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<span>Two-dimensional</span> </div> </a> <button aria-controls="toc-Two-dimensional-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Two-dimensional subsection</span> </button> <ul id="toc-Two-dimensional-sublist" class="vector-toc-list"> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Geometric properties</span> </div> </a> <ul id="toc-Geometric_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bounded_spirals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bounded_spirals"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Bounded spirals</span> </div> </a> <ul id="toc-Bounded_spirals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Three-dimensional" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Three-dimensional"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Three-dimensional</span> </div> </a> <button aria-controls="toc-Three-dimensional-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Three-dimensional subsection</span> </button> <ul id="toc-Three-dimensional-sublist" class="vector-toc-list"> <li id="toc-Conical_spirals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conical_spirals"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Conical spirals</span> </div> </a> <ul id="toc-Conical_spirals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spherical_spirals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spherical_spirals"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Spherical spirals</span> </div> </a> <ul id="toc-Spherical_spirals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_nature" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_nature"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>In nature</span> </div> </a> <ul id="toc-In_nature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_a_symbol" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#As_a_symbol"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>As a symbol</span> </div> </a> <ul id="toc-As_a_symbol-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_art" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_art"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>In art</span> </div> </a> <ul id="toc-In_art-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">7</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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_publications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_publications"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Related publications</span> </div> </a> <ul id="toc-Related_publications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Spiral</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 51 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-51" 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">51 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Spiraal" title="Spiraal – Afrikaans" lang="af" hreflang="af" data-title="Spiraal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%84%D8%B2%D9%88%D9%86_(%D9%87%D9%86%D8%AF%D8%B3%D8%A9)" title="حلزون (هندسة) – Arabic" lang="ar" hreflang="ar" data-title="حلزون (هندسة)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Spiral" title="Spiral – Azerbaijani" lang="az" hreflang="az" data-title="Spiral" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%B0%E0%A7%8D%E0%A6%AA%E0%A6%BF%E0%A6%B2" title="সর্পিল – Bangla" lang="bn" hreflang="bn" data-title="সর্পিল" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D0%BF%D1%96%D1%80%D0%B0%D0%BB%D1%8C" title="Спіраль – Belarusian" lang="be" hreflang="be" data-title="Спіраль" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D1%80%D0%B0%D0%BB%D0%B0" title="Спирала – Bulgarian" lang="bg" hreflang="bg" data-title="Спирала" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Spirala" title="Spirala – Bosnian" lang="bs" hreflang="bs" data-title="Spirala" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espiral" title="Espiral – Catalan" lang="ca" hreflang="ca" data-title="Espiral" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D1%80%D0%B0%D0%BB%D1%8C" title="Спираль – Chuvash" lang="cv" hreflang="cv" data-title="Спираль" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Spir%C3%A1la" title="Spirála – Czech" lang="cs" hreflang="cs" data-title="Spirála" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Sbiral" title="Sbiral – Welsh" lang="cy" hreflang="cy" data-title="Sbiral" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Spiral_(matematik)" title="Spiral (matematik) – Danish" lang="da" hreflang="da" data-title="Spiral (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Spirale" title="Spirale – German" lang="de" hreflang="de" data-title="Spirale" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Spiraal" title="Spiraal – Estonian" lang="et" hreflang="et" data-title="Spiraal" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espiral" title="Espiral – Spanish" lang="es" hreflang="es" data-title="Espiral" 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/Spiralo" title="Spiralo – Esperanto" lang="eo" hreflang="eo" data-title="Spiralo" 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/Kiribil" title="Kiribil – Basque" lang="eu" hreflang="eu" data-title="Kiribil" 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%85%D8%A7%D8%B1%D9%BE%DB%8C%DA%86" title="مارپیچ – Persian" lang="fa" hreflang="fa" data-title="مارپیچ" 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/Spirale" title="Spirale – French" lang="fr" hreflang="fr" data-title="Spirale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Spiraal_(wiskunde)" title="Spiraal (wiskunde) – Western Frisian" lang="fy" hreflang="fy" data-title="Spiraal (wiskunde)" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espiral" title="Espiral – Galician" lang="gl" hreflang="gl" data-title="Espiral" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%99%80%EC%84%A0" title="와선 – Korean" lang="ko" hreflang="ko" data-title="와선" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%B0%E0%A5%8D%E0%A4%AA%E0%A4%BF%E0%A4%B2" title="सर्पिल – Hindi" lang="hi" hreflang="hi" data-title="सर्पिल" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Spiralo" title="Spiralo – Ido" lang="io" hreflang="io" data-title="Spiralo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pilin" title="Pilin – Indonesian" lang="id" hreflang="id" data-title="Pilin" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Ku%C3%B0ungsferill" title="Kuðungsferill – Icelandic" lang="is" hreflang="is" data-title="Kuðungsferill" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spirale" title="Spirale – Italian" lang="it" hreflang="it" data-title="Spirale" 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%A1%D7%A4%D7%99%D7%A8%D7%9C%D7%94" title="ספירלה – Hebrew" lang="he" hreflang="he" data-title="ספירלה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A8%D0%B8%D1%8B%D1%80%D1%88%D1%8B%D2%9B" title="Шиыршық – Kazakh" lang="kk" hreflang="kk" data-title="Шиыршық" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Spiralis" title="Spiralis – Latin" lang="la" hreflang="la" data-title="Spiralis" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Spir%C3%A1l" title="Spirál – Hungarian" lang="hu" hreflang="hu" data-title="Spirál" 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/Spiraal_(wiskunde)" title="Spiraal (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Spiraal (wiskunde)" 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/%E6%B8%A6%E5%B7%BB" title="渦巻 – Japanese" lang="ja" hreflang="ja" data-title="渦巻" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Spiral" title="Spiral – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Spiral" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Spiral" title="Spiral – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Spiral" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Spirallar" title="Spirallar – Uzbek" lang="uz" hreflang="uz" data-title="Spirallar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espiral" title="Espiral – Portuguese" lang="pt" hreflang="pt" data-title="Espiral" 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%A1%D0%BF%D0%B8%D1%80%D0%B0%D0%BB%D1%8C" title="Спираль – Russian" lang="ru" hreflang="ru" data-title="Спираль" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Spiral" title="Spiral – Simple English" lang="en-simple" hreflang="en-simple" data-title="Spiral" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/%C5%A0pir%C3%A1la_(matematika)" title="Špirála (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Špirála (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Spirala" title="Spirala – Slovenian" lang="sl" hreflang="sl" data-title="Spirala" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%D9%88%D9%88%D9%84%D9%BE%DB%8E%DA%86_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="لوولپێچ (ماتماتیک) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="لوولپێچ (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Spirala" title="Spirala – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Spirala" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Spiraali" title="Spiraali – Finnish" lang="fi" hreflang="fi" data-title="Spiraali" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Spiral" title="Spiral – Swedish" lang="sv" hreflang="sv" data-title="Spiral" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Sinuso_(hugis)" title="Sinuso (hugis) – Tagalog" lang="tl" hreflang="tl" data-title="Sinuso (hugis)" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Spiral" title="Spiral – Turkish" lang="tr" hreflang="tr" data-title="Spiral" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BF%D1%96%D1%80%D0%B0%D0%BB%D1%8C" title="Спіраль – Ukrainian" lang="uk" hreflang="uk" data-title="Спіраль" 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/Xo%E1%BA%AFn_%E1%BB%91c" title="Xoắn ốc – Vietnamese" lang="vi" hreflang="vi" data-title="Xoắn ốc" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </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">Curve that winds around a central point</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">For other uses, see <a href="/wiki/Spiral_(disambiguation)" class="mw-disambig" title="Spiral (disambiguation)">Spiral (disambiguation)</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:NautilusCutawayLogarithmicSpiral.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/NautilusCutawayLogarithmicSpiral.jpg/220px-NautilusCutawayLogarithmicSpiral.jpg" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/NautilusCutawayLogarithmicSpiral.jpg/330px-NautilusCutawayLogarithmicSpiral.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/NautilusCutawayLogarithmicSpiral.jpg/440px-NautilusCutawayLogarithmicSpiral.jpg 2x" data-file-width="2240" data-file-height="1693" /></a><figcaption>Cutaway of a <a href="/wiki/Nautilus" title="Nautilus">nautilus</a> shell showing the chambers arranged in an approximately <a href="/wiki/Logarithmic_spiral" title="Logarithmic spiral">logarithmic spiral</a></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>spiral</b> is a <a href="/wiki/Curve" title="Curve">curve</a> which emanates from a point, moving further away as it revolves around the point.<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><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><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><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> It is a subtype of <a href="/wiki/Whorl" title="Whorl">whorled</a> patterns, a broad group that also includes <a href="/wiki/Concentric_objects" title="Concentric objects">concentric objects</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Helices">Helices <span class="anchor" id="Spiral_or_helix"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=1" title="Edit section: Helices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Schraube_und_archimedische_Spirale.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Schraube_und_archimedische_Spirale.png/220px-Schraube_und_archimedische_Spirale.png" decoding="async" width="220" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/72/Schraube_und_archimedische_Spirale.png 1.5x" data-file-width="254" data-file-height="178" /></a><figcaption>An Archimedean spiral (black), a helix (green), and a conical spiral (red)</figcaption></figure> <p>Two major definitions of "spiral" in the <a href="/wiki/American_Heritage_Dictionary" class="mw-redirect" title="American Heritage Dictionary">American Heritage Dictionary</a> are:<sup id="cite_ref-free_5-0" class="reference"><a href="#cite_note-free-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <ol><li>a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.</li> <li>a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a <a href="/wiki/Helix" title="Helix">helix</a>.</li></ol> <p>The first definition describes a <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">planar</a> curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a <a href="/wiki/Gramophone_record" class="mw-redirect" title="Gramophone record">gramophone record</a> closely approximates a plane spiral (and it is by the finite width and depth of the groove, but <i>not</i> by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loops <i>differ</i> in diameter. In another example, the "center lines" of the arms of a <a href="/wiki/Spiral_galaxy" title="Spiral galaxy">spiral galaxy</a> trace <a href="/wiki/Logarithmic_spiral" title="Logarithmic spiral">logarithmic spirals</a>. </p><p>The second definition includes two kinds of 3-dimensional relatives of spirals: </p> <ul><li>A conical or <a href="/wiki/Volute_spring" title="Volute spring">volute spring</a> (including the spring used to hold and make contact with the negative terminals of AA or AAA batteries in a <a href="/wiki/Battery_(electricity)" class="mw-redirect" title="Battery (electricity)">battery box</a>), and the <a href="/wiki/Vortex" title="Vortex">vortex</a> that is created when water is draining in a sink is often described as a spiral, or as a <a href="/wiki/Conical_helix" class="mw-redirect" title="Conical helix">conical helix</a>.