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class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Brownian_motion_on_the_Sierpiński_carpet" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Brownian_motion_on_the_Sierpiński_carpet"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Brownian motion on the Sierpiński carpet</span> </div> </a> <ul id="toc-Brownian_motion_on_the_Sierpiński_carpet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wallis_sieve" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Wallis_sieve"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Wallis sieve</span> </div> </a> <ul id="toc-Wallis_sieve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-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" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Sierpiński carpet</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 28 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-28" 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">28 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D8%B1%D8%A8%D9%8A%D8%A9_%D8%B3%D9%8A%D8%B1%D8%A8%D9%86%D8%B3%D9%83%D9%8A" title="زربية سيربنسكي – Arabic" lang="ar" hreflang="ar" data-title="زربية سيربنسكي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%8B%D0%B2%D0%B0%D0%BD_%D0%A1%D1%8F%D1%80%D0%BF%D1%96%D0%BD%D1%81%D0%BA%D0%B0%D0%B3%D0%B0" title="Дыван Сярпінскага – Belarusian" lang="be" hreflang="be" data-title="Дыван Сярпінскага" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Catifa_de_Sierpinski" title="Catifa de Sierpinski – Catalan" lang="ca" hreflang="ca" data-title="Catifa de Sierpinski" 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/Sierpi%C5%84sk%C3%A9ho_koberec" title="Sierpińského koberec – Czech" lang="cs" hreflang="cs" data-title="Sierpińského koberec" 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/Sierpinski-Teppich" title="Sierpinski-Teppich – German" lang="de" hreflang="de" data-title="Sierpinski-Teppich" 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%A7%CE%B1%CE%BB%CE%AF_%CF%84%CE%BF%CF%85_%CE%A3%CE%B9%CE%B5%CF%81%CF%80%CE%AF%CE%BD%CF%83%CE%BA%CE%B9" title="Χαλί του Σιερπίνσκι – Greek" lang="el" hreflang="el" data-title="Χαλί του Σιερπίνσκι" 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/Alfombra_de_Sierpinski" title="Alfombra de Sierpinski – Spanish" lang="es" hreflang="es" data-title="Alfombra de Sierpinski" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Tapi%C5%9Do_de_Sjerpinski" title="Tapiŝo de Sjerpinski – Esperanto" lang="eo" hreflang="eo" data-title="Tapiŝo de Sjerpinski" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Sierpinskiren_tapiza" title="Sierpinskiren tapiza – Basque" lang="eu" hreflang="eu" data-title="Sierpinskiren tapiza" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%A7%D9%84%DB%8C_%D8%B4%D8%B1%D9%BE%DB%8C%D9%86%D8%B3%DA%A9%DB%8C" title="قالی شرپینسکی – Persian" lang="fa" hreflang="fa" data-title="قالی شرپینسکی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Tapis_de_Sierpi%C5%84ski" title="Tapis de Sierpiński – French" lang="fr" hreflang="fr" data-title="Tapis de Sierpiński" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8B%9C%EC%97%90%EB%A5%B4%ED%95%80%EC%8A%A4%ED%82%A4_%EC%B9%B4%ED%8E%AB" title="시에르핀스키 카펫 – Korean" lang="ko" hreflang="ko" data-title="시에르핀스키 카펫" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Tepih_Sierpi%C5%84skog" title="Tepih Sierpińskog – Croatian" lang="hr" hreflang="hr" data-title="Tepih Sierpińskog" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Karpet_Sierpi%C5%84ski" title="Karpet Sierpiński – Indonesian" lang="id" hreflang="id" data-title="Karpet Sierpiński" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tappeto_di_Sierpinski" title="Tappeto di Sierpinski – Italian" lang="it" hreflang="it" data-title="Tappeto di Sierpinski" 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%A9%D7%98%D7%99%D7%97_%D7%A9%D7%A8%D7%A4%D7%99%D7%A0%D7%A1%D7%A7%D7%99" title="שטיח שרפינסקי – Hebrew" lang="he" hreflang="he" data-title="שטיח שרפינסקי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sierpi%C5%84ski-sz%C5%91nyeg" title="Sierpiński-szőnyeg – Hungarian" lang="hu" hreflang="hu" data-title="Sierpiński-szőnyeg" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tapijt_van_Sierpi%C5%84ski" title="Tapijt van Sierpiński – Dutch" lang="nl" hreflang="nl" data-title="Tapijt van Sierpiński" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B7%E3%82%A7%E3%83%AB%E3%83%94%E3%83%B3%E3%82%B9%E3%82%AD%E3%83%BC%E3%81%AE%E3%82%AB%E3%83%BC%E3%83%9A%E3%83%83%E3%83%88" title="シェルピンスキーのカーペット – Japanese" lang="ja" hreflang="ja" data-title="シェルピンスキーのカーペット" 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/Dywan_Sierpi%C5%84skiego" title="Dywan Sierpińskiego – Polish" lang="pl" hreflang="pl" data-title="Dywan Sierpińskiego" 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/Tapete_de_Sierpinski" title="Tapete de Sierpinski – Portuguese" lang="pt" hreflang="pt" data-title="Tapete