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id="toc-Heighway_dragon-sublist" class="vector-toc-list"> <li id="toc-Construction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Construction</span> </div> </a> <ul id="toc-Construction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Folding_the_dragon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Folding_the_dragon"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Folding the dragon</span> </div> </a> <ul id="toc-Folding_the_dragon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Twindragon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Twindragon"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Twindragon</span> </div> </a> <ul id="toc-Twindragon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Terdragon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Terdragon"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Terdragon</span> </div> </a> <ul id="toc-Terdragon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lévy_dragon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lévy_dragon"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Lévy dragon</span> </div> </a> <ul id="toc-Lévy_dragon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Occurrences_of_the_dragon_curve_in_solution_sets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Occurrences_of_the_dragon_curve_in_solution_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Occurrences of the dragon curve in solution sets</span> </div> </a> <ul id="toc-Occurrences_of_the_dragon_curve_in_solution_sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-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">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" 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Available in 19 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-19" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">19 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Corba_del_drac" title="Corba del drac – Catalan" lang="ca" hreflang="ca" data-title="Corba del drac" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dra%C4%8D%C3%AD_k%C5%99ivka" title="Dračí křivka – Czech" lang="cs" hreflang="cs" data-title="Dračí křivka" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Drachenkurve" title="Drachenkurve – German" lang="de" hreflang="de" data-title="Drachenkurve" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B1%CE%BC%CF%80%CF%8D%CE%BB%CE%B7_%CF%84%CE%BF%CF%85_%CE%B4%CF%81%CE%AC%CE%BA%CE%BF%CF%85" title="Καμπύλη του δράκου – Greek" lang="el" hreflang="el" data-title="Καμπύλη του δράκου" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Curva_del_drag%C3%B3n" title="Curva del dragón – Spanish" lang="es" hreflang="es" data-title="Curva del dragón" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Courbe_du_dragon" title="Courbe du dragon – French" lang="fr" hreflang="fr" data-title="Courbe du dragon" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Zmajolika_krivulja" title="Zmajolika krivulja – Croatian" lang="hr" hreflang="hr" data-title="Zmajolika krivulja" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Curva_del_drago_di_Heighway" title="Curva del drago di Heighway – Italian" lang="it" hreflang="it" data-title="Curva del drago di Heighway" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A7%D7%95%D7%9E%D7%AA_%D7%93%D7%A8%D7%A7%D7%95%D7%9F" title="עקומת דרקון – Hebrew" lang="he" hreflang="he" data-title="עקומת דרקון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Dragon_curve" title="Dragon curve – Swahili" lang="sw" hreflang="sw" data-title="Dragon curve" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%89%E3%83%A9%E3%82%B4%E3%83%B3%E6%9B%B2%E7%B7%9A" title="ドラゴン曲線 – Japanese" lang="ja" hreflang="ja" data-title="ドラゴン曲線" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Smok_Heighwaya" title="Smok Heighwaya – Polish" lang="pl" hreflang="pl" data-title="Smok Heighwaya" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Curva_de_drag%C3%A3o" title="Curva de dragão – Portuguese" lang="pt" hreflang="pt" data-title="Curva de dragão" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%B0%D1%8F_%D0%B4%D1%80%D0%B0%D0%BA%D0%BE%D0%BD%D0%B0" title="Кривая дракона – Russian" lang="ru" hreflang="ru" data-title="Кривая дракона" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Zmajolika_krivulja" title="Zmajolika krivulja – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Zmajolika krivulja" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Fractal constructible with L-systems</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Dragon fractal" redirects here. For the filled Julia sets, see <a href="/wiki/Douady_rabbit" title="Douady rabbit">Douady rabbit</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Twindragon" redirects here. For other uses, see <a href="/wiki/Twin_Dragon_(disambiguation)" class="mw-redirect mw-disambig" title="Twin Dragon (disambiguation)">Twin Dragon (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fractal_dragon_curve.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Fractal_dragon_curve.jpg/220px-Fractal_dragon_curve.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Fractal_dragon_curve.jpg/330px-Fractal_dragon_curve.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Fractal_dragon_curve.jpg/440px-Fractal_dragon_curve.jpg 2x" data-file-width="590" data-file-height="589" /></a><figcaption>Heighway dragon curve</figcaption></figure> <p>A <b>dragon curve</b> is any member of a family of <a href="/wiki/Self-similarity" title="Self-similarity">self-similar</a> <a href="/wiki/Fractal_curves" class="mw-redirect" title="Fractal curves">fractal curves</a>, which can be approximated by <a href="/wiki/Recursion" title="Recursion">recursive</a> methods such as <a href="/wiki/Lindenmayer_system" class="mw-redirect" title="Lindenmayer system">Lindenmayer systems</a>. The dragon curve is probably most commonly thought of as the shape that is generated from repeatedly folding a strip of paper in half, although there are other curves that are called dragon curves that are generated differently. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Heighway_dragon">Heighway dragon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dragon_curve&action=edit&section=1" title="Edit section: Heighway dragon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Heighway dragon</b> (also known as the <b>Harter–Heighway dragon</b> or the <b>Jurassic Park dragon</b>) was first investigated by <a href="/wiki/NASA" title="NASA">NASA</a> physicists John Heighway, Bruce Banks, and William Harter. It was described by <a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a> in his <a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a> column <i><a href="/wiki/Mathematical_Games_(column)" class="mw-redirect" title="Mathematical Games (column)">Mathematical Games</a></i> in 1967. Many of its properties were first published by <a href="/wiki/Chandler_Davis" title="Chandler Davis">Chandler Davis</a> and <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>. It appeared on the section title pages of the <a href="/wiki/Michael_Crichton" title="Michael Crichton">Michael Crichton</a> novel <i><a href="/wiki/Jurassic_Park_(novel)" title="Jurassic Park (novel)">Jurassic Park</a></i>.