/wiki/Fibonacci_sequence#Binet's_formula" title="Fibonacci sequence">Binet 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 S_{n-2}=a\alpha ^{n}+b\beta ^{n}+c\gamma ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n-2}=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/e810d2a1d5ee45e3ad7264d8c87e5266d2cd019c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.164ex; height:2.843ex;" alt="{\displaystyle S_{n-2}=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 59x^{3}+4x-1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>59</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 59x^{3}+4x-1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c834d93957b04ec5fc3a61ff0f21fe32cd82372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.952ex; height:2.843ex;" alt="{\displaystyle 59x^{3}+4x-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</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>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</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 =\varsigma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>ς</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\varsigma ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec72c0aa85ddf201bb6c1bbfa9f0f6c83effc07f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.175ex; height:2.009ex;" alt="{\displaystyle \alpha =\varsigma ,}"></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 S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{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\,\varsigma ^{n+2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\varsigma ^{n+2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ba78b103a243739e95b4147ffb166fd99827c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.554ex; height:3.009ex;" alt="{\displaystyle a\,\varsigma ^{n+2},}"></span> with <span class="texhtml"><i>n</i> ≥ 0</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=\varsigma /(2\varsigma ^{2}+3)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=\varsigma /(2\varsigma ^{2}+3)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d783fe06289107d8da3fdd0acc89af2246c8dfb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.886ex; height:3.176ex;" alt="{\displaystyle a=\varsigma /(2\varsigma ^{2}+3)=}"></span> <span style="white-space:nowrap">0.17327<span style="margin-left:0.25em">02315</span><span style="margin-left:0.25em">50408</span><span style="margin-left:0.25em">18074</span><span style="margin-left:0.25em">84794...</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 A_{n}=S_{n}+2S_{n-3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}=S_{n}+2S_{n-3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82420e2684b7e0a71cadb2529a6a6c5894694bb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.097ex; height:2.509ex;" alt="{\displaystyle A_{n}=S_{n}+2S_{n-3}.}"></span> </p><p>The first few terms are 3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008,... (sequence <span class="nowrap external"><a href="//oeis.org/A332647" class="extiw" title="oeis:A332647">A332647</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 third-order Pell-Lucas sequence 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 A_{p}\equiv A_{1}{\bmod {p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≡</mo> <msub> <mi>A</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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{p}\equiv A_{1}{\bmod {p}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f21c82063ff4912f1182a7b7ba46eceb018a404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.196ex; height:2.843ex;" alt="{\displaystyle A_{p}\equiv A_{1}{\bmod {p}}.}"></span> The converse does not hold, but the small number of odd <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 (A_{n}-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>n</mi> <mo>∣</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,n\mid (A_{n}-2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18960c4e3167e8d351fb98ebd7859b711fdfaa6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.493ex; height:2.843ex;" alt="{\displaystyle \,n\mid (A_{n}-2)}"></span> makes the sequence special. The 14 odd composite numbers below <span class="texhtml">10<sup>8</sup> </span> to pass the test are n = 3<sup>2</sup>, 5<sup>2</sup>, 5<sup>3</sup>, 315, 99297, 222443, 418625, 9122185, 3257<sup>2</sup>, 11889745, 20909625, 24299681, 64036831, 76917325.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Supersilver_Rauzy_aba.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Supersilver_Rauzy_aba.png/260px-Supersilver_Rauzy_aba.png" decoding="async" width="260" height="306" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Supersilver_Rauzy_aba.png/390px-Supersilver_Rauzy_aba.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Supersilver_Rauzy_aba.png/520px-Supersilver_Rauzy_aba.png 2x" data-file-width="1700" data-file-height="2000" /></a><figcaption><i>The Pilgrim</i>: a supersilver Rauzy fractal of type <span class="nowrap">a ↦ aba.</span> The three subtiles have areas in ratio <span class="texhtml">ς.