</li> <li>Quite explicitly, definition 2 also includes a cylindrical coil spring and a strand of <a href="/wiki/DNA" title="DNA">DNA</a>, both of which are fairly helical, so that "helix" is a more <i>useful</i> description than "spiral" for each of them. In general, "spiral" is seldom applied if successive "loops" of a curve have the same diameter.<sup id="cite_ref-free_5-1" class="reference"><a href="#cite_note-free-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <p>In the side picture, the black curve at the bottom is an <a href="/wiki/Archimedean_spiral" title="Archimedean spiral">Archimedean spiral</a>, while the green curve is a helix. The curve shown in red is a conical spiral. </p> <div class="mw-heading mw-heading2"><h2 id="Two-dimensional">Two-dimensional</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=2" title="Edit section: Two-dimensional"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_spirals" title="List of spirals">List of spirals</a></div> <p>A <a href="/wiki/Two-dimensional" class="mw-redirect" title="Two-dimensional">two-dimensional</a>, or plane, spiral may be easily described using <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a>, where the <a href="/wiki/Radius" title="Radius">radius</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is a <a href="/wiki/Monotonic" class="mw-redirect" title="Monotonic">monotonic</a> <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> of angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r(\varphi )\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r(\varphi )\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2337675d40502693aea9eb0fa44aaef43ae4153f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.817ex; height:2.843ex;" alt="{\displaystyle r=r(\varphi )\;.}"></span></li></ul> <p>The circle would be regarded as a <a href="/wiki/Degenerate_(mathematics)" class="mw-redirect" title="Degenerate (mathematics)">degenerate</a> case (the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> not being strictly monotonic, but rather <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a>). </p><p>In <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-coordinates</i> the curve has the parametric representation: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/990c66f4d0c9537cc5fbb4f75a601ba71e72ef4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.545ex; height:2.843ex;" alt="{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \;.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some of the most important sorts of two-dimensional spirals include: </p> <ul><li>The <a href="/wiki/Archimedean_spiral" title="Archimedean spiral">Archimedean spiral</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1da291348acd4ad9857d6d4f4c57e6ffc86ba547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.897ex; height:2.176ex;" alt="{\displaystyle r=a\varphi }"></span></li> <li>The <a href="/wiki/Hyperbolic_spiral" title="Hyperbolic spiral">hyperbolic spiral</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a/\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a/\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dee3f159394eef32e7c0d7fce6bd018d8db9d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.06ex; height:2.843ex;" alt="{\displaystyle r=a/\varphi }"></span></li> <li><a href="/wiki/Fermat%27s_spiral" title="Fermat's spiral">Fermat's spiral</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a\varphi ^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a\varphi ^{1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bdeedda974df62016bf773b55d48c7e090b4342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.595ex; height:3.343ex;" alt="{\displaystyle r=a\varphi ^{1/2}}"></span></li> <li>The <a href="/wiki/Lituus_(mathematics)" title="Lituus (mathematics)">lituus</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a\varphi ^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a\varphi ^{-1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/316930981eae57d16515613e51f5cff4d4dc7a1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.874ex; height:3.343ex;" alt="{\displaystyle r=a\varphi ^{-1/2}}"></span></li> <li>The <a href="/wiki/Logarithmic_spiral" title="Logarithmic spiral">logarithmic spiral</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=ae^{k\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>φ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=ae^{k\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1337d832eead5adda77daa3dc9c9dfb6fb30f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.624ex; height:2.676ex;" alt="{\displaystyle r=ae^{k\varphi }}"></span></li> <li>The <a href="/wiki/Cornu_spiral" class="mw-redirect" title="Cornu spiral">Cornu spiral</a> or <i>clothoid</i></li> <li>The <a href="/wiki/Fibonacci_spiral" class="mw-redirect" title="Fibonacci spiral">Fibonacci spiral</a> and <a href="/wiki/Golden_spiral" title="Golden spiral">golden spiral</a></li> <li>The <a href="/wiki/Spiral_of_Theodorus" title="Spiral of Theodorus">Spiral of Theodorus</a>: an approximation of the Archimedean spiral composed of contiguous right triangles</li> <li>The <a href="/wiki/Involute" title="Involute">involute</a> of a circle</li></ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Archimedean_spiral.svg" class="mw-file-description" title="Archimedean spiral"><img alt="Archimedean spiral" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Archimedean_spiral.svg/120px-Archimedean_spiral.svg.png" decoding="async" width="120" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Archimedean_spiral.svg/180px-Archimedean_spiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Archimedean_spiral.svg/240px-Archimedean_spiral.svg.png 2x" data-file-width="650" data-file-height="600" /></a></span></div> <div class="gallerytext">Archimedean spiral</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Hyperspiral.svg" class="mw-file-description" title="hyperbolic spiral"><img alt="hyperbolic spiral" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Hyperspiral.svg/120px-Hyperspiral.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Hyperspiral.svg/180px-Hyperspiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Hyperspiral.svg/240px-Hyperspiral.svg.png 2x" data-file-width="205" data-file-height="205" /></a></span></div> <div class="gallerytext">hyperbolic spiral</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Fermat%27s_spiral.svg" class="mw-file-description" title="Fermat's spiral"><img alt="Fermat's spiral" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Fermat%27s_spiral.svg/120px-Fermat%27s_spiral.svg.png" decoding="async" width="120" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Fermat%27s_spiral.svg/180px-Fermat%27s_spiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Fermat%27s_spiral.svg/240px-Fermat%27s_spiral.svg.png 2x" data-file-width="328" data-file-height="310" /></a></span></div> <div class="gallerytext">Fermat's spiral</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Lituus.svg" class="mw-file-description" title="lituus"><img alt="lituus" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Lituus.svg/120px-Lituus.svg.png" decoding="async" width="120" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Lituus.svg/180px-Lituus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Lituus.svg/240px-Lituus.svg.png 2x" data-file-width="539" data-file-height="310" /></a></span></div> <div class="gallerytext">lituus</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Logarithmic_Spiral_Pylab.svg" class="mw-file-description" title="logarithmic spiral"><img alt="logarithmic spiral" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Logarithmic_Spiral_Pylab.svg/120px-Logarithmic_Spiral_Pylab.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Logarithmic_Spiral_Pylab.svg/180px-Logarithmic_Spiral_Pylab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Logarithmic_Spiral_Pylab.svg/240px-Logarithmic_Spiral_Pylab.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="gallerytext">logarithmic spiral</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Cornu_Spiral.svg" class="mw-file-description" title="Cornu spiral"><img alt="Cornu spiral" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cornu_Spiral.svg/120px-Cornu_Spiral.svg.png" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cornu_Spiral.svg/180px-Cornu_Spiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cornu_Spiral.svg/240px-Cornu_Spiral.svg.png 2x" data-file-width="480" data-file-height="460" /></a></span></div> <div class="gallerytext">Cornu spiral</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Spiral_of_Theodorus.svg" class="mw-file-description" title="spiral of Theodorus"><img alt="spiral of Theodorus" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Spiral_of_Theodorus.svg/120px-Spiral_of_Theodorus.svg.png" decoding="async" width="120" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Spiral_of_Theodorus.svg/180px-Spiral_of_Theodorus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Spiral_of_Theodorus.svg/240px-Spiral_of_Theodorus.svg.png 2x" data-file-width="700" data-file-height="570" /></a></span></div> <div class="gallerytext">spiral of Theodorus</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Fibonacci_spiral.svg" class="mw-file-description" title="Fibonacci Spiral (golden spiral)"><img alt="Fibonacci Spiral (golden spiral)" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Fibonacci_spiral.svg/120px-Fibonacci_spiral.svg.png" decoding="async" width="120" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Fibonacci_spiral.svg/180px-Fibonacci_spiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Fibonacci_spiral.svg/240px-Fibonacci_spiral.svg.png 2x" data-file-width="915" data-file-height="580" /></a></span></div> <div class="gallerytext">Fibonacci Spiral (golden spiral)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Archimedean-involute-circle-spirals-comparison.svg" class="mw-file-description" title="The involute of a circle (black) is not identical to the Archimedean spiral (red)."><img alt="The involute of a circle (black) is not identical to the Archimedean spiral (red)." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Archimedean-involute-circle-spirals-comparison.svg/120px-Archimedean-involute-circle-spirals-comparison.svg.png" decoding="async" width="120" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Archimedean-involute-circle-spirals-comparison.svg/180px-Archimedean-involute-circle-spirals-comparison.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Archimedean-involute-circle-spirals-comparison.svg/240px-Archimedean-involute-circle-spirals-comparison.svg.png 2x" data-file-width="639" data-file-height="600" /></a></span></div> <div class="gallerytext">The involute of a circle (black) is not identical to the Archimedean spiral (red).</div> </li> </ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Schraublinie-hyp-spirale.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Schraublinie-hyp-spirale.svg/130px-Schraublinie-hyp-spirale.svg.png" decoding="async" width="130" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Schraublinie-hyp-spirale.svg/195px-Schraublinie-hyp-spirale.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Schraublinie-hyp-spirale.svg/260px-Schraublinie-hyp-spirale.svg.png 2x" data-file-width="228" data-file-height="393" /></a><figcaption>Hyperbolic spiral as central projection of a helix</figcaption></figure> <p>An <i>Archimedean spiral</i> is, for example, generated while coiling a carpet.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>A <i>hyperbolic spiral</i> appears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times called <i>reciproke</i> spiral, because it is the image of an Archimedean spiral with a circle-inversion (see below).<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><p>The name <i>logarithmic spiral</i> is due to 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 \varphi ={\tfrac {1}{k}}\cdot \ln {\tfrac {r}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>r</mi> <mi>a</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\tfrac {1}{k}}\cdot \ln {\tfrac {r}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d8ebb0ac367610efc5ca04a7a8fce7442b4af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.023ex; height:3.676ex;" alt="{\displaystyle \varphi ={\tfrac {1}{k}}\cdot \ln {\tfrac {r}{a}}}"></span>. Approximations of this are found in nature. </p><p>Spirals which do not fit into this scheme of the first 5 examples: </p><p>A <i>Cornu spiral</i> has two asymptotic points.<br /> The <i>spiral of Theodorus</i> is a polygon.<br /> The <i>Fibonacci Spiral</i> consists of a sequence of circle arcs.<br /> The <i>involute of a circle</i> looks like an Archimedean, but is not: see <a href="/wiki/Involute#Examples" title="Involute">Involute#Examples</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Geometric_properties">Geometric properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=4" title="Edit section: Geometric properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following considerations are dealing with spirals, which can be described by a polar 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 r=r(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/631f196b4d374c195c6f5bdfdd2ac0a33911150a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.525ex; height:2.843ex;" alt="{\displaystyle r=r(\varphi )}"></span>, especially for the cases <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(\varphi )=a\varphi ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\varphi )=a\varphi ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec85777b0b2223bd9bd1faa73e06847ae6a11dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.445ex; height:2.843ex;" alt="{\displaystyle r(\varphi )=a\varphi ^{n}}"></span> (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=ae^{k\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>φ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=ae^{k\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1337d832eead5adda77daa3dc9c9dfb6fb30f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.624ex; height:2.676ex;" alt="{\displaystyle r=ae^{k\varphi }}"></span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sektor-steigung-pk-def.