<sup id="cite_ref-tabachnikov_1-0" class="reference"><a href="#cite_note-tabachnikov-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Construction">Construction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dragon_curve&action=edit&section=2" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Dragon_Curve_unfolding_zoom_numbered.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Dragon_Curve_unfolding_zoom_numbered.gif/300px-Dragon_Curve_unfolding_zoom_numbered.gif" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/a0/Dragon_Curve_unfolding_zoom_numbered.gif 1.5x" data-file-width="352" data-file-height="352" /></a><figcaption>Recursive construction of the curve</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Dragon_Curve_adding_corners_trails_rectangular_numbered_R.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Dragon_Curve_adding_corners_trails_rectangular_numbered_R.gif/300px-Dragon_Curve_adding_corners_trails_rectangular_numbered_R.gif" decoding="async" width="300" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Dragon_Curve_adding_corners_trails_rectangular_numbered_R.gif/450px-Dragon_Curve_adding_corners_trails_rectangular_numbered_R.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/7/72/Dragon_Curve_adding_corners_trails_rectangular_numbered_R.gif 2x" data-file-width="504" data-file-height="341" /></a><figcaption>Recursive construction of the curve</figcaption></figure> <p>The Heighway dragon can be constructed from a base <a href="/wiki/Line_segment" title="Line segment">line segment</a> by repeatedly replacing each segment by two segments with a right angle and with a rotation of 45° alternatively to the right and to the left:<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> <figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/File:Dragon_curve_iterations_(2).svg" class="mw-file-description" title="The first 5 iterations and the 9th"><img alt="The first 5 iterations and the 9th" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Dragon_curve_iterations_%282%29.svg/700px-Dragon_curve_iterations_%282%29.svg.png" decoding="async" width="700" height="83" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Dragon_curve_iterations_%282%29.svg/1050px-Dragon_curve_iterations_%282%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Dragon_curve_iterations_%282%29.svg/1400px-Dragon_curve_iterations_%282%29.svg.png 2x" data-file-width="1574" data-file-height="186" /></a><figcaption>The first 5 iterations and the 9th</figcaption></figure> <p>The Heighway dragon is also the limit set of the following <a href="/wiki/Iterated_function_system" title="Iterated function system">iterated function system</a> in the complex plane: </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 f_{1}(z)={\frac {(1+i)z}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mi>z</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}(z)={\frac {(1+i)z}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73d58f24596de176e109dac3acc70b2efb954ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.728ex; height:5.676ex;" alt="{\displaystyle f_{1}(z)={\frac {(1+i)z}{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 f_{2}(z)=1-{\frac {(1-i)z}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mi>z</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{2}(z)=1-{\frac {(1-i)z}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b58506afa90881221f35ef2df83531a300c2176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.731ex; height:5.676ex;" alt="{\displaystyle f_{2}(z)=1-{\frac {(1-i)z}{2}}}"></span></dd></dl> <p>with the initial set of points <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}=\{0,1\}}"> <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> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{0}=\{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec695e918a52247f22f5ca81bac38d10d9e68a7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.261ex; height:2.843ex;" alt="{\displaystyle S_{0}=\{0,1\}}"></span>. </p><p>Using pairs of real numbers instead, this is the same as the two functions consisting of </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 f_{1}(x,y)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos 45^{\circ }&-\sin 45^{\circ }\\\sin 45^{\circ }&\cos 45^{\circ }\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}(x,y)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos 45^{\circ }&-\sin 45^{\circ }\\\sin 45^{\circ }&\cos 45^{\circ }\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b552c0b5bf4dd7cd75e120d63c73db65b535740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:42.247ex; height:6.509ex;" alt="{\displaystyle f_{1}(x,y)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos 45^{\circ }&-\sin 45^{\circ }\\\sin 45^{\circ }&\cos 45^{\circ }\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}}"></span></dd></dl> <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 f_{2}(x,y)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos 135^{\circ }&-\sin 135^{\circ }\\\sin 135^{\circ }&\cos 135^{\circ }\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+{\begin{pmatrix}1\\0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>135</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>135</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <msup> <mn>135</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>135</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{2}(x,y)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos 135^{\circ }&-\sin 135^{\circ }\\\sin 135^{\circ }&\cos 135^{\circ }\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+{\begin{pmatrix}1\\0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d69e488908d34fa9c5e9a11d75e35100dd27f54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:52.747ex; height:6.509ex;" alt="{\displaystyle f_{2}(x,y)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos 135^{\circ }&-\sin 135^{\circ }\\\sin 135^{\circ }&\cos 135^{\circ }\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+{\begin{pmatrix}1\\0\end{pmatrix}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Folding_the_dragon">Folding the dragon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dragon_curve&action=edit&section=3" title="Edit section: Folding the dragon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Heighway dragon curve can be constructed by <a href="/wiki/Regular_paperfolding_sequence" title="Regular paperfolding sequence">folding a strip of paper</a>, which is how it was originally discovered.