de Sierpinski" 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%D0%BE%D0%B2%D1%91%D1%80_%D0%A1%D0%B5%D1%80%D0%BF%D0%B8%D0%BD%D1%81%D0%BA%D0%BE%D0%B3%D0%BE" title="Ковёр Серпинского – Russian" lang="ru" hreflang="ru" data-title="Ковёр Серпинского" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%DB%95%DA%95%D8%B4%DB%8C_%D8%B3%DB%8C%D8%B1%D9%BE%DB%8C%D9%86%D8%B3%DA%A9%DB%8C" title="فەڕشی سیرپینسکی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="فەڕشی سیرپینسکی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BF%D0%B8%D1%85_%D0%A1%D1%98%D0%B5%D1%80%D0%BF%D0%B8%D1%9A%D1%81%D0%BA%D0%BE%D0%B3" title="Тепих Сјерпињског – Serbian" lang="sr" hreflang="sr" data-title="Тепих Сјерпињског" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Tepih_Sierpi%C5%84skog" title="Tepih Sierpińskog – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Tepih Sierpińskog" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%B8%D0%BB%D0%B8%D0%BC_%D0%A1%D0%B5%D1%80%D0%BF%D1%96%D0%BD%D1%81%D1%8C%D0%BA%D0%BE%D0%B3%D0%BE" title="Килим Серпінського – Ukrainian" lang="uk" hreflang="uk" data-title="Килим Серпінського" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BA%A5m_th%E1%BA%A3m_Sierpinski" title="Tấm thảm Sierpinski – Vietnamese" lang="vi" hreflang="vi" data-title="Tấm thảm Sierpinski" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%B0%A2%E5%B0%94%E5%AE%BE%E6%96%AF%E5%9F%BA%E5%9C%B0%E6%AF%AF" title="谢尔宾斯基地毯 – Chinese" lang="zh" hreflang="zh" data-title="谢尔宾斯基地毯" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q280081#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Sierpi%C5%84ski_carpet" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li 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.hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Sierpinski snowflake" redirects here. For other uses, see <a href="/wiki/Sierpi%C5%84ski_curve" title="Sierpiński curve">Sierpiński curve</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Animated_Sierpinski_carpet.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Animated_Sierpinski_carpet.gif/220px-Animated_Sierpinski_carpet.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Animated_Sierpinski_carpet.gif/330px-Animated_Sierpinski_carpet.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Animated_Sierpinski_carpet.gif/440px-Animated_Sierpinski_carpet.gif 2x" data-file-width="750" data-file-height="750" /></a><figcaption>6 steps of a Sierpiński carpet.</figcaption></figure> <p>The <b>Sierpiński carpet</b> is a plane <a href="/wiki/Fractal" title="Fractal">fractal</a> first described by <a href="/wiki/Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Wacław Sierpiński</a> in 1916. The carpet is a generalization of the <a href="/wiki/Cantor_set" title="Cantor set">Cantor set</a> to two dimensions; another such generalization is the <a href="/wiki/Cantor_dust" class="mw-redirect" title="Cantor dust">Cantor dust</a>. </p><p>The technique of <a href="/wiki/Rep-tile" title="Rep-tile">subdividing a shape into smaller copies of itself</a>, removing one or more copies, and continuing <a href="/wiki/Recursion" title="Recursion">recursively</a> can be extended to other shapes. For instance, subdividing an equilateral triangle into four equilateral triangles, removing the middle triangle, and recursing leads to the <a href="/wiki/Sierpi%C5%84ski_triangle" title="Sierpiński triangle">Sierpiński triangle</a>. In three dimensions, a similar construction based on cubes is known as the <a href="/wiki/Menger_sponge" title="Menger sponge">Menger sponge</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sierpi%C5%84ski_carpet&action=edit&section=1" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The construction of the Sierpiński carpet begins with a <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a>. The square is cut into 9 <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> subsquares in a 3-by-3 grid, and the central subsquare is removed. The same procedure is then applied <a href="/wiki/Recursion" title="Recursion">recursively</a> to the remaining 8 subsquares, <i>ad infinitum</i>. It can be realised as the set of points in the unit square whose coordinates written in base three do not both have a digit '1' in the same position, using the infinitesimal number representation 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 0.1111\dots =0.2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.1111</mn> <mo>⋯</mo> <mo>=</mo> <mn>0.2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.1111\dots =0.2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4354ed02fe19dcb1d9e338acdbcf1895cd6fb55b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.64ex; height:2.176ex;" alt="{\displaystyle 0.1111\dots =0.2}"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The process of recursively removing squares is an example of a <a href="/wiki/Finite_subdivision_rule" title="Finite subdivision rule">finite subdivision rule</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sierpi%C5%84ski_carpet&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Peano_Sierpinski_carpet_4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Peano_Sierpinski_carpet_4.svg/220px-Peano_Sierpinski_carpet_4.