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Sektor-steigung-pk-def.svg/220px-Sektor-steigung-pk-def.svg.png" decoding="async" width="220" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Sektor-steigung-pk-def.svg/330px-Sektor-steigung-pk-def.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Sektor-steigung-pk-def.svg/440px-Sektor-steigung-pk-def.svg.png 2x" data-file-width="262" data-file-height="215" /></a><figcaption>Definition of sector (light blue) and polar slope angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span></figcaption></figure> <dl><dt>Polar slope angle</dt></dl> <p>The angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> between the spiral tangent and the corresponding polar circle (see diagram) is called <i>angle of the polar slope and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c622cda1e123d1a43bffbc0b8c1f57530cfc4e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.234ex; height:2.009ex;" alt="{\displaystyle \tan \alpha }"></span> the </i>polar slope<i>.</i> </p><p>From <a href="/wiki/Polar_coordinate_system#Vector_calculus" title="Polar coordinate system">vector calculus in polar coordinates</a> one gets the formula </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 \tan \alpha ={\frac {r'}{r}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mo>′</mo> </msup> <mi>r</mi> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \alpha ={\frac {r'}{r}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf908b04f8709898d279755d02b1896dcbb4f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.13ex; height:5.509ex;" alt="{\displaystyle \tan \alpha ={\frac {r'}{r}}\ .}"></span></dd></dl> <p>Hence the slope of the spiral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r=a\varphi ^{n}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r=a\varphi ^{n}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e052f204af06b10f81ee684a6f8fd654d3938eb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.406ex; height:2.843ex;" alt="{\displaystyle \;r=a\varphi ^{n}\;}"></span> is </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \alpha ={\frac {n}{\varphi }}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>φ</mi> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \alpha ={\frac {n}{\varphi }}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f96ee4d54abadc7b13528751aff9dbe103b68337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.917ex; height:5.176ex;" alt="{\displaystyle \tan \alpha ={\frac {n}{\varphi }}\ .}"></span></li></ul> <p>In case of an <i>Archimedean spiral</i> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span>) the polar slope 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 \;\tan \alpha ={\tfrac {1}{\varphi }}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>φ</mi> </mfrac> </mstyle> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\tan \alpha ={\tfrac {1}{\varphi }}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a186e6b5e4e80854a3c2fb5895dd9e3e8d4516f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.116ex; height:3.676ex;" alt="{\displaystyle \;\tan \alpha ={\tfrac {1}{\varphi }}\ .}"></span> </p><p>In a <i>logarithmic spiral</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \tan \alpha =k\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mi>k</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \tan \alpha =k\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1686834ffc142a056f8b26e856247c9b3562167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.092ex; height:2.176ex;" alt="{\displaystyle \ \tan \alpha =k\ }"></span> is constant. </p> <dl><dt>Curvature</dt></dl> <p>The curvature <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span> of a curve with polar 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 r=r(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/631f196b4d374c195c6f5bdfdd2ac0a33911150a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.525ex; height:2.843ex;" alt="{\displaystyle r=r(\varphi )}"></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ={\frac {r^{2}+2(r')^{2}-r\;r''}{(r^{2}+(r')^{2})^{3/2}}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>r</mi> <mspace width="thickmathspace" /> <msup> <mi>r</mi> <mo>″</mo> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\frac {r^{2}+2(r')^{2}-r\;r''}{(r^{2}+(r')^{2})^{3/2}}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2004aef618f8c09e8f9b25e9fe3abb31bd08d9b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.924ex; height:6.843ex;" alt="{\displaystyle \kappa ={\frac {r^{2}+2(r')^{2}-r\;r''}{(r^{2}+(r')^{2})^{3/2}}}\ .}"></span></dd></dl> <p>For a spiral with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a\varphi ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a\varphi ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38c276b0426396d0348f35e634ead2e822a79935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.116ex; height:2.843ex;" alt="{\displaystyle r=a\varphi ^{n}}"></span> one gets </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa =\dotsb ={\frac {1}{a\varphi ^{n-1}}}{\frac {\varphi ^{2}+n^{2}+n}{(\varphi ^{2}+n^{2})^{3/2}}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa =\dotsb ={\frac {1}{a\varphi ^{n-1}}}{\frac {\varphi ^{2}+n^{2}+n}{(\varphi ^{2}+n^{2})^{3/2}}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a9ee52164b3d60b7debaa745427b26d31890860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.599ex; height:6.676ex;" alt="{\displaystyle \kappa =\dotsb ={\frac {1}{a\varphi ^{n-1}}}{\frac {\varphi ^{2}+n^{2}+n}{(\varphi ^{2}+n^{2})^{3/2}}}\ .}"></span></li></ul> <p>In case 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 n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span> <i>(Archimedean spiral)</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ={\tfrac {\varphi ^{2}+2}{a(\varphi ^{2}+1)^{3/2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\tfrac {\varphi ^{2}+2}{a(\varphi ^{2}+1)^{3/2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e895cb43074b50685ed1f4a5ace6a2cbdb9715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.596ex; height:5.509ex;" alt="{\displaystyle \kappa ={\tfrac {\varphi ^{2}+2}{a(\varphi ^{2}+1)^{3/2}}}}"></span>.<br /> Only 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 -1<n<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>n</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1<n<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e428e4c653495407431f87c633cff49ae66b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.725ex; height:2.343ex;" alt="{\displaystyle -1<n<0}"></span> the spiral has an <i>inflection point</i>. </p><p>The curvature of a <i>logarithmic spiral</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r=ae^{k\varphi }\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>φ</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r=ae^{k\varphi }\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f0272e71ddf4183186c5d609971ec1fea787f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.914ex; height:2.676ex;" alt="{\displaystyle \;r=ae^{k\varphi }\;}"></span> 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 \;\kappa ={\tfrac {1}{r{\sqrt {1+k^{2}}}}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\kappa ={\tfrac {1}{r{\sqrt {1+k^{2}}}}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5d3e44833498bbad80cedbaaf79f6b1439e8fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.384ex; height:5.509ex;" alt="{\displaystyle \;\kappa ={\tfrac {1}{r{\sqrt {1+k^{2}}}}}\;.}"></span> </p> <dl><dt>Sector area</dt></dl> <p>The area of a sector of a curve (see diagram) with polar 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 r=r(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/631f196b4d374c195c6f5bdfdd2ac0a33911150a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.525ex; height:2.843ex;" alt="{\displaystyle r=r(\varphi )}"></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\;d\varphi \ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mi>d</mi> <mi>φ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\;d\varphi \ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dae169a3082334737ebaaa58d206497af2a3aa49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.258ex; height:6.343ex;" alt="{\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\;d\varphi \ .}"></span></dd></dl> <p>For a spiral with 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 r=a\varphi ^{n}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a\varphi ^{n}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1019858eeea0c8bfd801d410869906201b68b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.761ex; height:2.843ex;" alt="{\displaystyle r=a\varphi ^{n}\;}"></span> one gets </p> <ul><li><span class="mwe-math-element"><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={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi ^{2n}\;d\varphi ={\frac {a^{2}}{2(2n+1)}}{\big (}\varphi _{2}^{2n+1}-\varphi _{1}^{2n+1}{\big )}\ ,\quad {\text{if}}\quad n\neq -{\frac {1}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mspace width="1em" /> <mi>n</mi> <mo>≠</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi ^{2n}\;d\varphi ={\frac {a^{2}}{2(2n+1)}}{\big (}\varphi _{2}^{2n+1}-\varphi _{1}^{2n+1}{\big )}\ ,\quad {\text{if}}\quad n\neq -{\frac {1}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22edb4c1bcfc34fb9e37a6d5bfbff072e0176b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:67.767ex; height:6.676ex;" alt="{\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi ^{2n}\;d\varphi ={\frac {a^{2}}{2(2n+1)}}{\big (}\varphi _{2}^{2n+1}-\varphi _{1}^{2n+1}{\big )}\ ,\quad {\text{if}}\quad n\neq -{\frac {1}{2}},}"></span></li></ul> <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 A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}{\frac {a^{2}}{\varphi }}\;d\varphi ={\frac {a^{2}}{2}}(\ln \varphi _{2}-\ln \varphi _{1})\ ,\quad {\text{if}}\quad n=-{\frac {1}{2}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> </mfrac> </mrow> <mspace width="thickmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mspace width="1em" /> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}{\frac {a^{2}}{\varphi }}\;d\varphi ={\frac {a^{2}}{2}}(\ln \varphi _{2}-\ln \varphi _{1})\ ,\quad {\text{if}}\quad n=-{\frac {1}{2}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f391d61642234c719ee7912321b0aa53a86d861c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:56.534ex; height:6.676ex;" alt="{\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}{\frac {a^{2}}{\varphi }}\;d\varphi ={\frac {a^{2}}{2}}(\ln \varphi _{2}-\ln \varphi _{1})\ ,\quad {\text{if}}\quad n=-{\frac {1}{2}}\ .}"></span></dd></dl> <p>The formula for a <i>logarithmic spiral</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r=ae^{k\varphi }\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>φ</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r=ae^{k\varphi }\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f0272e71ddf4183186c5d609971ec1fea787f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.914ex; height:2.676ex;" alt="{\displaystyle \;r=ae^{k\varphi }\;}"></span> 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 \ A={\tfrac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2})}{4k}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mi>k</mi> </mrow> </mfrac> </mstyle> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ A={\tfrac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2})}{4k}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b858cced7a712b2fe4b768338a92a82a9a81635c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.922ex; height:4.676ex;" alt="{\displaystyle \ A={\tfrac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2})}{4k}}\ .}"></span> </p> <dl><dt>Arc length</dt></dl> <p>The length of an arc of a curve with polar 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 r=r(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/631f196b4d374c195c6f5bdfdd2ac0a33911150a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.525ex; height:2.843ex;" alt="{\displaystyle r=r(\varphi )}"></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi \ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi \ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c700e397690c25848b0145ceb20606c851a9eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:30.654ex; height:9.343ex;" alt="{\displaystyle L=\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi \ .}"></span></dd></dl> <p>For the spiral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a\varphi ^{n}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a\varphi ^{n}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1019858eeea0c8bfd801d410869906201b68b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.761ex; height:2.843ex;" alt="{\displaystyle r=a\varphi ^{n}\;}"></span> the length is </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {n^{2}r^{2}}{\varphi ^{2}}}+r^{2}}}\;d\varphi =a\int \limits _{\varphi _{1}}^{\varphi _{2}}\varphi ^{n-1}{\sqrt {n^{2}+\varphi ^{2}}}d\varphi \ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thickmathspace" /> <mi>d</mi> <mi>φ</mi> <mo>=</mo> <mi>a</mi> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </munderover> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>d</mi> <mi>φ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {n^{2}r^{2}}{\varphi ^{2}}}+r^{2}}}\;d\varphi =a\int \limits _{\varphi _{1}}^{\varphi _{2}}\varphi ^{n-1}{\sqrt {n^{2}+\varphi ^{2}}}d\varphi \ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e7de46597f811a2e437f67d14e8e596106e7fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:52.436ex; height:9.343ex;" alt="{\displaystyle L=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {n^{2}r^{2}}{\varphi ^{2}}}+r^{2}}}\;d\varphi =a\int \limits _{\varphi _{1}}^{\varphi _{2}}\varphi ^{n-1}{\sqrt {n^{2}+\varphi ^{2}}}d\varphi \ .