<sup id="cite_ref-tabachnikov_1-1" class="reference"><a href="#cite_note-tabachnikov-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Take a strip of paper and fold it in half to the right. Fold it in half again to the right. If the strip was opened out now, unbending each fold to become a 90-degree turn, the turn sequence would be RRL, i.e. the second iteration of the Heighway dragon. Fold the strip in half again to the right, and the turn sequence of the unfolded strip is now RRLRRLL – the third iteration of the Heighway dragon. Continuing folding the strip in half to the right to create further iterations of the Heighway dragon (in practice, the strip becomes too thick to fold sharply after four or five iterations). </p> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Dragon_curve_paper_strip.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Dragon_curve_paper_strip.png/800px-Dragon_curve_paper_strip.png" decoding="async" width="800" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Dragon_curve_paper_strip.png/1200px-Dragon_curve_paper_strip.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Dragon_curve_paper_strip.png/1600px-Dragon_curve_paper_strip.png 2x" data-file-width="1714" data-file-height="400" /></a><figcaption></figcaption></figure> <p>The folding patterns of this sequence of paper strips, as sequences of right (R) and left (L) folds, are: </p> <ul><li>1st iteration: R</li> <li>2nd iteration: <b>R</b> R <b>L</b></li> <li>3rd iteration: <b>R</b> R <b>L</b> R <b>R</b> L <b>L</b></li> <li>4th iteration: <b>R</b> R <b>L</b> R <b>R</b> L <b>L</b> R <b>R</b> R <b>L</b> L <b>R</b> L <b>L</b>.</li></ul> <p>Each iteration can be found by copying the previous iteration, then an R, then a second copy of the previous iteration in reverse order with the L and R letters swapped.<sup id="cite_ref-tabachnikov_1-2" class="reference"><a href="#cite_note-tabachnikov-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dragon_curve&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Many <b>self-similarities</b> can be seen in the Heighway dragon curve. The most obvious is the repetition of the same pattern tilted by 45° and with a reduction ratio 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 \textstyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5287e0a313d8c5505fcb11e3fb36d5026ff2ca3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle \textstyle {\sqrt {2}}}"></span>. Based on these self-similarities, many of its lengths are simple rational numbers.</li></ul> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tnone center"><div class="thumbinner multiimageinner" style="width:792px;max-width:792px"><div class="trow"><div class="tsingle" style="width:451px;max-width:451px"><div class="thumbimage" style="height:241px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Dimensions_fractale_dragon.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Dimensions_fractale_dragon.png/449px-Dimensions_fractale_dragon.png" decoding="async" width="449" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Dimensions_fractale_dragon.png/674px-Dimensions_fractale_dragon.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d7/Dimensions_fractale_dragon.png 2x" data-file-width="849" data-file-height="456" /></a></span></div><div class="thumbcaption">Lengths</div></div><div class="tsingle" style="width:337px;max-width:337px"><div class="thumbimage" style="height:241px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Auto-similarity_dragon_curve.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Auto-similarity_dragon_curve.png/335px-Auto-similarity_dragon_curve.png" decoding="async" width="335" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Auto-similarity_dragon_curve.png/503px-Auto-similarity_dragon_curve.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Auto-similarity_dragon_curve.png/670px-Auto-similarity_dragon_curve.png 2x" data-file-width="808" data-file-height="582" /></a></span></div><div class="thumbcaption">Self-similarities</div></div></div></div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Full_tiling_dragon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Full_tiling_dragon.svg/300px-Full_tiling_dragon.svg.png" decoding="async" width="300" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Full_tiling_dragon.svg/450px-Full_tiling_dragon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Full_tiling_dragon.svg/600px-Full_tiling_dragon.svg.png 2x" data-file-width="1435" data-file-height="866" /></a><figcaption>Tiling of the plane by dragon curves</figcaption></figure> <ul><li>The dragon curve can <a href="/wiki/Tessellation" title="Tessellation">tile the plane</a>. One possible tiling replaces each edge of a <a href="/wiki/Square_tiling" title="Square tiling">square tiling</a> with a dragon curve, using the recursive definition of the dragon starting from a line segment. The initial direction to expand each segment can be determined from a checkerboard coloring of a square tiling, expanding vertical segments into black tiles and out of white tiles, and expanding horizontal segments into white tiles and out of black ones.<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></li> <li>As a <a href="/wiki/Space-filling_curve" title="Space-filling curve">space-filling curve</a>, the dragon curve has <a href="/wiki/Fractal_dimension" title="Fractal dimension">fractal dimension</a> exactly 2. For a dragon curve with initial segment length 1, its area is 1/2, as can be seen from its tilings of the plane.<sup id="cite_ref-tabachnikov_1-3" class="reference"><a href="#cite_note-tabachnikov-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>The boundary of the set covered by the dragon curve has infinite length, with fractal dimension <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\log _{2}\lambda \approx 1.523627086202492,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>λ</mi> <mo>≈</mo> <mn>1.523627086202492</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\log _{2}\lambda \approx 1.523627086202492,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a9b9e5686aa8217eccc7195a610a8dcccbf36b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.309ex; height:2.676ex;" alt="{\displaystyle 2\log _{2}\lambda \approx 1.523627086202492,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {1+(28-3{\sqrt {87}})^{1/3}+(28+3{\sqrt {87}})^{1/3}}{3}}\approx 1.69562076956}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>28</mn> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>87</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>28</mn> <mo>+</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>87</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>≈</mo> <mn>1.69562076956</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {1+(28-3{\sqrt {87}})^{1/3}+(28+3{\sqrt {87}})^{1/3}}{3}}\approx 1.69562076956}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af26bb2e7bd5112d2a1f5385d486fe54cb533449" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.02ex; height:6.009ex;" alt="{\displaystyle \lambda ={\frac {1+(28-3{\sqrt {87}})^{1/3}+(28+3{\sqrt {87}})^{1/3}}{3}}\approx 1.69562076956}"></span> is the 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 \lambda ^{3}-\lambda ^{2}-2=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{3}-\lambda ^{2}-2=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6622979e803ab024609ce7a7e1517a55ecb59b80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.57ex; height:2.843ex;" alt="{\displaystyle \lambda ^{3}-\lambda ^{2}-2=0.