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Peano_Sierpinski_carpet_4.svg/330px-Peano_Sierpinski_carpet_4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Peano_Sierpinski_carpet_4.svg/440px-Peano_Sierpinski_carpet_4.svg.png 2x" data-file-width="1868" data-file-height="1868" /></a><figcaption>Variant of the <a href="/wiki/Peano_curve" title="Peano curve">Peano curve</a> with the middle line erased creates a Sierpiński carpet</figcaption></figure> <p>The area of the carpet is zero (in standard <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>). </p> <dl><dd><b>Proof:</b> Denote as <span class="texhtml mvar" style="font-style:italic;">a<sub>i</sub></span> the area of iteration <span class="texhtml mvar" style="font-style:italic;">i</span>. Then <span class="texhtml"><i>a</i><sub><i>i</i> + 1</sub> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">8</span><span class="sr-only">/</span><span class="den">9</span></span>⁠</span><i>a<sub>i</sub></i></span>. So <span class="texhtml"><i>a<sub>i</sub></i> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">8</span><span class="sr-only">/</span><span class="den">9</span></span>⁠</span>)<sup><i>i</i></sup></span>, which tends to 0 as <span class="texhtml mvar" style="font-style:italic;">i</span> goes to infinity.</dd></dl> <p>The <a href="/wiki/Interior_(topology)" title="Interior (topology)">interior</a> of the carpet is empty. </p> <dl><dd><b>Proof:</b> Suppose by contradiction that there is a point <span class="texhtml mvar" style="font-style:italic;">P</span> in the interior of the carpet. Then there is a square centered at <span class="texhtml mvar" style="font-style:italic;">P</span> which is entirely contained in the carpet. This square contains a smaller square whose coordinates are multiples of <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3<sup><i>k</i></sup></span></span>⁠</span></span> for some <span class="texhtml mvar" style="font-style:italic;">k</span>. But, if this square has not been previously removed, it must have been holed in iteration <span class="texhtml"><i>k</i> + 1</span>, so it cannot be contained in the carpet – a contradiction.</dd></dl> <p>The <a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a> of the carpet is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\log 8}{\log 3}}\approx 1.8928}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>log</mi> <mo>⁡</mo> <mn>8</mn> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>1.8928</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\log 8}{\log 3}}\approx 1.8928}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cc61ff79faba59d7ef726401bb80c0ac0a78c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.915ex; height:5.843ex;" alt="{\displaystyle {\frac {\log 8}{\log 3}}\approx 1.8928}"></span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Sierpiński demonstrated that his carpet is a universal plane curve.<sup id="cite_ref-sierpinski_3-0" class="reference"><a href="#cite_note-sierpinski-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> That is: the Sierpiński carpet is a compact subset of the plane with <a href="/wiki/Lebesgue_covering_dimension" title="Lebesgue covering dimension">Lebesgue covering dimension</a> 1, and every subset of the plane with these properties is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to some subset of the Sierpiński carpet. </p><p>This "universality" of the Sierpiński carpet is not a true universal property in the sense of category theory: it does not uniquely characterize this space up to homeomorphism. For example, the disjoint union of a Sierpiński carpet and a circle is also a universal plane curve. However, in 1958 <a href="/wiki/Gordon_Whyburn" title="Gordon Whyburn">Gordon Whyburn</a><sup id="cite_ref-whyburn_4-0" class="reference"><a href="#cite_note-whyburn-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> uniquely characterized the Sierpiński carpet as follows: any curve that is <a href="/wiki/Locally_connected" class="mw-redirect" title="Locally connected">locally connected</a> and has no 'local cut-points' is homeomorphic to the Sierpiński carpet. Here a <b>local cut-point</b> is a point <span class="texhtml mvar" style="font-style:italic;">p</span> for which some connected neighborhood <span class="texhtml mvar" style="font-style:italic;">U</span> of <span class="texhtml mvar" style="font-style:italic;">p</span> has the property that <span class="texhtml"><i>U</i> − {<i>p</i>} </span> is not connected. So, for example, any point of the circle is a local cut point. </p><p>In the same paper Whyburn gave another characterization of the Sierpiński carpet. Recall that a <a href="/wiki/Continuum_(topology)" title="Continuum (topology)">continuum</a> is a nonempty connected compact metric space. Suppose <span class="texhtml mvar" style="font-style:italic;">X</span> is a continuum embedded in the plane. Suppose its complement in the plane has countably many connected components <span class="texhtml"><i>C</i><sub>1</sub>, <i>C</i><sub>2</sub>, <i>C</i><sub>3</sub>, ...