}"></span></li></ul> <p>Not all these integrals can be solved by a suitable table. In case of a Fermat's spiral, the integral can be expressed by <a href="/wiki/Elliptic_integral" title="Elliptic integral">elliptic integrals</a> only. </p><p>The arc length of a <i>logarithmic spiral</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r=ae^{k\varphi }\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>φ</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r=ae^{k\varphi }\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f0272e71ddf4183186c5d609971ec1fea787f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.914ex; height:2.676ex;" alt="{\displaystyle \;r=ae^{k\varphi }\;}"></span> 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 \ L={\tfrac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> <mi>k</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ L={\tfrac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c05cd0cf8c2f5ea582a4655fc740c7326d66e9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.592ex; height:5.676ex;" alt="{\displaystyle \ L={\tfrac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}\ .}"></span> </p> <dl><dt>Circle inversion</dt></dl> <p>The <a href="/wiki/Circle_inversion" class="mw-redirect" title="Circle inversion">inversion at the unit circle</a> has in polar coordinates the simple description: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b91ce2aeeb184564cbf6cb23cb0523dba862d22d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.209ex; height:3.343ex;" alt="{\displaystyle \ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ }"></span>. </p> <ul><li>The image of a spiral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ r=a\varphi ^{n}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ r=a\varphi ^{n}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e270789e592689f82dcecee1c6bb68b08914dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.277ex; height:2.843ex;" alt="{\displaystyle \ r=a\varphi ^{n}\ }"></span> under the inversion at the unit circle is the spiral with polar 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 \ r={\tfrac {1}{a}}\varphi ^{-n}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mstyle> </mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ r={\tfrac {1}{a}}\varphi ^{-n}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15440d8ca26fd87ff1a18aa8abf39d730758bad0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.031ex; height:3.343ex;" alt="{\displaystyle \ r={\tfrac {1}{a}}\varphi ^{-n}\ }"></span>. For example: The inverse of an Archimedean spiral is a hyperbolic spiral.</li></ul> <dl><dd>A logarithmic spiral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r=ae^{k\varphi }\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>φ</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r=ae^{k\varphi }\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f0272e71ddf4183186c5d609971ec1fea787f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.914ex; height:2.676ex;" alt="{\displaystyle \;r=ae^{k\varphi }\;}"></span> is mapped onto the logarithmic spiral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r={\tfrac {1}{a}}e^{-k\varphi }\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mstyle> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> <mi>φ</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r={\tfrac {1}{a}}e^{-k\varphi }\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d16af39fb83d4e96818bd5b73efb88ef56890290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.316ex; height:3.343ex;" alt="{\displaystyle \;r={\tfrac {1}{a}}e^{-k\varphi }\;.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Bounded_spirals">Bounded spirals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=5" title="Edit section: Bounded spirals"><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:Spiral-arctan-1-2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Spiral-arctan-1-2.svg/310px-Spiral-arctan-1-2.svg.png" decoding="async" width="310" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Spiral-arctan-1-2.svg/465px-Spiral-arctan-1-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Spiral-arctan-1-2.svg/620px-Spiral-arctan-1-2.svg.png 2x" data-file-width="498" data-file-height="251" /></a><figcaption>Bounded spirals:<br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a\arctan(k\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a\arctan(k\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/156f84cb23d19c2c5921d0ed78e84dbac047aa95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.771ex; height:2.843ex;" alt="{\displaystyle r=a\arctan(k\varphi )}"></span> (left), <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a(\arctan(k\varphi )+\pi /2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a(\arctan(k\varphi )+\pi /2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d55f205fa2008e32c2504fc576fec254231783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.69ex; height:2.843ex;" alt="{\displaystyle r=a(\arctan(k\varphi )+\pi /2)}"></span> (right)</figcaption></figure> <p>The function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa420a7788c8a1fe01979a97723606512b0017d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle r(\varphi )}"></span> of a spiral is usually strictly monotonic, continuous and un<a href="/wiki/Bounded_function" title="Bounded function">bounded</a>. For the standard spirals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa420a7788c8a1fe01979a97723606512b0017d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle r(\varphi )}"></span> is either a power function or an exponential function. If one chooses 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 r(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa420a7788c8a1fe01979a97723606512b0017d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle r(\varphi )}"></span> a <i>bounded</i> function, the spiral is bounded, too. A suitable bounded function is the <a href="/wiki/Arctan" class="mw-redirect" title="Arctan">arctan</a> function: </p> <dl><dt>Example 1</dt></dl> <p>Setting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r=a\arctan(k\varphi )\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r=a\arctan(k\varphi )\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e4860ff38d3291ac1dbc8b30dd6ab89fb2270b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.061ex; height:2.843ex;" alt="{\displaystyle \;r=a\arctan(k\varphi )\;}"></span> and the choice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;k=0.1,a=4,\;\varphi \geq 0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>k</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>φ</mi> <mo>≥</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;k=0.1,a=4,\;\varphi \geq 0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03851d8509999297030ab2c32f7e457d2989ce6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.557ex; height:2.676ex;" alt="{\displaystyle \;k=0.1,a=4,\;\varphi \geq 0\;}"></span> gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r=a\pi /2\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r=a\pi /2\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac24ad73e43a7c5dd3e0804b1e78253e69032603" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.324ex; height:2.843ex;" alt="{\displaystyle \;r=a\pi /2\;}"></span> (diagram, left). </p> <dl><dt>Example 2</dt></dl> <p>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 \;r=a(\arctan(k\varphi )+\pi /2)\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r=a(\arctan(k\varphi )+\pi /2)\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae812ff7efdbf7fe7e509fd174662301553455f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.981ex; height:2.843ex;" alt="{\displaystyle \;r=a(\arctan(k\varphi )+\pi /2)\;}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;k=0.2,a=2,\;-\infty <\varphi <\infty \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>k</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo><</mo> <mi>φ</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;k=0.2,a=2,\;-\infty <\varphi <\infty \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd34f73c089eaf7b23ac8fde1a09274d72fe7275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.948ex; height:2.676ex;" alt="{\displaystyle \;k=0.2,a=2,\;-\infty <\varphi <\infty \;}"></span> one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r=a\pi \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mi>π</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r=a\pi \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591edd485f0743da24e76f944845f03cca688842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.999ex; height:1.676ex;" alt="{\displaystyle \;r=a\pi \;}"></span> (diagram, right). </p> <div class="mw-heading mw-heading2"><h2 id="Three-dimensional">Three-dimensional</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=6" title="Edit section: Three-dimensional"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Space spiral" redirects here. For the building, see <a href="/wiki/Space_Spiral" title="Space Spiral">Space Spiral</a>.</div> <p>Two well-known spiral <a href="/wiki/Space_curve" class="mw-redirect" title="Space curve">space curves</a> are <i>conical spirals</i> and <i>spherical spirals</i>, defined below. Another instance of space spirals is the <i>toroidal spiral</i>.<sup id="cite_ref-von_Seggern_1994_p._241_8-0" class="reference"><a href="#cite_note-von_Seggern_1994_p._241-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> A spiral wound around a helix,<sup id="cite_ref-Wolfram_MathWorld_2002_9-0" class="reference"><a href="#cite_note-Wolfram_MathWorld_2002-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> also known as <i>double-twisted helix</i>,<sup id="cite_ref-Ugajin_Ishimoto_Kuroki_Hirata_2001_pp._437–451_10-0" class="reference"><a href="#cite_note-Ugajin_Ishimoto_Kuroki_Hirata_2001_pp._437–451-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> represents objects such as <a href="/wiki/Coiled_coil_filament" class="mw-redirect" title="Coiled coil filament">coiled coil filaments</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Conical_spirals">Conical spirals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=7" title="Edit section: Conical spirals"><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:Spiral-cone-arch-s.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Spiral-cone-arch-s.svg/180px-Spiral-cone-arch-s.svg.png" decoding="async" width="180" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Spiral-cone-arch-s.svg/270px-Spiral-cone-arch-s.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Spiral-cone-arch-s.svg/360px-Spiral-cone-arch-s.svg.png 2x" data-file-width="228" data-file-height="254" /></a><figcaption>Conical spiral with Archimedean spiral as floor plan</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Conical_spiral" title="Conical spiral">conical spiral</a></div> <p>If in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-plane a spiral with parametric representation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113fa49e9011194717010c20dd3027936d0e5def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.253ex; height:2.843ex;" alt="{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi }"></span></dd></dl> <p>is given, then there can be added a third coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b1cd0a3b2c929c890cc62109913f331cda54846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle z(\varphi )}"></span>, such that the now space curve lies on the <a href="/wiki/Cone" title="Cone">cone</a> with 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 \;m(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mi>m</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;m(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae6974765c3594ece32358620088d6bead3ef501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.102ex; height:3.176ex;" alt="{\displaystyle \;m(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;}"></span>: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \ ,\qquad \color {red}{z=z_{0}+mr(\varphi )}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \ ,\qquad \color {red}{z=z_{0}+mr(\varphi )}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d26defb57252548819e48c874fa8a92f55dc9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.322ex; height:2.843ex;" alt="{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \ ,\qquad \color {red}{z=z_{0}+mr(\varphi )}\ .