}"></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></li></ul> <div class="mw-heading mw-heading2"><h2 id="Twindragon">Twindragon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dragon_curve&action=edit&section=5" title="Edit section: Twindragon"><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:Twindragon.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Twindragon.png/220px-Twindragon.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Twindragon.png/330px-Twindragon.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Twindragon.png/440px-Twindragon.png 2x" data-file-width="800" data-file-height="600" /></a><figcaption>Twindragon curve constructed from two Heighway dragons</figcaption></figure><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Complex-base_system#Base_−1_±_i" title="Complex-base system">Complex-base system § Base −1 ± i</a></div> <p>The <b>twindragon</b> (also known as the <b>Davis–Knuth dragon</b>) can be constructed by placing two Heighway dragon curves back to back (after flipping the original dragon curve vertically and horizontally). It is also the limit set of the following iterated function system: </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 f_{1}(z)={\frac {(1+i)z}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mi>z</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}(z)={\frac {(1+i)z}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73d58f24596de176e109dac3acc70b2efb954ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.728ex; height:5.676ex;" alt="{\displaystyle f_{1}(z)={\frac {(1+i)z}{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 f_{2}(z)=1-{\frac {(1+i)z}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mi>z</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{2}(z)=1-{\frac {(1+i)z}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db1550806fcc8f58e97d6be2100bf6f7c2c244a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.731ex; height:5.676ex;" alt="{\displaystyle f_{2}(z)=1-{\frac {(1+i)z}{2}}}"></span></dd></dl> <p>where the initial shape is defined by the following set <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}=\{0,1,1-i\}}"> <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> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{0}=\{0,1,1-i\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce41850070bb2a202c2d32adf9aa455ed5ebc94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.101ex; height:2.843ex;" alt="{\displaystyle S_{0}=\{0,1,1-i\}}"></span>. </p><p>It can be also written as a <a href="/wiki/Lindenmayer_system" class="mw-redirect" title="Lindenmayer system">Lindenmayer system</a> – it only needs adding another section in the initial string: </p> <ul><li>angle 90°</li> <li>initial string <i>FX+FX+</i></li> <li>string rewriting rules <ul><li><i>X</i> ↦ <i>X</i>+<i>YF</i></li> <li><i>Y</i> ↦ <i>FX</i>−<i>Y</i>.</li></ul></li></ul> <p>It is also the locus of points in the complex plane with the same integer part when written in <a href="/wiki/Complex-base_system#Base_−1_±_i" title="Complex-base system">base <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\pm i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>±</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1\pm i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f063495834321148d570eda169b00162348a96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.423ex; height:2.843ex;" alt="{\displaystyle (-1\pm i)}"></span></a>.<sup id="cite_ref-Knuth2_5-0" class="reference"><a href="#cite_note-Knuth2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Terdragon">Terdragon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dragon_curve&action=edit&section=6" title="Edit section: Terdragon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Terdragon.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Terdragon.png/200px-Terdragon.png" decoding="async" width="200" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Terdragon.png/300px-Terdragon.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e3/Terdragon.png 2x" data-file-width="400" data-file-height="600" /></a><figcaption>Terdragon curve.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Developing_Terdragon_Curve.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Developing_Terdragon_Curve.jpg/300px-Developing_Terdragon_Curve.jpg" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Developing_Terdragon_Curve.jpg/450px-Developing_Terdragon_Curve.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Developing_Terdragon_Curve.jpg/600px-Developing_Terdragon_Curve.jpg 2x" data-file-width="2171" data-file-height="2171" /></a><figcaption> A sculpture depicting multiple iterations of the Lindenmayer system that generates the terdragon curve.<br />by <a href="/wiki/Henry_Segerman" title="Henry Segerman">Henry Segerman</a></figcaption></figure> <p>The <b>terdragon</b> can be written as a <a href="/wiki/Lindenmayer_system" class="mw-redirect" title="Lindenmayer system">Lindenmayer system</a>: </p> <ul><li>angle 120°</li> <li>initial string <i>F</i></li> <li>string rewriting rules <ul><li><i>F</i> ↦ <i>F+F−F</i>.</li></ul></li></ul> <p>It is the limit set of the following iterated function system: </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 f_{1}(z)=\lambda z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}(z)=\lambda z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/603fac378d1eb5c620333e8eaed25ec52cef119c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.633ex; height:2.843ex;" alt="{\displaystyle f_{1}(z)=\lambda z}"></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 f_{2}(z)={\frac {i}{\sqrt {3}}}z+\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mi>z</mi> <mo>+</mo> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{2}(z)={\frac {i}{\sqrt {3}}}z+\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/420effe65ca3654fcd760805b67519ffe66d4d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.408ex; height:6.176ex;" alt="{\displaystyle f_{2}(z)={\frac {i}{\sqrt {3}}}z+\lambda }"></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 f_{3}(z)=\lambda z+\lambda ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>z</mi> <mo>+</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{3}(z)=\lambda z+\lambda ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/201d93421e33daad1dd32367f21d231bec6ae848" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.883ex; height:2.843ex;" alt="{\displaystyle f_{3}(z)=\lambda z+\lambda ^{*}}"></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 {\mbox{where }}\lambda ={\frac {1}{2}}-{\frac {i}{2{\sqrt {3}}}}{\text{ and }}\lambda ^{*}={\frac {1}{2}}+{\frac {i}{2{\sqrt {3}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>where </mtext> </mstyle> </mrow> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{where }}\lambda ={\frac {1}{2}}-{\frac {i}{2{\sqrt {3}}}}{\text{ and }}\lambda ^{*}={\frac {1}{2}}+{\frac {i}{2{\sqrt {3}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bee375501597480f1b53f929ced70ac594b3940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.917ex; height:6.176ex;" alt="{\displaystyle {\mbox{where }}\lambda ={\frac {1}{2}}-{\frac {i}{2{\sqrt {3}}}}{\text{ and }}\lambda ^{*}={\frac {1}{2}}+{\frac {i}{2{\sqrt {3}}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Lévy_dragon"><span id="L.C3.A9vy_dragon"></span>Lévy dragon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dragon_curve&action=edit&section=7" title="Edit section: Lévy dragon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/L%C3%A9vy_C_curve" title="Lévy C curve">Lévy C curve</a> is sometimes known as the <b>Lévy dragon</b>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <table> <tbody><tr> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:L%C3%A9vy%27s_C-curve_(IFS).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/L%C3%A9vy%27s_C-curve_%28IFS%29.jpg/200px-L%C3%A9vy%27s_C-curve_%28IFS%29.jpg" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/L%C3%A9vy%27s_C-curve_%28IFS%29.jpg/300px-L%C3%A9vy%27s_C-curve_%28IFS%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/L%C3%A9vy%27s_C-curve_%28IFS%29.jpg/400px-L%C3%A9vy%27s_C-curve_%28IFS%29.jpg 2x" data-file-width="589" data-file-height="590" /></a><figcaption>Lévy C curve.</figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Occurrences_of_the_dragon_curve_in_solution_sets">Occurrences of the dragon curve in solution sets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dragon_curve&action=edit&section=8" title="Edit section: Occurrences of the dragon curve in solution sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Having obtained the set of solutions to a linear differential equation, any linear combination of the solutions will, because of the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</a>, also obey the original equation. In other words, new solutions are obtained by applying a function to the set of existing solutions. This is similar to how an iterated function system produces new points in a set, though not all IFS are linear functions. In a conceptually similar vein, a set of <a href="/wiki/Littlewood_polynomial" title="Littlewood polynomial">Littlewood polynomials</a> can be arrived at by such iterated applications of a set of functions. </p><p>A Littlewood polynomial is a polynomial: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=\sum _{i=0}^{n}a_{i}x^{i}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)=\sum _{i=0}^{n}a_{i}x^{i}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cded56614f521734e88b3fee88352dded3e5e9f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:15.784ex; height:6.843ex;" alt="{\displaystyle p(x)=\sum _{i=0}^{n}a_{i}x^{i}\,}"></span> where all <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_{i}=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=\pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f88e23fcacf840db6d5ec37b0e5d9a20277b69e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.099ex; height:2.509ex;" alt="{\displaystyle a_{i}=\pm 1}"></span>. </p><p>For some <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |w|<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd52cdea4952598ac35dd1ad3709c6669538122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.219ex; height:2.843ex;" alt="{\displaystyle |w|<1}"></span> we define the following functions: </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 f_{+}(z)=1+wz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>w</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{+}(z)=1+wz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b937f81c89118829e6cd3b8fa75fb0838a35d472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.401ex; height:2.843ex;" alt="{\displaystyle f_{+}(z)=1+wz}"></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 f_{-}(z)=1-wz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>w</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{-}(z)=1-wz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3cd943adf53bc7182f27e8c0d1d064d2fc303e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.401ex; height:2.843ex;" alt="{\displaystyle f_{-}(z)=1-wz}"></span></dd></dl> <p>Starting at z=0 we can generate all Littlewood polynomials of degree d using these functions iteratively d+1 times.<sup id="cite_ref-ncafe_7-0" class="reference"><a href="#cite_note-ncafe-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For instance: <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 f_{+}(f_{-}(f_{-}(0)))=1+(1-w)w=1+1w-1w^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mi>w</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mi>w</mi> <mo>−</mo> <mn>1</mn> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{+}(f_{-}(f_{-}(0)))=1+(1-w)w=1+1w-1w^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c553836b7a888777d415606cc7789eebc2870a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.431ex; height:3.176ex;" alt="{\displaystyle f_{+}(f_{-}(f_{-}(0)))=1+(1-w)w=1+1w-1w^{2}}"></span> </p><p>It can be seen that 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 w=(1+i)/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=(1+i)/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9114ee9cf02e8888774f5d208d7127553005f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.702ex; height:2.843ex;" alt="{\displaystyle w=(1+i)/2}"></span>, the above pair of functions is equivalent to the IFS formulation of the Heighway dragon. That is, the Heighway dragon, iterated to a certain iteration, describe the set of all Littlewood polynomials up to a certain degree, evaluated at the point <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+i)/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=(1+i)/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9114ee9cf02e8888774f5d208d7127553005f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.702ex; height:2.843ex;" alt="{\displaystyle w=(1+i)/2}"></span>. Indeed, when plotting a sufficiently high number of roots of the Littlewood polynomials, structures similar to the dragon curve appear at points close to these coordinates.