</span> and suppose: </p> <ul><li>the diameter of <span class="texhtml mvar" style="font-style:italic;">C<sub>i</sub></span> goes to zero as <span class="texhtml"><i>i</i> → ∞</span>;</li> <li>the boundary of <span class="texhtml mvar" style="font-style:italic;">C<sub>i</sub></span> and the boundary of <span class="texhtml mvar" style="font-style:italic;">C<sub>j</sub></span> are disjoint if <span class="texhtml"><i>i</i> ≠ <i>j</i></span>;</li> <li>the boundary of <span class="texhtml mvar" style="font-style:italic;">C<sub>i</sub></span> is a simple closed curve for each <span class="texhtml mvar" style="font-style:italic;">i</span>;</li> <li>the union of the boundaries of the sets <span class="texhtml mvar" style="font-style:italic;">C<sub>i</sub></span> is dense in <span class="texhtml mvar" style="font-style:italic;">X</span>.</li></ul> <p>Then <span class="texhtml mvar" style="font-style:italic;">X</span> is homeomorphic to the Sierpiński carpet. </p> <div class="mw-heading mw-heading2"><h2 id="Brownian_motion_on_the_Sierpiński_carpet"><span id="Brownian_motion_on_the_Sierpi.C5.84ski_carpet"></span>Brownian motion on the Sierpiński carpet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sierpi%C5%84ski_carpet&action=edit&section=3" title="Edit section: Brownian motion on the Sierpiński carpet"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The topic of <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> on the Sierpiński carpet has attracted interest in recent years.<sup id="cite_ref-barlow_5-0" class="reference"><a href="#cite_note-barlow-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Martin Barlow and Richard Bass have shown that a <a href="/wiki/Random_walk" title="Random walk">random walk</a> on the Sierpiński carpet diffuses at a slower rate than an unrestricted random walk in the plane. The latter reaches a mean distance proportional to <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>n</i></span></span></span> after <span class="texhtml mvar" style="font-style:italic;">n</span> steps, but the random walk on the discrete Sierpiński carpet reaches only a mean distance proportional to <span class="texhtml"><span class="nowrap"><sup style="margin-right: -0.5em; vertical-align: 0.8em;"><i>β</i></sup>√<span style="border-top:1px solid; padding:0 0.1em;"><i>n</i></span></span></span> for some <span class="texhtml"><i>β</i> > 2</span>. They also showed that this random walk satisfies stronger <a href="/wiki/Large_deviations_theory" title="Large deviations theory">large deviation</a> inequalities (so called "sub-Gaussian inequalities") and that it satisfies the elliptic <a href="/wiki/Harnack_inequality" class="mw-redirect" title="Harnack inequality">Harnack inequality</a> without satisfying the parabolic one. The existence of such an example was an open problem for many years. </p> <div class="mw-heading mw-heading2"><h2 id="Wallis_sieve">Wallis sieve</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sierpi%C5%84ski_carpet&action=edit&section=4" title="Edit section: Wallis sieve"><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:Wallis_sieve_iteration_3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Wallis_sieve_iteration_3.png/220px-Wallis_sieve_iteration_3.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Wallis_sieve_iteration_3.png/330px-Wallis_sieve_iteration_3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Wallis_sieve_iteration_3.png/440px-Wallis_sieve_iteration_3.png 2x" data-file-width="2016" data-file-height="2016" /></a><figcaption>Third iteration of the Wallis sieve</figcaption></figure> <p>A variation of the Sierpiński carpet, called the <b>Wallis sieve</b>, starts in the same way, by subdividing the unit square into nine smaller squares and removing the middle of them. At the next level of subdivision, it subdivides each of the squares into 25 smaller squares and removes the middle one, and it continues at the <span class="texhtml mvar" style="font-style:italic;">i</span>th step by subdividing each square into <span class="texhtml">(2<i>i</i> + 1)<sup>2</sup></span> (the <a href="/wiki/Square_number" title="Square number">odd squares</a><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>) smaller squares and removing the middle one. By the <a href="/wiki/Wallis_product" title="Wallis product">Wallis product</a>, the area of the resulting set is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span>, unlike the standard Sierpiński carpet which has zero limiting area. Although the Wallis sieve has positive <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>, no subset that is a <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of two sets of real numbers has this property, so its <a href="/wiki/Jordan_measure" class="mw-redirect" title="Jordan measure">Jordan measure</a> is zero.