</span></figcaption></figure> <p>The third-order Pell numbers are obtained as integral powers <span class="texhtml"><i>n</i> > 3</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 \varsigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1fab64d12c6ee8fbf378b73072d0cf13c8c4f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:0.942ex; height:1.843ex;" alt="{\displaystyle \varsigma }"></span>⁠</span> </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}2&0&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>2</mn> </mtd> <mtd> <mn>0</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}2&0&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/48880754e51e3c1308963134fae3742f1aa9981b" 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}2&0&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}S_{n}&S_{n-2}&S_{n-1}\\S_{n-1}&S_{n-3}&S_{n-2}\\S_{n-2}&S_{n-4}&S_{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>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</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}S_{n}&S_{n-2}&S_{n-1}\\S_{n-1}&S_{n-3}&S_{n-2}\\S_{n-2}&S_{n-4}&S_{n-3}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0af21cac58dc62a816e9c983bf1876ea15e0f28c" class="mwe-math-fallback-image-inline mw-invert skin-invert" id="matrix" aria-hidden="true" style="vertical-align: -4.171ex; width:29.85ex; height:9.509ex;" alt="{\displaystyle Q^{n}={\begin{pmatrix}S_{n}&S_{n-2}&S_{n-1}\\S_{n-1}&S_{n-3}&S_{n-2}\\S_{n-2}&S_{n-4}&S_{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 above <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_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e427e4647a786b35696f185ad91b5acecc3261d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.608ex; height:2.509ex;" alt="{\displaystyle A_{n}.}"></span>⁠</span> </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 \;aab\\b\;\mapsto \;c\\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>a</mi> <mi>a</mi> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> <mspace width="thickmathspace" /> <mo stretchy="false">↦</mo> <mspace width="thickmathspace" /> <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 \;aab\\b\;\mapsto \;c\\c\;\mapsto \;a\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec5dbbc96f0fa4ba15633c95bf0478877fb7842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:12.409ex; height:8.509ex;" alt="{\displaystyle {\begin{cases}a\;\mapsto \;aab\\b\;\mapsto \;c\\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}=b}"> <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>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{0}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/744aa310f7ff83fe30b191ff84d0dd31dfe836b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.814ex; height:2.509ex;" alt="{\displaystyle w_{0}=b}"></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 third-order Pell numbers. The lengths of these words are given by <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})=S_{n-2}+S_{n-3}+S_{n-4}.}"> <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>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>4</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l(w_{n})=S_{n-2}+S_{n-3}+S_{n-4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2aa1aade3b039e734d5057ef6b7e95c2d1a411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.043ex; height:2.843ex;" alt="{\displaystyle l(w_{n})=S_{n-2}+S_{n-3}+S_{n-4}.}"></span><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Associated to this string rewriting process is a <a href="/wiki/Compact_space" title="Compact space">compact</a> set composed of <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 three-letter sequence.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Supersilver_rectangle">Supersilver rectangle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supersilver_ratio&action=edit&section=4" title="Edit section: Supersilver 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:Supersilver_ratio.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Supersilver_ratio.svg/330px-Supersilver_ratio.svg.png" decoding="async" width="330" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Supersilver_ratio.svg/495px-Supersilver_ratio.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Supersilver_ratio.svg/660px-Supersilver_ratio.svg.png 2x" data-file-width="740" data-file-height="350" /></a><figcaption>Powers of <span class="texhtml">ς</span> within a supersilver rectangle.</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 \varsigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1fab64d12c6ee8fbf378b73072d0cf13c8c4f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:0.942ex; height:1.843ex;" alt="{\displaystyle \varsigma }"></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 \varsigma {\sqrt {\varsigma -1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ς</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma {\sqrt {\varsigma -1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79ab6f7e457a71c55286286b6ef2cbf0d70390f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.469ex; height:3.009ex;" alt="{\displaystyle \varsigma {\sqrt {\varsigma -1}}.