}"></span></li></ul> <p>Spirals based on this procedure are called <b>conical spirals</b>. </p> <dl><dt>Example</dt></dl> <p>Starting with an <i>archimedean spiral</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;r(\varphi )=a\varphi \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>r</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>φ</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;r(\varphi )=a\varphi \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf4b4e5247d29a2bca680ad5dd67ee9562df9ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.517ex; height:2.843ex;" alt="{\displaystyle \;r(\varphi )=a\varphi \;}"></span> one gets the conical spiral (see diagram) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mi>φ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mi>φ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mi>z</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>m</mi> <mi>a</mi> <mi>φ</mi> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mi>φ</mi> <mo>≥</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a99db3d4281727b1dd564fdf0ca30b3057f3ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.155ex; height:2.676ex;" alt="{\displaystyle x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Spherical_spirals">Spherical spirals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=8" title="Edit section: Spherical spirals"><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:Kugel-spirale-1-2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kugel-spirale-1-2.svg/260px-Kugel-spirale-1-2.svg.png" decoding="async" width="260" height="129" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kugel-spirale-1-2.svg/390px-Kugel-spirale-1-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kugel-spirale-1-2.svg/520px-Kugel-spirale-1-2.svg.png 2x" data-file-width="759" data-file-height="377" /></a><figcaption>Clelia curve with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0977619356f0caa405a5f40070bed06c655db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=8}"></span></figcaption></figure> <p>Any <a href="/wiki/Cylindrical_map_projection" class="mw-redirect" title="Cylindrical map projection">cylindrical map projection</a> can be used as the basis for a <b>spherical spiral</b>: draw a straight line on the map and find its inverse projection on the sphere, a kind of <a href="/wiki/Spherical_curve" class="mw-redirect" title="Spherical curve">spherical curve</a>. </p><p>One of the most basic families of spherical spirals is the <a href="/wiki/Clelia_curve" class="mw-redirect" title="Clelia curve">Clelia curves</a>, which project to straight lines on an <a href="/wiki/Equirectangular_projection" title="Equirectangular projection">equirectangular projection</a>. These are curves for which <a href="/wiki/Longitude" title="Longitude">longitude</a> and <a href="/wiki/Colatitude" title="Colatitude">colatitude</a> are in a linear relationship, analogous to Archimedean spirals in the plane; under the <a href="/wiki/Azimuthal_equidistant_projection" title="Azimuthal equidistant projection">azimuthal equidistant projection</a> a Clelia curve projects to a planar Archimedean spiral. </p><p>If one represents a unit sphere by <a href="/wiki/Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\sin \theta \,\cos \varphi ,\quad y=\sin \theta \,\sin \varphi ,\quad z=\cos \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mspace width="1em" /> <mi>y</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mspace width="1em" /> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\sin \theta \,\cos \varphi ,\quad y=\sin \theta \,\sin \varphi ,\quad z=\cos \theta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd3549b7711e68d92714912524fedcc7dbabdfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.813ex; height:2.676ex;" alt="{\displaystyle x=\sin \theta \,\cos \varphi ,\quad y=\sin \theta \,\sin \varphi ,\quad z=\cos \theta ,}"></span></dd></dl> <p>then setting the linear dependency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =c\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>c</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =c\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a365cf05a4238676a4e7e910703829da0fa09d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.716ex; height:2.676ex;" alt="{\displaystyle \varphi =c\theta }"></span> for the angle coordinates gives a <a href="/wiki/Parametric_curve" class="mw-redirect" title="Parametric curve">parametric curve</a> in terms of parameter <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>⁠</span>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <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 {\bigl (}\sin \theta \,\cos c\theta ,\,\sin \theta \,\sin c\theta ,\,\cos \theta \,{\bigr )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>c</mi> <mi>θ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>c</mi> <mi>θ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}\sin \theta \,\cos c\theta ,\,\sin \theta \,\sin c\theta ,\,\cos \theta \,{\bigr )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bfe964ce6c7426524a66c0e77a1ff49e8556f9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.132ex; height:3.176ex;" alt="{\displaystyle {\bigl (}\sin \theta \,\cos c\theta ,\,\sin \theta \,\sin c\theta ,\,\cos \theta \,{\bigr )}.}"></span></dd></dl> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:KUGSPI-5_Archimedische_Kugelspirale.gif" class="mw-file-description" title="Clelia curve"><img alt="Clelia curve" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/KUGSPI-5_Archimedische_Kugelspirale.gif/120px-KUGSPI-5_Archimedische_Kugelspirale.gif" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/KUGSPI-5_Archimedische_Kugelspirale.gif/180px-KUGSPI-5_Archimedische_Kugelspirale.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/b/bb/KUGSPI-5_Archimedische_Kugelspirale.gif 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">Clelia curve</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:KUGSPI-9_Loxodrome.gif" class="mw-file-description" title="Loxodrome"><img alt="Loxodrome" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/KUGSPI-9_Loxodrome.gif/120px-KUGSPI-9_Loxodrome.gif" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/KUGSPI-9_Loxodrome.gif/180px-KUGSPI-9_Loxodrome.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6e/KUGSPI-9_Loxodrome.gif 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">Loxodrome</div> </li> </ul> <p>Another family of spherical spirals is the <a href="/wiki/Rhumb_line" title="Rhumb line">rhumb lines</a> or loxodromes, that project to straight lines on the <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a>. These are the trajectories traced by a ship traveling with constant <a href="/wiki/Bearing_(navigation)" title="Bearing (navigation)">bearing</a>. Any loxodrome (except for the meridians and parallels) spirals infinitely around either pole, closer and closer each time, unlike a Clelia curve which maintains uniform spacing in colatitude. Under <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a>, a loxodrome projects to a logarithmic spiral in the plane. </p> <div class="mw-heading mw-heading2"><h2 id="In_nature">In nature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=9" title="Edit section: In nature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The study of spirals in <a href="/wiki/Nature" title="Nature">nature</a> has a long history. <a href="/wiki/Christopher_Wren" title="Christopher Wren">Christopher Wren</a> observed that many <a href="/wiki/Exoskeleton" title="Exoskeleton">shells</a> form a <a href="/wiki/Logarithmic_spiral" title="Logarithmic spiral">logarithmic spiral</a>; <a href="/wiki/Jan_Swammerdam" title="Jan Swammerdam">Jan Swammerdam</a> observed the common mathematical characteristics of a wide range of shells from <i><a href="/wiki/Helix_(genus)" class="mw-redirect" title="Helix (genus)">Helix</a></i> to <i><a href="/wiki/Spirula" title="Spirula">Spirula</a></i>; and <a href="/wiki/Henry_Nottidge_Moseley" title="Henry Nottidge Moseley">Henry Nottidge Moseley</a> described the mathematics of <a href="/wiki/Univalve" class="mw-redirect" title="Univalve">univalve</a> shells. <a href="/wiki/D%27Arcy_Wentworth_Thompson" title="D'Arcy Wentworth Thompson">D’Arcy Wentworth Thompson</a>'s <i><a href="/wiki/On_Growth_and_Form" title="On Growth and Form">On Growth and Form</a></i> gives extensive treatment to these spirals. He describes how shells are formed by rotating a closed curve around a fixed axis: the <a href="/wiki/Shape" title="Shape">shape</a> of the curve remains fixed, but its size grows in a <a href="/wiki/Geometric_progression" title="Geometric progression">geometric progression</a>. In some shells, such as <i><a href="/wiki/Nautilus" title="Nautilus">Nautilus</a></i> and <a href="/wiki/Ammonite" class="mw-redirect" title="Ammonite">ammonites</a>, the generating curve revolves in a plane perpendicular to the axis and the shell will form a planar discoid shape. In others it follows a skew path forming a <a href="/wiki/Helix" title="Helix">helico</a>-spiral pattern. Thompson also studied spirals occurring in <a href="/wiki/Horn_(anatomy)" title="Horn (anatomy)">horns</a>, <a href="/wiki/Teeth" class="mw-redirect" title="Teeth">teeth</a>, <a href="/wiki/Claw" title="Claw">claws</a> and <a href="/wiki/Plant" title="Plant">plants</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>A model for the pattern of <a href="/wiki/Floret" class="mw-redirect" title="Floret">florets</a> in the head of a <a href="/wiki/Sunflower" class="mw-redirect" title="Sunflower">sunflower</a><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> was proposed by H. Vogel. This has the form </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 \theta =n\times 137.5^{\circ },\ r=c{\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>n</mi> <mo>×</mo> <msup> <mn>137.5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mi>r</mi> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =n\times 137.5^{\circ },\ r=c{\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58ef90014a2f275adab8b24b5319157ab3c3db7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.874ex; height:3.009ex;" alt="{\displaystyle \theta =n\times 137.5^{\circ },\ r=c{\sqrt {n}}}"></span></dd></dl> <p>where <i>n</i> is the index number of the floret and <i>c</i> is a constant scaling factor, and is a form of <a href="/wiki/Fermat%27s_spiral" title="Fermat's spiral">Fermat's spiral</a>. The angle 137.5° is the <a href="/wiki/Golden_angle" title="Golden angle">golden angle</a> which is related to the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> and gives a close packing of florets.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Spirals in plants and animals are frequently described as <a href="/wiki/Whorl_(botany)" title="Whorl (botany)">whorls</a>. This is also the name given to spiral shaped <a href="/wiki/Fingerprint" title="Fingerprint">fingerprints</a>. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 190px;"><span typeof="mw:File"><a href="/wiki/File:Milky_Way_2008.jpg" class="mw-file-description" title="An artist's rendering of a spiral galaxy."><img alt="An artist's rendering of a spiral galaxy." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Milky_Way_2008.jpg/160px-Milky_Way_2008.jpg" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Milky_Way_2008.jpg/240px-Milky_Way_2008.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Milky_Way_2008.jpg/320px-Milky_Way_2008.jpg 2x" data-file-width="5600" data-file-height="5600" /></a></span></div> <div class="gallerytext">An artist's rendering of a spiral galaxy.</div> </li> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 190px;"><span typeof="mw:File"><a href="/wiki/File:Helianthus_whorl.jpg" class="mw-file-description" title="Sunflower head displaying florets in spirals of 34 and 55 around the outside."><img alt="Sunflower head displaying florets in spirals of 34 and 55 around the outside." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Helianthus_whorl.jpg/213px-Helianthus_whorl.jpg" decoding="async" width="213" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Helianthus_whorl.jpg/320px-Helianthus_whorl.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Helianthus_whorl.jpg/427px-Helianthus_whorl.jpg 2x" data-file-width="640" data-file-height="480" /></a></span></div> <div class="gallerytext">Sunflower head displaying florets in spirals of 34 and 55 around the outside.</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="As_a_symbol">As a symbol</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=10" title="Edit section: As a symbol"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A spiral like form has been found in <a href="/wiki/Mezine" title="Mezine">Mezine</a>, <a href="/wiki/Ukraine" title="Ukraine">Ukraine</a>, as part of a decorative object dated to 10,000 BCE.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2019)">citation needed</span></a></i>]</sup> Spiral and <a href="/wiki/Triple_spiral" class="mw-redirect" title="Triple spiral">triple spiral</a> motifs served as <a href="/wiki/Neolithic" title="Neolithic">Neolithic</a> symbols in Europe (<a href="/wiki/Megalithic_Temples_of_Malta" title="Megalithic Temples of Malta">Megalithic Temples of Malta</a>). The <a href="/wiki/Celts" title="Celts">Celtic</a> triple-spiral is in fact a pre-Celtic symbol.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> It is carved into the rock of a stone lozenge near the main entrance of the prehistoric <a href="/wiki/Newgrange" title="Newgrange">Newgrange</a> monument in <a href="/wiki/County_Meath" title="County Meath">County Meath</a>, <a href="/wiki/Republic_of_Ireland" title="Republic of Ireland">Ireland</a>. Newgrange was built around 3200 BCE, predating the Celts; triple spirals were carved at least 2,500 years before the Celts reached Ireland, but have long since become part of Celtic culture.<sup id="cite_ref-knowth.com_16-0" class="reference"><a href="#cite_note-knowth.com-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Triskelion" title="Triskelion">triskelion</a> symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures: examples include <a href="/wiki/Mycenaean_Greece" title="Mycenaean Greece">Mycenaean</a> vessels, coinage from <a href="/wiki/Lycia" title="Lycia">Lycia</a>, <a href="/wiki/Stater" title="Stater">staters</a> of <a href="/wiki/Pamphylia" title="Pamphylia">Pamphylia</a> (at <a href="/wiki/Aspendos" title="Aspendos">Aspendos</a>, 370–333 BC) and <a href="/wiki/Pisidia" title="Pisidia">Pisidia</a>, as well as the <a href="/wiki/Heraldic" class="mw-redirect" title="Heraldic">heraldic</a> emblem on warriors' shields depicted on Greek pottery.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Spirals occur commonly in pre-Columbian art in Latin and Central America. The more than 1,400 <a href="/wiki/Petroglyphs" class="mw-redirect" title="Petroglyphs">petroglyphs</a> (rock engravings) in <a href="/w/index.php?title=Las_Plazuelas&action=edit&redlink=1" class="new" title="Las Plazuelas (page does not exist)">Las Plazuelas</a>, <a href="/wiki/Guanajuato" title="Guanajuato">Guanajuato</a> <a href="/wiki/Mexico" title="Mexico">Mexico</a>, dating 750-1200 AD, predominantly depict spirals, dot figures and scale models.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In Colombia, monkeys, frog and lizard-like figures depicted in petroglyphs or as gold offering-figures frequently include spirals, for example on the palms of hands.