<sup id="cite_ref-ncafe_7-1" class="reference"><a href="#cite_note-ncafe-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><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><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> </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=Dragon_curve&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_fractals_by_Hausdorff_dimension" title="List of fractals by Hausdorff dimension">List of fractals by Hausdorff dimension</a></li> <li><a href="/wiki/Complex-base_system" title="Complex-base system">Complex-base system</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dragon_curve&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-tabachnikov-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-tabachnikov_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-tabachnikov_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-tabachnikov_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-tabachnikov_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output 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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="CITEREFTabachnikov2014" class="citation cs2">Tabachnikov, Sergei (2014), "Dragon curves revisited", <i>The Mathematical Intelligencer</i>, <b>36</b> (1): <span class="nowrap">13–</span>17, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00283-013-9428-y">10.1007/s00283-013-9428-y</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3166985">3166985</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14420269">14420269</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+Intelligencer&rft.atitle=Dragon+curves+revisited&rft.volume=36&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E13-%3C%2Fspan%3E17&rft.date=2014&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3166985%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14420269%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00283-013-9428-y&rft.aulast=Tabachnikov&rft.aufirst=Sergei&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADragon+curve" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdgar2008" class="citation cs2">Edgar, Gerald (2008), "Heighway's Dragon", in Edgar, Gerald (ed.), <i>Measure, Topology, and Fractal Geometry</i>, Undergraduate Texts in Mathematics (2nd ed.), New York: Springer, pp. <span class="nowrap">20–</span>22, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-74749-1">10.1007/978-0-387-74749-1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-74748-4" title="Special:BookSources/978-0-387-74748-4"><bdi>978-0-387-74748-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2356043">2356043</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Heighway%27s+Dragon&rft.btitle=Measure%2C+Topology%2C+and+Fractal+Geometry&rft.place=New+York&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E20-%3C%2Fspan%3E22&rft.edition=2nd&rft.pub=Springer&rft.date=2008&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2356043%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-0-387-74749-1&rft.isbn=978-0-387-74748-4&rft.aulast=Edgar&rft.aufirst=Gerald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADragon+curve" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFEdgar2008">Edgar (2008)</a>, "Heighway’s Dragon Tiles the Plane", pp. 74–75.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFEdgar2008">Edgar (2008)</a>, "Heighway Dragon Boundary", pp. 194–195.</span> </li> <li id="cite_note-Knuth2-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Knuth2_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1998" class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1998). "Positional Number Systems". <i>The art of computer programming</i>. Vol. 2 (3rd ed.). Boston: Addison-Wesley. p. 206. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-89684-2" title="Special:BookSources/0-201-89684-2"><bdi>0-201-89684-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/48246681">48246681</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Positional+Number+Systems&rft.btitle=The+art+of+computer+programming&rft.place=Boston&rft.pages=206&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=1998&rft_id=info%3Aoclcnum%2F48246681&rft.isbn=0-201-89684-2&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADragon+curve" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaileyKimStrichartz2002" class="citation cs2">Bailey, Scott; Kim, Theodore; Strichartz, Robert S. (2002), "Inside the Lévy dragon", <i><a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>, <b>109</b> (8): <span class="nowrap">689–</span>703, <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%2F3072395">10.2307/3072395</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/3072395">3072395</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1927621">1927621</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Inside+the+L%C3%A9vy+dragon&rft.volume=109&rft.issue=8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E689-%3C%2Fspan%3E703&rft.date=2002&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1927621%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3072395%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3072395&rft.aulast=Bailey&rft.aufirst=Scott&rft.au=Kim%2C+Theodore&rft.au=Strichartz%2C+Robert+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADragon+curve" class="Z3988"></span>.</span> </li> <li id="cite_note-ncafe-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-ncafe_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ncafe_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://golem.ph.utexas.edu/category/2009/12/this_weeks_finds_in_mathematic_46.html">"The n-Category Café"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+n-Category+Caf%C3%A9&rft_id=http%3A%2F%2Fgolem.ph.utexas.edu%2Fcategory%2F2009%2F12%2Fthis_weeks_finds_in_mathematic_46.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADragon+curve" class="Z3988"></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week285.html">"Week285"</a>.</cite><span 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Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DragonCurve.html">"Dragon Curve"</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=Dragon+Curve&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDragonCurve.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADragon+curve" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=v678Em6qyzk">Knuth on the Dragon Curve</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output 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title="Minkowski–Bouligand dimension">Box-counting</a> <ul><li><a href="/wiki/Higuchi_dimension" title="Higuchi dimension">Higuchi</a></li></ul></li> <li><a href="/wiki/Correlation_dimension" title="Correlation dimension">Correlation</a></li> <li><a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff</a></li> <li><a href="/wiki/Packing_dimension" title="Packing dimension">Packing</a></li> <li><a href="/wiki/Lebesgue_covering_dimension" title="Lebesgue covering dimension">Topological</a></li></ul></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Self-similarity" title="Self-similarity">Self-similarity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Iterated_function_system" title="Iterated function system">Iterated function <br />system</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barnsley_fern" title="Barnsley fern">Barnsley fern</a></li> <li><a href="/wiki/Cantor_set" title="Cantor set">Cantor set</a></li> <li><a href="/wiki/Koch_snowflake" title="Koch snowflake">Koch