}"></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 {\varsigma -1}}\,;}"> <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>ς</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {\varsigma -1}}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/760aa88565a6332df775791ad09453754fba3f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.239ex; height:3.343ex;" alt="{\displaystyle 1/{\sqrt {\varsigma -1}}\,;}"></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 \varsigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76c2a0a2022c2265c76ad9e4929a629ec705a98d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.025ex; height:2.843ex;" alt="{\displaystyle \varsigma ^{2}}"></span>⁠</span>. </p><p>On the right-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="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+1/\varsigma ^{2}:1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1/\varsigma ^{2}:1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e7628c58522c8f9513143a927d233ac2abaa26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.453ex; height:3.176ex;" alt="{\displaystyle 1+1/\varsigma ^{2}:1}"></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 \varsigma =2+1/\varsigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma =2+1/\varsigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb0e7c62ba9c67e2c9a6ffbeec6079ce2563446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.393ex; height:3.176ex;" alt="{\displaystyle \varsigma =2+1/\varsigma ^{2}}"></span>). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>The parent supersilver 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 \varsigma :\varsigma -1:1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> <mo>:</mo> <mi>ς</mi> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma :\varsigma -1:1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5261a64ed5a85714b83d735fd5bcb8ed526fcad0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.57ex; height:2.343ex;" alt="{\displaystyle \varsigma :\varsigma -1:1.}"></span> The areas of the rectangles opposite the diagonal are both equal 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 (\varsigma -1)/\varsigma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ς</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ς</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varsigma -1)/\varsigma ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b589535fb3dde5105f50c6db786267d0e692679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.505ex; height:2.843ex;" alt="{\displaystyle (\varsigma -1)/\varsigma ,}"></span> with aspect 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 \varsigma (\varsigma -1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> <mo stretchy="false">(</mo> <mi>ς</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma (\varsigma -1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f3e7e91aff8f9139fd3b04425eb3cacff70d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.696ex; height:2.843ex;" alt="{\displaystyle \varsigma (\varsigma -1)}"></span> (below) 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 \varsigma /(\varsigma -1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>ς</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma /(\varsigma -1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54eeac33bc9c9b5d96465c4e9d65a6f2d9512e54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.858ex; height:2.843ex;" alt="{\displaystyle \varsigma /(\varsigma -1)}"></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 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 \varsigma ^{2}+1:\varsigma ^{2}:\varsigma ^{2}-1:\varsigma +1:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>:</mo> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <msup> <mi>ς</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mi>ς</mi> <mo>+</mo> <mn>1</mn> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma ^{2}+1:\varsigma ^{2}:\varsigma ^{2}-1:\varsigma +1:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ae6a7e4d080f16366bbc64c71db026e948ac91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.13ex; height:2.843ex;" alt="{\displaystyle \varsigma ^{2}+1:\varsigma ^{2}:\varsigma ^{2}-1:\varsigma +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 \,\varsigma (\varsigma -1):\varsigma :2/(\varsigma -1):1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>ς</mi> <mo stretchy="false">(</mo> <mi>ς</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mi>ς</mi> <mo>:</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>ς</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\varsigma (\varsigma -1):\varsigma :2/(\varsigma -1):1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e4320631a375d65f93ae18a23f7fd7a2734b775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.724ex; height:2.843ex;" alt="{\displaystyle \,\varsigma (\varsigma -1):\varsigma :2/(\varsigma -1):1.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Supersilver_spiral">Supersilver spiral</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supersilver_ratio&action=edit&section=5" title="Edit section: Supersilver spiral"><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:Supersilver_spiral.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Supersilver_spiral.svg/330px-Supersilver_spiral.svg.png" decoding="async" width="330" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Supersilver_spiral.svg/495px-Supersilver_spiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Supersilver_spiral.svg/660px-Supersilver_spiral.svg.png 2x" data-file-width="1320" data-file-height="650" /></a><figcaption>Supersilver spirals with different initial angles on a <span class="texhtml">ς</span>− rectangle.</figcaption></figure> <p>A supersilver 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 \varsigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1fab64d12c6ee8fbf378b73072d0cf13c8c4f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:0.942ex; height:1.843ex;" alt="{\displaystyle \varsigma }"></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</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>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(\varsigma )}{\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>ς</mi> <mo stretchy="false">)</mo> </mrow> <mi>π</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {2\ln(\varsigma )}{\pi }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac819396d7e565c63e8a240b9770e5e0e9cd551" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.033ex; height:5.676ex;" alt="{\displaystyle k={\frac {2\ln(\varsigma )}{\pi }}.