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In Lower Central America, spirals along with circles, wavy lines, crosses and points are universal petroglyph characters.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Spirals also appear among the <a href="/wiki/Nazca_Lines" class="mw-redirect" title="Nazca Lines">Nazca Lines</a> in the coastal desert of Peru, dating from 200 BC to 500 AD. The <a href="/wiki/Geoglyphs" class="mw-redirect" title="Geoglyphs">geoglyphs</a> number in the thousands and depict animals, plants and geometric motifs, including spirals.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Spiral shapes, including the <a href="/wiki/Swastika" title="Swastika">swastika</a>, <a href="/wiki/Triskele" class="mw-redirect" title="Triskele">triskele</a>, etc., have often been interpreted as <a href="/wiki/Solar_symbol" title="Solar symbol">solar symbols</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2016)">citation needed</span></a></i>]</sup> Roof tiles dating back to the <a href="/wiki/Tang_dynasty" title="Tang dynasty">Tang dynasty</a> with this symbol have been found west of the ancient city of <a href="/wiki/Chang%27an" title="Chang'an">Chang'an</a> (modern-day Xi'an).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2016)">citation needed</span></a></i>]</sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers" title="Wikipedia:Manual of Style/Dates and numbers"><span title="Need the year this event took place (November 2016)">year needed</span></a></i>]</sup> </p><p>Spirals are also a symbol of <a href="/wiki/Hypnosis" title="Hypnosis">hypnosis</a>, stemming from the <a href="/wiki/Clich%C3%A9" title="Cliché">cliché</a> of people and cartoon characters being hypnotized by staring into a spinning spiral (one example being <a href="/wiki/Kaa" title="Kaa">Kaa</a> in Disney's <a href="/wiki/The_Jungle_Book_(1967_film)" title="The Jungle Book (1967 film)"> <i>The Jungle Book</i></a>). They are also used as a symbol of <a href="/wiki/Dizziness" title="Dizziness">dizziness</a>, where the eyes of a cartoon character, especially in <a href="/wiki/Anime" title="Anime">anime</a> and <a href="/wiki/Manga" title="Manga">manga</a>, will turn into spirals to suggest that they are dizzy or dazed. The spiral is also found in structures as small as the <a href="/wiki/Double_helix" class="mw-redirect" title="Double helix">double helix</a> of <a href="/wiki/DNA" title="DNA">DNA</a> and as large as a <a href="/wiki/Spiral_galaxy" title="Spiral galaxy">galaxy</a>. Due to this frequent natural occurrence, the spiral is the official symbol of the <a href="/wiki/World_Pantheist_Movement" title="World Pantheist Movement">World Pantheist Movement</a>.<sup id="cite_ref-WPM_22-0" class="reference"><a href="#cite_note-WPM-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The spiral is also a symbol of the <a href="/wiki/Dialectic" title="Dialectic">dialectic</a> process and of <a href="/wiki/Dialectical_monism" title="Dialectical monism">Dialectical monism</a>. </p> <blockquote> <p>The spiral is a frequent symbol for <a href="/wiki/Spiritual_experience" class="mw-redirect" title="Spiritual experience"> spiritual</a> purification, both within <a href="/wiki/Christianity" title="Christianity">Christianity</a> and beyond (one thinks of the spiral as the <a href="/wiki/Neoplatonism" title="Neoplatonism"> neo-Platonist</a> symbol for prayer and contemplation, circling around a subject and ascending at the same time, and as a <a href="/wiki/Buddhist" class="mw-redirect" title="Buddhist">Buddhist</a> symbol for the gradual process on the Path to <a href="/wiki/Enlightenment_in_Buddhism" title="Enlightenment in Buddhism"> Enlightenment</a>). [...] while a helix is repetitive, a spiral expands and thus epitomizes <a href="/wiki/Exponential_growth" title="Exponential growth"> growth</a> - conceptually <i>ad infinitum</i>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> </blockquote> <ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 258px"> <div class="thumb" style="width: 256px;"><span typeof="mw:File"><a href="/wiki/File:%E5%BA%93%E5%BA%93%E7%89%B9%E5%B0%BC%E9%99%B6%E7%A2%97%E9%99%B6%E7%BD%90.JPG" class="mw-file-description" title="Cucuteni Culture spirals on a bowl on stand, a vessel on stand, and an amphora, 4300-4000 BCE, ceramic, Palace of Culture, Iași, Romania"><img alt="Cucuteni Culture spirals on a bowl on stand, a vessel on stand, and an amphora, 4300-4000 BCE, ceramic, Palace of Culture, Iași, Romania" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/%E5%BA%93%E5%BA%93%E7%89%B9%E5%B0%BC%E9%99%B6%E7%A2%97%E9%99%B6%E7%BD%90.JPG/384px-%E5%BA%93%E5%BA%93%E7%89%B9%E5%B0%BC%E9%99%B6%E7%A2%97%E9%99%B6%E7%BD%90.JPG" decoding="async" width="256" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/%E5%BA%93%E5%BA%93%E7%89%B9%E5%B0%BC%E9%99%B6%E7%A2%97%E9%99%B6%E7%BD%90.JPG/577px-%E5%BA%93%E5%BA%93%E7%89%B9%E5%B0%BC%E9%99%B6%E7%A2%97%E9%99%B6%E7%BD%90.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/%E5%BA%93%E5%BA%93%E7%89%B9%E5%B0%BC%E9%99%B6%E7%A2%97%E9%99%B6%E7%BD%90.JPG/768px-%E5%BA%93%E5%BA%93%E7%89%B9%E5%B0%BC%E9%99%B6%E7%A2%97%E9%99%B6%E7%BD%90.JPG 2x" data-file-width="3684" data-file-height="2160" /></a></span></div> <div class="gallerytext"><a href="/wiki/Cucuteni_Culture" class="mw-redirect" title="Cucuteni Culture">Cucuteni Culture</a> spirals on a bowl on stand, a vessel on stand, and an amphora, 4300-4000 BCE, ceramic, <a href="/wiki/Palace_of_Culture_(Ia%C8%99i)" title="Palace of Culture (Iași)">Palace of Culture</a>, <a href="/wiki/Ia%C8%99i" title="Iași">Iași</a>, <a href="/wiki/Romania" title="Romania">Romania</a></div> </li> <li class="gallerybox" style="width: 227.33333333333px"> <div class="thumb" style="width: 225.33333333333px;"><span typeof="mw:File"><a href="/wiki/File:Newgrange_Entrance_Stone.jpg" class="mw-file-description" title="Neolithic spirals on the Newgrange entrance slab, unknown sculptor or architect, 3rd millennium BC"><img alt="Neolithic spirals on the Newgrange entrance slab, unknown sculptor or architect, 3rd millennium BC" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Newgrange_Entrance_Stone.jpg/338px-Newgrange_Entrance_Stone.jpg" decoding="async" width="226" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/62/Newgrange_Entrance_Stone.jpg 1.5x" data-file-width="450" data-file-height="300" /></a></span></div> <div class="gallerytext"><a href="/wiki/Neolithic_Europe" title="Neolithic Europe">Neolithic</a> spirals on the <a href="/wiki/Newgrange" title="Newgrange">Newgrange</a> entrance slab, unknown sculptor or architect, 3rd millennium BC</div> </li> <li class="gallerybox" style="width: 107.33333333333px"> <div class="thumb" style="width: 105.33333333333px;"><span typeof="mw:File"><a href="/wiki/File:Mycenaean_funerary_stele_at_the_National_Archaeological_Museum_of_Athens_on_October_6,_2021.jpg" class="mw-file-description" title="Mycenaean spirals on a burial stela, Grave Circle A, c.1550 BC, stone, National Archaeological Museum, Athens, Greece"><img alt="Mycenaean spirals on a burial stela, Grave Circle A, c.1550 BC, stone, National Archaeological Museum, Athens, Greece" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Mycenaean_funerary_stele_at_the_National_Archaeological_Museum_of_Athens_on_October_6%2C_2021.jpg/158px-Mycenaean_funerary_stele_at_the_National_Archaeological_Museum_of_Athens_on_October_6%2C_2021.jpg" decoding="async" width="106" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Mycenaean_funerary_stele_at_the_National_Archaeological_Museum_of_Athens_on_October_6%2C_2021.jpg/238px-Mycenaean_funerary_stele_at_the_National_Archaeological_Museum_of_Athens_on_October_6%2C_2021.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Mycenaean_funerary_stele_at_the_National_Archaeological_Museum_of_Athens_on_October_6%2C_2021.jpg/316px-Mycenaean_funerary_stele_at_the_National_Archaeological_Museum_of_Athens_on_October_6%2C_2021.jpg 2x" data-file-width="3295" data-file-height="4684" /></a></span></div> <div class="gallerytext"><a href="/wiki/Mycenaean_Greece" title="Mycenaean Greece">Mycenaean</a> spirals on a burial stela, Grave Circle A, <abbr title="circa">c.</abbr>1550 BC, stone, <a href="/wiki/National_Archaeological_Museum,_Athens" title="National Archaeological Museum, Athens">National Archaeological Museum</a>, <a href="/wiki/Athens" title="Athens">Athens</a>, Greece</div> </li> <li class="gallerybox" style="width: 114.66666666667px"> <div class="thumb" style="width: 112.66666666667px;"><span typeof="mw:File"><a href="/wiki/File:Temple_of_Amun_alley_of_rams_(4)_(34143965175).jpg" class="mw-file-description" title="Meroitic spirals on a ram of the alley of the Amun Temple of Naqa, unknown sculptor, 1st century AD, stone, in situ"><img alt="Meroitic spirals on a ram of the alley of the Amun Temple of Naqa, unknown sculptor, 1st century AD, stone, in situ" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Temple_of_Amun_alley_of_rams_%284%29_%2834143965175%29.jpg/169px-Temple_of_Amun_alley_of_rams_%284%29_%2834143965175%29.jpg" decoding="async" width="113" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Temple_of_Amun_alley_of_rams_%284%29_%2834143965175%29.jpg/253px-Temple_of_Amun_alley_of_rams_%284%29_%2834143965175%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Temple_of_Amun_alley_of_rams_%284%29_%2834143965175%29.jpg/337px-Temple_of_Amun_alley_of_rams_%284%29_%2834143965175%29.jpg 2x" data-file-width="2805" data-file-height="3740" /></a></span></div> <div class="gallerytext"><a href="/wiki/Mero%C3%AB" title="Meroë">Meroitic</a> spirals on a ram of the alley of the <a href="/wiki/Amun" title="Amun">Amun</a> Temple of <a href="/wiki/Naqa" title="Naqa">Naqa</a>, unknown sculptor, 1st century AD, stone, <a href="/wiki/In_situ" title="In situ">in situ</a></div> </li> <li class="gallerybox" style="width: 222px"> <div class="thumb" style="width: 220px;"><span typeof="mw:File"><a href="/wiki/File:Samarra,_Iraq_(25270211056)_edited.jpg" class="mw-file-description" title="Islamic spiral design of the Great Mosque of Samarra, Samarra, Iraq, unknown architect, c. 851"><img alt="Islamic spiral design of the Great Mosque of Samarra, Samarra, Iraq, unknown architect, c. 851" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Samarra%2C_Iraq_%2825270211056%29_edited.jpg/330px-Samarra%2C_Iraq_%2825270211056%29_edited.jpg" decoding="async" width="220" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Samarra%2C_Iraq_%2825270211056%29_edited.jpg/495px-Samarra%2C_Iraq_%2825270211056%29_edited.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Samarra%2C_Iraq_%2825270211056%29_edited.jpg/659px-Samarra%2C_Iraq_%2825270211056%29_edited.jpg 2x" data-file-width="2639" data-file-height="1802" /></a></span></div> <div class="gallerytext"><a href="/wiki/Islamic_architecture" title="Islamic architecture">Islamic</a> spiral design of the <a href="/wiki/Great_Mosque_of_Samarra" title="Great Mosque of Samarra">Great Mosque of Samarra</a>, <a href="/wiki/Samarra" title="Samarra">Samarra</a>, <a href="/wiki/Iraq" title="Iraq">Iraq</a>, unknown architect, <abbr title="circa">c.</abbr> 851</div> </li> <li class="gallerybox" style="width: 114.66666666667px"> <div class="thumb" style="width: 112.66666666667px;"><span typeof="mw:File"><a href="/wiki/File:Nantes_Maison_compagnonnage_Clocher_tors.jpg" class="mw-file-description" title="Gothic Revival spiralling bell-tower of the Maison des compagnons du tour de France, Nantes, unknown architect, c. 1910"><img alt="Gothic Revival spiralling bell-tower of the Maison des compagnons du tour de France, Nantes, unknown architect, c. 1910" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Nantes_Maison_compagnonnage_Clocher_tors.jpg/169px-Nantes_Maison_compagnonnage_Clocher_tors.jpg" decoding="async" width="113" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Nantes_Maison_compagnonnage_Clocher_tors.jpg/253px-Nantes_Maison_compagnonnage_Clocher_tors.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Nantes_Maison_compagnonnage_Clocher_tors.jpg/337px-Nantes_Maison_compagnonnage_Clocher_tors.jpg 2x" data-file-width="1944" data-file-height="2592" /></a></span></div> <div class="gallerytext"><a href="/wiki/Gothic_Revival" class="mw-redirect" title="Gothic Revival">Gothic Revival</a> spiralling bell-tower of the Maison des compagnons du tour de France, <a href="/wiki/Nantes" title="Nantes">Nantes</a>, unknown architect, <abbr title="circa">c.</abbr> 1910</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="In_art">In art</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=11" title="Edit section: In art"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The spiral has inspired artists throughout the ages. Among the most famous of spiral-inspired art is <a href="/wiki/Robert_Smithson" title="Robert Smithson">Robert Smithson</a>'s <a href="/wiki/Earthworks_(art)" class="mw-redirect" title="Earthworks (art)">earthwork</a>, "<a href="/wiki/Spiral_Jetty" title="Spiral Jetty">Spiral Jetty</a>", at the <a href="/wiki/Great_Salt_Lake" title="Great Salt Lake">Great Salt Lake</a> in Utah.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The spiral theme is also present in David Wood's Spiral Resonance Field at the <a href="/wiki/Anderson-Abruzzo_Albuquerque_International_Balloon_Museum" title="Anderson-Abruzzo Albuquerque International Balloon Museum">Balloon Museum</a> in Albuquerque, as well as in the critically acclaimed <a href="/wiki/Nine_Inch_Nails" title="Nine Inch Nails">Nine Inch Nails</a> 1994 concept album <i><a href="/wiki/The_Downward_Spiral" title="The Downward Spiral">The Downward Spiral</a></i>. The Spiral is also a prominent theme in the anime <i><a href="/wiki/Gurren_Lagann" title="Gurren Lagann">Gurren Lagann</a></i>, where it represents a philosophy and way of life. It also central in Mario Merz and Andy Goldsworthy's work. The spiral is the central theme of the horror manga <i><a href="/wiki/Uzumaki" title="Uzumaki">Uzumaki</a></i> by <a href="/wiki/Junji_Ito" title="Junji Ito">Junji Ito</a>, where a small coastal town is afflicted by a curse involving spirals. <i>2012 A Piece of Mind By Wayne A Beale</i> also depicts a large spiral in this book of dreams and images.