snowflake</a></li> <li><a href="/wiki/Menger_sponge" title="Menger sponge">Menger sponge</a></li> <li><a href="/wiki/Sierpi%C5%84ski_carpet" title="Sierpiński carpet">Sierpiński carpet</a></li> <li><a href="/wiki/Sierpi%C5%84ski_triangle" title="Sierpiński triangle">Sierpiński triangle</a></li> <li><a href="/wiki/Apollonian_gasket" title="Apollonian gasket">Apollonian gasket</a></li> <li><a href="/wiki/Fibonacci_word_fractal" title="Fibonacci word fractal">Fibonacci word</a></li> <li><a href="/wiki/Space-filling_curve" title="Space-filling curve">Space-filling curve</a> <ul><li><a href="/wiki/Blancmange_curve" title="Blancmange curve">Blancmange curve</a></li> <li><a href="/wiki/De_Rham_curve" title="De Rham curve">De Rham curve</a> <ul><li><a href="/wiki/Minkowski_sausage" title="Minkowski sausage">Minkowski</a></li></ul></li> <li><a class="mw-selflink selflink">Dragon curve</a></li> <li><a href="/wiki/Hilbert_curve" title="Hilbert curve">Hilbert curve</a></li> <li><a href="/wiki/Koch_snowflake" title="Koch snowflake">Koch curve</a></li> <li><a href="/wiki/L%C3%A9vy_C_curve" title="Lévy C curve">Lévy C curve</a></li> <li><a href="/wiki/Moore_curve" title="Moore curve">Moore curve</a></li> <li><a href="/wiki/Peano_curve" title="Peano curve">Peano curve</a></li> <li><a href="/wiki/Sierpi%C5%84ski_curve" title="Sierpiński curve">Sierpiński curve</a></li> <li><a href="/wiki/Z-order_curve" title="Z-order curve">Z-order curve</a></li></ul></li> <li><a href="/wiki/Fractal_string" title="Fractal string">String</a></li> <li><a href="/wiki/T-square_(fractal)" title="T-square (fractal)">T-square</a></li> <li><a href="/wiki/N-flake" title="N-flake">n-flake</a></li> <li><a href="/wiki/Vicsek_fractal" title="Vicsek fractal">Vicsek fractal</a></li> <li><a href="/wiki/Gosper_curve" title="Gosper curve">Gosper curve</a></li> <li><a href="/wiki/Pythagoras_tree_(fractal)" title="Pythagoras tree (fractal)">Pythagoras tree</a></li> <li><a href="/wiki/Weierstrass_function" title="Weierstrass function">Weierstrass function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Attractor#Strange_attractor" title="Attractor">Strange attractor</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multifractal_system" title="Multifractal system">Multifractal system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L-system" title="L-system">L-system</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fractal_canopy" title="Fractal canopy">Fractal canopy</a></li> <li><a href="/wiki/Space-filling_curve" title="Space-filling curve">Space-filling curve</a> <ul><li><a href="/wiki/H_tree" title="H tree">H tree</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fractal#Common_techniques_for_generating_fractals" title="Fractal">Escape-time <br />fractals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burning_Ship_fractal" title="Burning Ship fractal">Burning Ship fractal</a></li> <li><a href="/wiki/Julia_set" title="Julia set">Julia set</a> <ul><li><a href="/wiki/Filled_Julia_set" title="Filled Julia set">Filled</a></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Douady_rabbit" title="Douady rabbit">Douady rabbit</a></li></ul></li> <li><a href="/wiki/Lyapunov_fractal" title="Lyapunov fractal">Lyapunov fractal</a></li> <li><a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a> <ul><li><a href="/wiki/Misiurewicz_point" title="Misiurewicz point">Misiurewicz point</a></li></ul></li> <li><a href="/wiki/Multibrot_set" title="Multibrot set">Multibrot set</a></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Tricorn_(mathematics)" title="Tricorn (mathematics)">Tricorn</a></li> <li><a href="/wiki/Mandelbox" title="Mandelbox">Mandelbox</a></li> <li><a href="/wiki/Mandelbulb" title="Mandelbulb">Mandelbulb</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a> techniques</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Buddhabrot" title="Buddhabrot">Buddhabrot</a></li> <li><a href="/wiki/Orbit_trap" title="Orbit trap">Orbit trap</a></li> <li><a href="/wiki/Pickover_stalk" title="Pickover stalk">Pickover stalk</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Chaos_game" title="Chaos game">Random</a> fractals</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> <ul><li><a href="/wiki/Diffusion-limited_aggregation" title="Diffusion-limited aggregation">Brownian tree</a></li> <li><a href="/wiki/Brownian_motor" title="Brownian motor">Brownian motor</a></li></ul></li> <li><a href="/wiki/Fractal_landscape" title="Fractal landscape">Fractal landscape</a></li> <li><a href="/wiki/L%C3%A9vy_flight" title="Lévy flight">Lévy flight</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation theory</a></li> <li><a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding walk</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Michael_Barnsley" title="Michael Barnsley">Michael Barnsley</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Bill_Gosper" title="Bill Gosper">Bill Gosper</a></li> <li><a href="/wiki/Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a></li> <li><a href="/wiki/Desmond_Paul_Henry" title="Desmond Paul Henry">Desmond Paul Henry</a></li> <li><a href="/wiki/Gaston_Julia" title="Gaston Julia">Gaston Julia</a></li> <li><a href="/wiki/Niels_Fabian_Helge_von_Koch" title="Niels Fabian Helge von Koch">Niels Fabian Helge von Koch</a></li> <li><a href="/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Paul Lévy</a></li> <li><a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a></li> <li><a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a></li> <li><a href="/wiki/Hamid_Naderi_Yeganeh" title="Hamid Naderi Yeganeh">Hamid Naderi Yeganeh</a></li> <li><a href="/wiki/Lewis_Fry_Richardson" title="Lewis Fry Richardson">Lewis Fry Richardson</a></li> <li><a href="/wiki/Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Wacław Sierpiński</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coastline_paradox" title="Coastline paradox">Coastline paradox</a></li> <li><a href="/wiki/Fractal_art" title="Fractal art">Fractal art</a></li> <li><a href="/wiki/List_of_fractals_by_Hausdorff_dimension" title="List of fractals by Hausdorff dimension">List of fractals by Hausdorff dimension</a></li> <li><i><a href="/wiki/The_Fractal_Geometry_of_Nature" title="The Fractal Geometry of Nature">The Fractal Geometry of Nature</a></i> (1982 book)</li> <li><i><a href="/wiki/The_Beauty_of_Fractals" title="The Beauty of Fractals">The Beauty of Fractals</a></i> (1986 book)</li> <li><i><a href="/wiki/Chaos:_Making_a_New_Science" title="Chaos: Making a New Science">Chaos: Making a New Science</a></i> (1987 book)</li> <li><a href="/wiki/Kaleidoscope" title="Kaleidoscope">Kaleidoscope</a></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Mathematics_of_paper_folding110" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematics_of_paper_folding" title="Template:Mathematics of paper folding"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematics_of_paper_folding" title="Template talk:Mathematics of paper