}"></span> If drawn on a supersilver rectangle, 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="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 \varsigma (\varsigma -1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ς</mi> <mo stretchy="false">(</mo> <mi>ς</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsigma (\varsigma -1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f3e7e91aff8f9139fd3b04425eb3cacff70d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.696ex; height:2.843ex;" alt="{\displaystyle \varsigma (\varsigma -1)}"></span> which are orthogonally aligned and successively scaled by a factor <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/\varsigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ς</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\varsigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35e6feb0c979c82feadf3aeca74fc8dabfc9ff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.914ex; height:2.843ex;" alt="{\displaystyle 1/\varsigma .}"></span> </p><p><br /> </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=Supersilver_ratio&action=edit&section=6" 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}=2x^{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> <mn>2</mn> <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}=2x^{2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0542beb2e75e872e0742ab14d5787b33ae1a755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.032ex; height:2.843ex;" alt="{\displaystyle x^{3}=2x^{2}+1}"></span>: <ul><li><a href="/wiki/Silver_ratio" title="Silver ratio">Silver 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}=2x+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> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=2x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e83a267194866ce5e876544b9c23a01d420c8618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.977ex; height:2.843ex;" alt="{\displaystyle x^{2}=2x+1}"></span></li> <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="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supersilver_ratio&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap 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_"A272874"" 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/A272874">"Sequence A272874"</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>. 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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%3ASupersilver+ratio" class="Z3988"></span></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">(sequence <span class="nowrap external"><a href="//oeis.org/A008998" class="extiw" title="oeis:A008998">A008998</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-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="CITEREFMahonHoradam1990" class="citation journal cs1">Mahon, Br. J. M.; Horadam, A. F. (1990). "Third-order diagonal functions of Pell polynomials". <i><a href="/wiki/The_Fibonacci_Quarterly" class="mw-redirect" title="The Fibonacci Quarterly">The Fibonacci Quarterly</a></i>. <b>28</b> (1): 3–10. <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%2F00150517.1990.12429513">10.1080/00150517.1990.12429513</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Fibonacci+Quarterly&rft.atitle=Third-order+diagonal+functions+of+Pell+polynomials&rft.volume=28&rft.issue=1&rft.pages=3-10&rft.date=1990&rft_id=info%3Adoi%2F10.1080%2F00150517.1990.12429513&rft.aulast=Mahon&rft.aufirst=Br.+J.+M.&rft.au=Horadam%2C+A.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASupersilver+ratio" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Only one of these is a 'restricted pseudoprime' as defined in: <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/10.1090%2FS0025-5718-1982-0658231-9">"Strong primality tests that are not sufficient"</a>. <i><a href="/wiki/Mathematics_of_Computation" title="Mathematics of Computation">Mathematics of Computation</a></i>. <b>39</b> (159). <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>: 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/10.1090%2FS0025-5718-1982-0658231-9">10.1090/S0025-5718-1982-0658231-9</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=Mathematics+of+Computation&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%2F10.1090%2FS0025-5718-1982-0658231-9&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%2F10.1090%252FS0025-5718-1982-0658231-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASupersilver+ratio" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">for n ≥ 2 (sequence <span class="nowrap external"><a href="//oeis.org/A193641" class="extiw" title="oeis:A193641">A193641</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-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</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%3ASupersilver+ratio" class="Z3988"></span></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">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%3ASupersilver+ratio" 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Supersilver ratio - Wikipedia