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (December 2018)">full citation needed</span></a></i>]</sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability"><span title="The material near this tag needs to be fact-checked with the cited source(s). (December 2018)">verification needed</span></a></i>]</sup> The coiled spiral is a central image in Australian artist Tanja Stark's <a href="/wiki/Suburban_Gothic" title="Suburban Gothic">Suburban Gothic</a> iconography, that incorporates spiral <a href="/wiki/Stove" title="Stove">electric stove top elements</a> as symbols of domestic alchemy and spirituality.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<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=Spiral&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Celtic_maze" title="Celtic maze">Celtic maze</a> (straight-line spiral)</li> <li><a href="/wiki/Concentric_circles" class="mw-redirect" title="Concentric circles">Concentric circles</a></li> <li><a href="/wiki/DNA" title="DNA">DNA</a></li> <li><a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci number</a></li> <li><a href="/wiki/Hypogeum_of_%C4%A6al-Saflieni" class="mw-redirect" title="Hypogeum of Ħal-Saflieni">Hypogeum of Ħal-Saflieni</a></li> <li><a href="/wiki/Megalithic_Temples_of_Malta" title="Megalithic Temples of Malta">Megalithic Temples of Malta</a></li> <li><a href="/wiki/Patterns_in_nature" title="Patterns in nature">Patterns in nature</a></li> <li><a href="/wiki/Seashell_surface" title="Seashell surface">Seashell surface</a></li> <li><a href="/wiki/Spirangle" title="Spirangle">Spirangle</a></li> <li><a href="/wiki/Spiral_vegetable_slicer" title="Spiral vegetable slicer">Spiral vegetable slicer</a></li> <li><a href="/wiki/Spiral_stairs" class="mw-redirect" title="Spiral stairs">Spiral stairs</a></li> <li><a href="/wiki/Triskelion" title="Triskelion">Triskelion</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=Spiral&action=edit&section=13" 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 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Retrieved <span class="nowrap">2020-10-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Franklin+Institute&rft.atitle=Math+Patterns+in+Nature&rft.date=2017-06-01&rft_id=https%3A%2F%2Fwww.fi.edu%2Fmath-patterns-nature&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-free-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-free_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-free_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="http://www.thefreedictionary.com/spiral">Spiral</a>, <i>American Heritage Dictionary of the English Language</i>, Houghton Mifflin Company, Fourth Edition, 2009.</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="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ArchimedeanSpiral.html">"Archimedean Spiral"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-10-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Archimedean+Spiral&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FArchimedeanSpiral.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" 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="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HyperbolicSpiral.html">"Hyperbolic Spiral"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-10-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Hyperbolic+Spiral&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FHyperbolicSpiral.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-von_Seggern_1994_p._241-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-von_Seggern_1994_p._241_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Seggern1994" class="citation book cs1">von Seggern, D.H. (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PVKXqob2dhAC&pg=PA241"><i>Practical Handbook of Curve Design and Generation</i></a>. Taylor & Francis. p. 241. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-8916-0" title="Special:BookSources/978-0-8493-8916-0"><bdi>978-0-8493-8916-0</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-03-03</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practical+Handbook+of+Curve+Design+and+Generation&rft.pages=241&rft.pub=Taylor+%26+Francis&rft.date=1994&rft.isbn=978-0-8493-8916-0&rft.aulast=von+Seggern&rft.aufirst=D.H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPVKXqob2dhAC%26pg%3DPA241&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-Wolfram_MathWorld_2002-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wolfram_MathWorld_2002_9-0">^</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="https://mathworld.wolfram.com/Slinky.html">"Slinky -- from Wolfram MathWorld"</a>. <i>Wolfram MathWorld</i>. 2002-09-13<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-03-03</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Slinky+--+from+Wolfram+MathWorld&rft.date=2002-09-13&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSlinky.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-Ugajin_Ishimoto_Kuroki_Hirata_2001_pp._437–451-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ugajin_Ishimoto_Kuroki_Hirata_2001_pp._437–451_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUgajinIshimotoKurokiHirata2001" class="citation journal cs1">Ugajin, R.; Ishimoto, C.; Kuroki, Y.; Hirata, S.; Watanabe, S. (2001). "Statistical analysis of a multiply-twisted helix". <i>Physica A: Statistical Mechanics and Its Applications</i>. <b>292</b> (1–4). Elsevier BV: 437–451. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001PhyA..292..437U">2001PhyA..292..437U</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0378-4371%2800%2900572-0">10.1016/s0378-4371(00)00572-0</a>. <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/0378-4371">0378-4371</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+A%3A+Statistical+Mechanics+and+Its+Applications&rft.atitle=Statistical+analysis+of+a+multiply-twisted+helix&rft.volume=292&rft.issue=1%E2%80%934&rft.pages=437-451&rft.date=2001&rft.issn=0378-4371&rft_id=info%3Adoi%2F10.1016%2Fs0378-4371%2800%2900572-0&rft_id=info%3Abibcode%2F2001PhyA..292..437U&rft.aulast=Ugajin&rft.aufirst=R.&rft.au=Ishimoto%2C+C.&rft.au=Kuroki%2C+Y.&rft.au=Hirata%2C+S.&rft.au=Watanabe%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Kuno Fladt: <i>Analytische Geometrie spezieller Flächen und Raumkurven</i>, Springer-Verlag, 2013, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3322853659" title="Special:BookSources/3322853659">3322853659</a>, 9783322853653, S. 132</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThompson1942" class="citation book cs1">Thompson, D'Arcy (1942) [1917]. <a rel="nofollow" class="external text" href="https://archive.org/details/ongrowthform00thom"><i>On Growth and Form</i></a>. Cambridge : University Press ; New York : Macmillan. pp. 748–933.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+Growth+and+Form&rft.pages=748-933&rft.pub=Cambridge+%3A+University+Press+%3B+New+York+%3A+Macmillan&rft.date=1942&rft.aulast=Thompson&rft.aufirst=D%27Arcy&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fongrowthform00thom&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBen_Sparks" class="citation web cs1">Ben Sparks. <a rel="nofollow" class="external text" href="https://www.geogebra.org/m/B4C9bbuy">"Geogebra: Sunflowers are Irrationally Pretty"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Geogebra%3A+Sunflowers+are+Irrationally+Pretty&rft.au=Ben+Sparks&rft_id=https%3A%2F%2Fwww.geogebra.org%2Fm%2FB4C9bbuy&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrusinkiewiczLindenmayer,_Aristid1990" class="citation book cs1"><a href="/wiki/Przemyslaw_Prusinkiewicz" class="mw-redirect" title="Przemyslaw Prusinkiewicz">Prusinkiewicz, Przemyslaw</a>; <a href="/wiki/Aristid_Lindenmayer" title="Aristid Lindenmayer">Lindenmayer, Aristid</a> (1990). <a rel="nofollow" class="external text" href="https://archive.org/details/algorithmicbeaut0000prus/page/101"><i>The Algorithmic Beauty of Plants</i></a>. Springer-Verlag. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/algorithmicbeaut0000prus/page/101">101–107</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97297-8" title="Special:BookSources/978-0-387-97297-8"><bdi>978-0-387-97297-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Algorithmic+Beauty+of+Plants&rft.pages=101-107&rft.pub=Springer-Verlag&rft.date=1990&rft.isbn=978-0-387-97297-8&rft.aulast=Prusinkiewicz&rft.aufirst=Przemyslaw&rft.au=Lindenmayer%2C+Aristid&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgorithmicbeaut0000prus%2Fpage%2F101&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Anthony Murphy and Richard Moore, <i>Island of the Setting Sun: In Search of Ireland's Ancient Astronomers,</i> 2nd ed., Dublin: The Liffey Press, 2008, pp. 168-169</span> </li> <li id="cite_note-knowth.com-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-knowth.com_16-0">^</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://knowth.com/newgrange.htm">"Newgrange Ireland - Megalithic Passage Tomb - World Heritage Site"</a>. 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Retrieved <span class="nowrap">2013-08-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Newgrange+Ireland+-+Megalithic+Passage+Tomb+-+World+Heritage+Site&rft.pub=Knowth.com&rft.date=2007-12-21&rft_id=http%3A%2F%2Fknowth.com%2Fnewgrange.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">For example, the trislele on <a href="/wiki/Achilles" title="Achilles">Achilles</a>' round shield on an Attic late sixth-century <i><a href="/wiki/Hydria" title="Hydria">hydria</a></i> at the <a href="/wiki/Boston_Museum_of_Fine_Arts" class="mw-redirect" title="Boston Museum of Fine Arts">Boston Museum of Fine Arts</a>, illustrated in John Boardman, Jasper Griffin and Oswyn Murray, <i>Greece and the Hellenistic World</i> (Oxford History of the Classical World) vol. I (1988), p. 50.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf">"Rock Art Of Latin America & The Caribbean"</a> <span class="cs1-format">(PDF)</span>. International Council on Monuments & Sites. June 2006. p. 5. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140105032613/http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 5 January 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">4 January</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Rock+Art+Of+Latin+America+%26+The+Caribbean&rft.pages=5&rft.pub=International+Council+on+Monuments+%26+Sites&rft.date=2006-06&rft_id=http%3A%2F%2Fwww.icomos.org%2Fstudies%2Frockart-latinamerica%2Ffulltext.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf">"Rock Art Of Latin America & The Caribbean"</a> <span class="cs1-format">(PDF)</span>. International Council on Monuments & Sites. 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Retrieved <span class="nowrap">4 January</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Rock+Art+Of+Latin+America+%26+The+Caribbean&rft.pages=99&rft.pub=International+Council+on+Monuments+%26+Sites&rft.date=2006-06&rft_id=http%3A%2F%2Fwww.icomos.org%2Fstudies%2Frockart-latinamerica%2Ffulltext.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf">"Rock Art Of Latin America & The Caribbean"</a> <span class="cs1-format">(PDF)</span>. International Council on Monuments & Sites. 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"The Exchange of Natures and the Nature(s) of Time and Silence". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_2V4i07PNzkC"><i>Images and Ideas in Modern French Piano Music: The Extra-musical Subtext in Piano Works by Ravel, Debussy, and Messiaen</i></a>. Aesthetics in music, ISSN 1062-404X, number 6. Stuyvesant, New York: Pendragon Press. p. 353<span class="reference-accessdate">. Retrieved <span class="nowrap">30 June</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Exchange+of+Natures+and+the+Nature%28s%29+of+Time+and+Silence&rft.btitle=Images+and+Ideas+in+Modern+French+Piano+Music%3A+The+Extra-musical+Subtext+in+Piano+Works+by+Ravel%2C+Debussy%2C+and+Messiaen&rft.place=Stuyvesant%2C+New+York&rft.series=Aesthetics+in+music%2C+ISSN+1062-404X%2C+number+6&rft.pages=353&rft.pub=Pendragon+Press&rft.date=1997&rft.aulast=Bruhn&rft.aufirst=Siglind&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_2V4i07PNzkC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span> </span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIsrael,_Nico2015" class="citation book cs1">Israel, Nico (2015). <i>Spirals : the whirled image in twentieth-century literature and art</i>. New York Columbia University Press. pp. 161–186. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-231-15302-7" title="Special:BookSources/978-0-231-15302-7"><bdi>978-0-231-15302-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spirals+%3A+the+whirled+image+in+twentieth-century+literature+and+art&rft.pages=161-186&rft.pub=New+York+Columbia+University+Press&rft.date=2015&rft.isbn=978-0-231-15302-7&rft.au=Israel%2C+Nico&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">2012 A Piece of Mind By Wayne A Beale</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www.blurb.com/distribution?id=573100/#/project/573100/project-details/edit">http://www.blurb.com/distribution?id=573100/#/project/573100/project-details/edit</a> <span style="font-size:0.95em; font-size:95%; color: var( --color-subtle, #555 )">(subscription required)</span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStark2012" class="citation web cs1">Stark, Tanja (4 July 2012). <a rel="nofollow" class="external text" href="https://tanjastark.com/exhibition-notes/">"Spiral Journeys : Turning and Returning"</a>. <i>tanjastark.