folding"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematics_of_paper_folding" title="Special:EditPage/Template:Mathematics of paper folding"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematics_of_paper_folding110" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematics_of_paper_folding" title="Mathematics of paper folding">Mathematics of paper folding</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Flat folding</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Big-little-big_lemma" title="Big-little-big lemma">Big-little-big lemma</a></li> <li><a href="/wiki/Crease_pattern" title="Crease pattern">Crease pattern</a></li> <li><a href="/wiki/Huzita%E2%80%93Hatori_axioms" title="Huzita–Hatori axioms">Huzita–Hatori axioms</a></li> <li><a href="/wiki/Kawasaki%27s_theorem" title="Kawasaki's theorem">Kawasaki's theorem</a></li> <li><a href="/wiki/Maekawa%27s_theorem" title="Maekawa's theorem">Maekawa's theorem</a></li> <li><a href="/wiki/Map_folding" title="Map folding">Map folding</a></li> <li><a href="/wiki/Napkin_folding_problem" title="Napkin folding problem">Napkin folding problem</a></li> <li><a href="/wiki/Pureland_origami" title="Pureland origami">Pureland origami</a></li> <li><a href="/wiki/Yoshizawa%E2%80%93Randlett_system" title="Yoshizawa–Randlett system">Yoshizawa–Randlett system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Strip folding</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Dragon curve</a></li> <li><a href="/wiki/Flexagon" title="Flexagon">Flexagon</a></li> <li><a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a></li> <li><a href="/wiki/Regular_paperfolding_sequence" title="Regular paperfolding sequence">Regular paperfolding sequence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">3d structures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Miura_fold" title="Miura fold">Miura fold</a></li> <li><a href="/wiki/Modular_origami" title="Modular origami">Modular origami</a></li> <li><a href="/wiki/Paper_bag_problem" title="Paper bag problem">Paper bag problem</a></li> <li><a href="/wiki/Rigid_origami" title="Rigid origami">Rigid origami</a></li> <li><a href="/wiki/Schwarz_lantern" title="Schwarz lantern">Schwarz lantern</a></li> <li><a href="/wiki/Sonobe" title="Sonobe">Sonobe</a></li> <li><a href="/wiki/Yoshimura_buckling" title="Yoshimura buckling">Yoshimura buckling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Polyhedra</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexandrov%27s_uniqueness_theorem" title="Alexandrov's uniqueness theorem">Alexandrov's uniqueness theorem</a></li> <li><a href="/wiki/Flexible_polyhedron" title="Flexible polyhedron">Flexible polyhedron</a> (<a href="/wiki/Bricard_octahedron" title="Bricard octahedron">Bricard octahedron</a>, <a href="/wiki/Steffen%27s_polyhedron" title="Steffen's polyhedron">Steffen's polyhedron</a>)</li> <li><a href="/wiki/Net_(polyhedron)" title="Net (polyhedron)">Net</a> <ul><li><a href="/wiki/Blooming_(geometry)" title="Blooming (geometry)">Blooming</a></li> <li><a href="/wiki/Common_net" title="Common net">Common net</a></li> <li><a href="/wiki/Source_unfolding" title="Source unfolding">Source unfolding</a></li> <li><a href="/wiki/Star_unfolding" title="Star unfolding">Star unfolding</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fold-and-cut_theorem" title="Fold-and-cut theorem">Fold-and-cut theorem</a></li> <li><a href="/wiki/Lill%27s_method" title="Lill's method">Lill's method</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Publications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Geometric_Exercises_in_Paper_Folding" title="Geometric Exercises in Paper Folding">Geometric Exercises in Paper Folding</a></i></li> <li><i><a href="/wiki/Geometric_Folding_Algorithms" title="Geometric Folding Algorithms">Geometric Folding Algorithms</a></i></li> <li><i><a href="/wiki/Geometric_Origami" title="Geometric Origami">Geometric Origami</a></i></li> <li><i><a href="/wiki/A_History_of_Folding_in_Mathematics" title="A History of Folding in Mathematics">A History of Folding in Mathematics</a></i></li> <li><i><a href="/wiki/Origami_Polyhedra_Design" title="Origami Polyhedra Design">Origami Polyhedra Design</a></i></li> <li><i><a href="/wiki/Origamics" title="Origamics">Origamics</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Roger_C._Alperin" title="Roger C. Alperin">Roger C. Alperin</a></li> <li><a href="/wiki/Margherita_Piazzola_Beloch" title="Margherita Piazzola Beloch">Margherita Piazzola Beloch</a></li> <li><a href="/wiki/Yan_Chen_(mechanical_engineer)" title="Yan Chen (mechanical engineer)">Yan Chen</a></li> <li><a href="/wiki/Robert_Connelly" title="Robert Connelly">Robert Connelly</a></li> <li><a href="/wiki/Erik_Demaine" title="Erik Demaine">Erik Demaine</a></li> <li><a href="/wiki/Martin_Demaine" title="Martin Demaine">Martin Demaine</a></li> <li><a href="/wiki/Rona_Gurkewitz" title="Rona Gurkewitz">Rona Gurkewitz</a></li> <li><a href="/wiki/David_A._Huffman" title="David A. Huffman">David A. Huffman</a></li> <li><a href="/wiki/Tom_Hull_(mathematician)" title="Tom Hull (mathematician)">Tom Hull</a></li> <li><a href="/wiki/K%C3%B4di_Husimi" title="Kôdi Husimi">Kôdi Husimi</a></li> <li><a href="/wiki/Humiaki_Huzita" title="Humiaki Huzita">Humiaki Huzita</a></li> <li><a href="/wiki/Toshikazu_Kawasaki" title="Toshikazu Kawasaki">Toshikazu Kawasaki</a></li> <li><a href="/wiki/Robert_J._Lang" title="Robert J. Lang">Robert J. Lang</a></li> <li><a href="/wiki/Anna_Lubiw" title="Anna Lubiw">Anna Lubiw</a></li> <li><a href="/wiki/Jun_Maekawa" title="Jun Maekawa">Jun Maekawa</a></li> <li><a href="/wiki/K%C5%8Dry%C5%8D_Miura" title="Kōryō Miura">Kōryō Miura</a></li> <li><a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">Joseph O'Rourke</a></li> <li><a href="/wiki/Tomohiro_Tachi" title="Tomohiro Tachi">Tomohiro Tachi</a></li> <li><a href="/wiki/Eve_Torrence" title="Eve Torrence">Eve Torrence</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Dragon_curve&oldid=1259574434">https://en.wikipedia.org/w/index.php?title=Dragon_curve&oldid=1259574434</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Fractal_curves" title="Category:Fractal curves">Fractal curves</a></li><li><a href="/wiki/Category:Paper_folding" title="Category:Paper folding">Paper folding</a></li><li><a href="/wiki/Category:L-systems" title="Category:L-systems">L-systems</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_May_2019" title="Category:Use dmy dates from May 2019">Use dmy dates from May 2019</a></li><li><a href="/wiki/Category:Pages_using_multiple_image_with_auto_scaled_images" title="Category:Pages using multiple image with auto scaled images">Pages using multiple image with auto scaled images</a></li><li><a href="/wiki/Category:Commons_link_from_Wikidata" title="Category:Commons link from Wikidata">Commons link from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 25 November 2024, at 22:12<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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