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sierpi%C5%84ski_carpet&action=edit&section=5" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mobile phone and <a href="/wiki/Wi-Fi" title="Wi-Fi">Wi-Fi</a> <a href="/wiki/Fractal_antenna" title="Fractal antenna">fractal antennas</a> have been produced in the form of few iterations of the Sierpiński carpet. Due to their <a href="/wiki/Self-similarity" title="Self-similarity">self-similarity</a> and <a href="/wiki/Scale_invariance" title="Scale invariance">scale invariance</a>, they easily accommodate multiple frequencies. They are also easy to fabricate and smaller than conventional antennas of similar performance, thus being optimal for pocket-sized mobile phones.<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><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <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=Sierpi%C5%84ski_carpet&action=edit&section=6" 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/Menger_sponge" title="Menger sponge">Menger sponge</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=Sierpi%C5%84ski_carpet&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAlloucheShallit2003" class="citation book cs1">Allouche, Jean-Paul; <a href="/wiki/Jeffrey_Shallit" title="Jeffrey Shallit">Shallit, Jeffrey</a> (2003). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/automaticsequenc00jpal"><i>Automatic Sequences: Theory, Applications, Generalizations</i></a></span>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/automaticsequenc00jpal/page/n422">405</a>–406. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-82332-6" title="Special:BookSources/978-0-521-82332-6"><bdi>978-0-521-82332-6</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1086.11015">1086.11015</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Automatic+Sequences%3A+Theory%2C+Applications%2C+Generalizations&rft.pages=405-406&rft.pub=Cambridge+University+Press&rft.date=2003&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1086.11015%23id-name%3DZbl&rft.isbn=978-0-521-82332-6&rft.aulast=Allouche&rft.aufirst=Jean-Paul&rft.au=Shallit%2C+Jeffrey&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fautomaticsequenc00jpal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASierpi%C5%84ski+carpet" 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="CITEREFSemmes2001" class="citation book cs1"><a href="/wiki/Stephen_Semmes" title="Stephen Semmes">Semmes, Stephen</a> (2001). <i>Some Novel Types of Fractal Geometry</i>. Oxford Mathematical Monographs. Oxford University Press. p. 31. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850806-9" title="Special:BookSources/0-19-850806-9"><bdi>0-19-850806-9</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0970.28001">0970.28001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Some+Novel+Types+of+Fractal+Geometry&rft.series=Oxford+Mathematical+Monographs&rft.pages=31&rft.pub=Oxford+University+Press&rft.date=2001&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0970.28001%23id-name%3DZbl&rft.isbn=0-19-850806-9&rft.aulast=Semmes&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASierpi%C5%84ski+carpet" class="Z3988"></span></span> </li> <li id="cite_note-sierpinski-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-sierpinski_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSierpiński1916" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Sierpiński, Wacław</a> (1916). "Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée". <i>C. R. Acad. Sci. Paris</i> (in French). <b>162</b>: 629–632. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-4036">0001-4036</a>. <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:46.0295.02">46.0295.02</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=C.+R.+Acad.+Sci.+Paris&rft.atitle=Sur+une+courbe+cantorienne+qui+contient+une+image+biunivoque+et+continue+de+toute+courbe+donn%C3%A9e&rft.volume=162&rft.pages=629-632&rft.date=1916&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A46.0295.02%23id-name%3DJFM&rft.issn=0001-4036&rft.aulast=Sierpi%C5%84ski&rft.aufirst=Wac%C5%82aw&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASierpi%C5%84ski+carpet" class="Z3988"></span></span> </li> <li id="cite_note-whyburn-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-whyburn_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhyburn1958" class="citation journal cs1"><a href="/wiki/Gordon_Whyburn" title="Gordon Whyburn">Whyburn, Gordon</a> (1958). <a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Ffm-45-1-320-324">"Topological chcracterization of the Sierpinski curve"</a>. <i>Fund. Math</i>. <b>45</b>: 320–324. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Ffm-45-1-320-324">10.4064/fm-45-1-320-324</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fund.+Math.&rft.atitle=Topological+chcracterization+of+the+Sierpinski+curve&rft.volume=45&rft.pages=320-324&rft.date=1958&rft_id=info%3Adoi%2F10.4064%2Ffm-45-1-320-324&rft.aulast=Whyburn&rft.aufirst=Gordon&rft_id=https%3A%2F%2Fdoi.org%2F10.4064%252Ffm-45-1-320-324&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASierpi%C5%84ski+carpet" class="Z3988"></span></span> </li> <li id="cite_note-barlow-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-barlow_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarlowBass" class="citation cs2">Barlow, Martin; Bass, Richard, <a rel="nofollow" class="external text" href="https://www.math.ubc.ca/~barlow/preprints/61_sch_pub.pdf"><i>Brownian motion and harmonic analysis on Sierpiński carpets</i></a> <span class="cs1-format">(PDF)</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Brownian+motion+and+harmonic+analysis+on+Sierpi%C5%84ski+carpets&rft.aulast=Barlow&rft.aufirst=Martin&rft.au=Bass%2C+Richard&rft_id=https%3A%2F%2Fwww.math.ubc.ca%2F~barlow%2Fpreprints%2F61_sch_pub.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASierpi%C5%84ski+carpet" 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="CITEREFSloane_"A016754"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A016754">"Sequence A016754 (Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.)"</a>. <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA016754%26%23x20%3B%28Odd+squares%3A+a%28n%29+%3D+%282n%2B1%29%5E2.+Also+centered+octagonal+numbers.%29&rft_id=https%3A%2F%2Foeis.org%2FA016754&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASierpi%C5%84ski+carpet" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRummler1993" class="citation journal cs1">Rummler, Hansklaus (1993). "Squaring the circle with holes". <i><a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>. <b>100</b> (9): 858–860. <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%2F2324662">10.2307/2324662</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/2324662">2324662</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=1247533">1247533</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=Squaring+the+circle+with+holes&rft.volume=100&rft.issue=9&rft.pages=858-860&rft.date=1993&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1247533%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2324662%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2324662&rft.aulast=Rummler&rft.aufirst=Hansklaus&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASierpi%C5%84ski+carpet" 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">N. A. Saidatul, A. A. H. Azremi, R. B. Ahmad, P. J. Soh and F. Malek, "A development of Fractal PIFA (planar inverted F antenna) with bandwidth enhancement for mobile phone applications," 2009 <i>Loughborough Antennas & Propagation Conference</i>, Loughborough, UK, 2009, pp. 113-116, doi: 10.1109/LAPC.2009.5352584.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">T. Kalaimani, P. M. Venkatesh, R. Mohanamurali and T. Shanmuganantham, "A modified Sierpinski carpet fractal antenna for wireless applications," 2013 <i>International Conference on Communication and Signal Processing</i>, Melmaruvathur, India, 2013, pp. 722-725, doi: 10.1109/iccsp.2013.6577150.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">W. -L. Chen, G. -M. Wang and C. -X. Zhang, "Small-Size Microstrip Patch Antennas Combining Koch and Sierpinski Fractal-Shapes," in <i>IEEE Antennas and Wireless Propagation Letters</i>, vol. 7, pp. 738-741, 2008, doi: 10.1109/LAWP.2008.2002808.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sierpi%C5%84ski_carpet&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style 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href="/wiki/Template:Fractals" title="Template:Fractals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Fractals" title="Template talk:Fractals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Fractals" title="Special:EditPage/Template:Fractals"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Fractals" style="font-size:114%;margin:0 4em"><a href="/wiki/Fractal" title="Fractal">Fractals</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Characteristics</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/Fractal_dimension" title="Fractal dimension">Fractal dimensions</a> <ul><li><a href="/wiki/Assouad_dimension" title="Assouad dimension">Assouad</a></li> <li><a href="/wiki/Minkowski%E2%80%93Bouligand_dimension" 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 class="mw-selflink selflink">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 href="/wiki/Dragon_curve" title="Dragon curve">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><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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