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=tanjastark.com&rft.atitle=Spiral+Journeys+%3A+Turning+and+Returning&rft.date=2012-07-04&rft.aulast=Stark&rft.aufirst=Tanja&rft_id=https%3A%2F%2Ftanjastark.com%2Fexhibition-notes%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStark" class="citation web cs1">Stark, Tanja. <a rel="nofollow" class="external text" href="https://www.jungsocietymelbourne.com/march-2022">"Lecture : Spiralling Undercurrents: Archetypal Symbols of Hurt, Hope and Healing"</a>. <i>Jung Society Melbourne</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Jung+Society+Melbourne&rft.atitle=Lecture+%3A+Spiralling+Undercurrents%3A+Archetypal+Symbols+of+Hurt%2C+Hope+and+Healing&rft.aulast=Stark&rft.aufirst=Tanja&rft_id=https%3A%2F%2Fwww.jungsocietymelbourne.com%2Fmarch-2022&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Related_publications">Related publications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spiral&action=edit&section=14" title="Edit section: Related publications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Cook, T., 1903. <i>Spirals in nature and art</i>. Nature 68 (1761), 296.</li> <li>Cook, T., 1979. <i>The curves of life</i>. Dover, New York.</li> <li>Habib, Z., Sakai, M., 2005. <i>Spiral transition curves and their applications</i>. Scientiae Mathematicae Japonicae 61 (2), 195 – 206.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDimulyoHabibSakai2009" class="citation journal cs1">Dimulyo, Sarpono; Habib, Zulfiqar; Sakai, Manabu (2009). "Fair cubic transition between two circles with one circle inside or tangent to the other". <i>Numerical Algorithms</i>. <b>51</b> (4): 461–476. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009NuAlg..51..461D">2009NuAlg..51..461D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11075-008-9252-1">10.1007/s11075-008-9252-1</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:22532724">22532724</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Numerical+Algorithms&rft.atitle=Fair+cubic+transition+between+two+circles+with+one+circle+inside+or+tangent+to+the+other&rft.volume=51&rft.issue=4&rft.pages=461-476&rft.date=2009&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A22532724%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs11075-008-9252-1&rft_id=info%3Abibcode%2F2009NuAlg..51..461D&rft.aulast=Dimulyo&rft.aufirst=Sarpono&rft.au=Habib%2C+Zulfiqar&rft.au=Sakai%2C+Manabu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></li> <li>Harary, G., Tal, A., 2011. <i>The natural 3D spiral</i>. Computer Graphics Forum 30 (2), 237 – 246 <a rel="nofollow" class="external autonumber" href="http://webee.technion.ac.il/~ayellet/Ps/11-HararyTal.pdf">[1]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151122013249/http://webee.technion.ac.il/~ayellet/Ps/11-HararyTal.pdf">Archived</a> 2015-11-22 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li>Xu, L., Mould, D., 2009. <i>Magnetic curves: curvature-controlled aesthetic curves using magnetic fields</i>. In: Deussen, O., Hall, P. (Eds.), Computational Aesthetics in Graphics, Visualization, and Imaging. The Eurographics Association <a rel="nofollow" class="external autonumber" href="http://gigl.scs.carleton.ca/sites/default/files/ling_xu/artn-cae.pdf">[2]</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWangZhaoZhangXu2004" class="citation journal cs1">Wang, Yulin; Zhao, Bingyan; Zhang, Luzou; Xu, Jiachuan; Wang, Kanchang; Wang, Shuchun (2004). "Designing fair curves using monotone curvature pieces". <i>Computer Aided Geometric Design</i>. <b>21</b> (5): 515–527. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cagd.2004.04.001">10.1016/j.cagd.2004.04.001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Aided+Geometric+Design&rft.atitle=Designing+fair+curves+using+monotone+curvature+pieces&rft.volume=21&rft.issue=5&rft.pages=515-527&rft.date=2004&rft_id=info%3Adoi%2F10.1016%2Fj.cagd.2004.04.001&rft.aulast=Wang&rft.aufirst=Yulin&rft.au=Zhao%2C+Bingyan&rft.au=Zhang%2C+Luzou&rft.au=Xu%2C+Jiachuan&rft.au=Wang%2C+Kanchang&rft.au=Wang%2C+Shuchun&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKurnosenko2010" class="citation journal cs1">Kurnosenko, A. (2010). "Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data". <i>Computer Aided Geometric Design</i>. <b>27</b> (3): 262–280. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0902.4834">0902.4834</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cagd.2009.12.004">10.1016/j.cagd.2009.12.004</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:14476206">14476206</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Aided+Geometric+Design&rft.atitle=Applying+inversion+to+construct+planar%2C+rational+spirals+that+satisfy+two-point+G2+Hermite+data&rft.volume=27&rft.issue=3&rft.pages=262-280&rft.date=2010&rft_id=info%3Aarxiv%2F0902.4834&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14476206%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.cagd.2009.12.004&rft.aulast=Kurnosenko&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></li> <li>A. Kurnosenko. <i>Two-point G2 Hermite interpolation with spirals by inversion of hyperbola</i>. Computer Aided Geometric Design, 27(6), 474–481, 2010.</li> <li>Miura, K.T., 2006. <i>A general equation of aesthetic curves and its self-affinity</i>. Computer-Aided Design and Applications 3 (1–4), 457–464 <a rel="nofollow" class="external autonumber" href="http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/pdf/KTMiura-CAD06Final.pdf">[3]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130628000547/http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/pdf/KTMiura-CAD06Final.pdf">Archived</a> 2013-06-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li>Miura, K., Sone, J., Yamashita, A., Kaneko, T., 2005. <i>Derivation of a general formula of aesthetic curves</i>. In: 8th International Conference on Humans and Computers (HC2005). Aizu-Wakamutsu, Japan, pp. 166 – 171 <a rel="nofollow" class="external autonumber" href="http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/pdf/acurveHC0.pdf">[4]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130628051506/http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/pdf/acurveHC0.pdf">Archived</a> 2013-06-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeekWalton1989" class="citation journal cs1">Meek, D.S.; Walton, D.J. (1989). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0377-0427%2889%2990076-9">"The use of Cornu spirals in drawing planar curves of controlled curvature"</a>. <i>Journal of Computational and Applied Mathematics</i>. <b>25</b>: 69–78. <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/10.1016%2F0377-0427%2889%2990076-9">10.1016/0377-0427(89)90076-9</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+and+Applied+Mathematics&rft.atitle=The+use+of+Cornu+spirals+in+drawing+planar+curves+of+controlled+curvature&rft.volume=25&rft.pages=69-78&rft.date=1989&rft_id=info%3Adoi%2F10.1016%2F0377-0427%2889%2990076-9&rft.aulast=Meek&rft.aufirst=D.S.&rft.au=Walton%2C+D.J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0377-0427%252889%252990076-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas2017" class="citation journal cs1">Thomas, Sunil (2017). "Potassium sulfate forms a spiral structure when dissolved in solution". <i>Russian Journal of Physical Chemistry B</i>. <b>11</b> (1): 195–198. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017RJPCB..11..195T">2017RJPCB..11..195T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2FS1990793117010328">10.1134/S1990793117010328</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:99162341">99162341</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Russian+Journal+of+Physical+Chemistry+B&rft.atitle=Potassium+sulfate+forms+a+spiral+structure+when+dissolved+in+solution&rft.volume=11&rft.issue=1&rft.pages=195-198&rft.date=2017&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A99162341%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1134%2FS1990793117010328&rft_id=info%3Abibcode%2F2017RJPCB..11..195T&rft.aulast=Thomas&rft.aufirst=Sunil&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpiral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFarin2006" class="citation journal cs1">Farin, Gerald (2006). 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Computer-Aided Design and Applications 4 (9–10), 477–486 <a rel="nofollow" class="external autonumber" href="http://www.yoshida-lab.net/aesthetic/cad07yoshida.pdf">[6]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303205632/http://www.yoshida-lab.net/aesthetic/cad07yoshida.pdf">Archived</a> 2016-03-03 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li>Ziatdinov, R., Yoshida, N., Kim, T., 2012. <i>Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions</i>. Computer Aided Geometric Design 29 (2), 129—140 <a rel="nofollow" class="external autonumber" href="https://www.sciencedirect.com/science/article/abs/pii/S0167839611001452">[7]</a>.</li> <li>Ziatdinov, R., Yoshida, N., Kim, T., 2012. <i>Fitting G2 multispiral transition curve joining two straight lines</i>, Computer-Aided Design 44(6), 591—596 <a rel="nofollow" class="external autonumber" href="https://www.sciencedirect.com/science/article/pii/S001044851200019X">[8]</a>.</li> <li>Ziatdinov, R., 2012. <i>Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function</i>. Computer Aided Geometric Design 29(7): 510–518, 2012 <a rel="nofollow" class="external autonumber" href="https://www.sciencedirect.com/science/article/abs/pii/S0167839612000325">[9]</a>.</li> <li>Ziatdinov, R., Miura K.T., 2012. <i>On the Variety of Planar Spirals and Their Applications in Computer Aided Design</i>. European Researcher 27(8–2), 1227—1232 <a rel="nofollow" class="external autonumber" href="http://www.erjournal.ru/pdf.html?n=1345307278.pdf">[10]</a>.</li></ul> <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=Spiral&action=edit&section=15" 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 data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Spirals" class="extiw" title="commons:Category:Spirals">Spiral</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external autonumber" href="http://www.mathe.tu-freiberg.de/~hebisch/cafe/jamnitzer/galerie7g.html">[11]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210702004420/http://www.mathe.tu-freiberg.de/~hebisch/cafe/jamnitzer/galerie7g.html">Archived</a> 2021-07-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://oeis.org/A202407">Archimedes' spiral transforms into Galileo's spiral. 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class="navbox-group" style="width:1%">Curves</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/Algebraic_curve" title="Algebraic curve">Algebraic</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Gallery_of_curves" title="Gallery of curves">Gallery</a></li> <li><a href="/wiki/List_of_curves" title="List of curves">List</a></li> <li><a href="/wiki/List_of_curves_topics" title="List of curves topics">Topics</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Golden_spiral_in_triangles.png/60px-Golden_spiral_in_triangles.png" decoding="async" width="60" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Golden_spiral_in_triangles.png/90px-Golden_spiral_in_triangles.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Golden_spiral_in_triangles.png/120px-Golden_spiral_in_triangles.png 2x" data-file-width="284" data-file-height="329" /></span></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Helices</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/Helix_angle" title="Helix angle">Angle</a></li> <li><a href="/wiki/Helical_antenna" title="Helical antenna">Antenna</a></li> <li><a href="/wiki/Boerdijk%E2%80%93Coxeter_helix" title="Boerdijk–Coxeter helix">Boerdijk–Coxeter</a></li> <li><a href="/wiki/Hemihelix" title="Hemihelix">Hemi</a></li> <li><a href="/wiki/Helical_symmetry" class="mw-redirect" title="Helical symmetry">Symmetry</a></li> <li><a href="/wiki/Triple_helix" title="Triple helix">Triple</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Biochemistry" scope="row" class="navbox-group" style="width:1%;text-align: center;">Biochemistry</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/310_helix" title="310 helix">3<sub>10</sub></a></li> <li><a href="/wiki/Alpha_helix" title="Alpha helix">Alpha</a></li> <li><a href="/wiki/Beta_helix" title="Beta helix">Beta</a></li> <li><a href="/wiki/Nucleic_acid_double_helix" title="Nucleic acid double helix">Double</a></li> <li><a href="/wiki/Pi_helix" title="Pi helix">Pi</a></li> <li><a href="/wiki/Polyproline_helix" title="Polyproline helix">Polyproline</a></li> <li><a href="/wiki/Superhelix" title="Superhelix">Super</a></li> <li><a href="/wiki/Triple_helix" title="Triple helix">Triple</a> <ul><li><a href="/wiki/Collagen_helix" title="Collagen helix">Collagen</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spirals</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/Archimedean_spiral" title="Archimedean spiral">Archimedean</a></li> <li><a href="/wiki/Cotes%27s_spiral" title="Cotes's spiral">Cotes's</a> <ul><li><a href="/wiki/Epispiral" title="Epispiral">Epispiral</a></li> <li><a href="/wiki/Hyperbolic_spiral" title="Hyperbolic spiral">Hyperbolic</a></li> <li><a href="/wiki/Poinsot%27s_spirals" title="Poinsot's spirals">Poinsot's</a></li></ul></li> <li><a href="/wiki/Doyle_spiral" title="Doyle spiral">Doyle</a></li> <li><a href="/wiki/Euler_spiral" title="Euler spiral">Euler</a></li> <li><a href="/wiki/Fermat%27s_spiral" title="Fermat's spiral">Fermat's</a></li> <li><a href="/wiki/Involute" title="Involute">Involute</a></li> <li><a href="/wiki/List_of_spirals" title="List of spirals">List</a></li> <li><a href="/wiki/Logarithmic_spiral" title="Logarithmic spiral">Logarithmic</a> <ul><li><a href="/wiki/Golden_spiral" title="Golden spiral">Golden</a></li></ul></li> <li><i><a href="/wiki/On_Spirals" title="On Spirals">On Spirals</a></i></li> <li><a href="/wiki/Padovan_cuboid_spiral" title="Padovan cuboid spiral">Padovan</a></li> <li><a href="/wiki/Pitch_angle_of_a_spiral" title="Pitch angle of a spiral">Pitch angle</a></li> <li><a href="/wiki/Spiral_of_Theodorus" title="Spiral of Theodorus">Theodorus</a></li> <li><a href="/wiki/Spirangle" title="Spirangle">Spirangle</a></li> <li><a href="/wiki/Ulam_spiral" title="Ulam spiral">Ulam</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"><style 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