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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extension_to_multiple_states"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Extension to multiple states</span> </div> </a> <ul id="toc-Extension_to_multiple_states-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extension_to_multiple_ants" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extension_to_multiple_ants"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Extension to multiple ants</span> </div> </a> <ul id="toc-Extension_to_multiple_ants-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-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">9</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">Langton's ant</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 16 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-16" 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">16 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Formiga_de_Langton" title="Formiga de Langton – Catalan" lang="ca" hreflang="ca" data-title="Formiga de Langton" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ameise_(Turingmaschine)" title="Ameise (Turingmaschine) – German" lang="de" hreflang="de" data-title="Ameise (Turingmaschine)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Hormiga_de_Langton" title="Hormiga de Langton – Spanish" lang="es" hreflang="es" data-title="Hormiga de Langton" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%88%D8%B1%DA%86%D9%87_%D9%84%D9%86%DA%AF%D8%AA%D9%88%D9%86" title="مورچه لنگتون – Persian" lang="fa" hreflang="fa" data-title="مورچه لنگتون" 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/Fourmi_de_Langton" title="Fourmi de Langton – French" lang="fr" hreflang="fr" data-title="Fourmi de Langton" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%9E%AD%ED%84%B4%EC%9D%98_%EA%B0%9C%EB%AF%B8" title="랭턴의 개미 – Korean" lang="ko" hreflang="ko" data-title="랭턴의 개미" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BC%D5%A1%D5%B6%D5%A3%D5%BF%D5%B8%D5%B6%D5%AB_%D5%B4%D6%80%D5%BB%D5%B5%D5%B8%D6%82%D5%B6" title="Լանգտոնի մրջյուն – Armenian" lang="hy" hreflang="hy" data-title="Լանգտոնի մրջյուն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Formica_di_Langton" title="Formica di Langton – Italian" lang="it" hreflang="it" data-title="Formica di Langton" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A0%D7%9E%D7%9C%D7%94_%D7%A9%D7%9C_%D7%9C%D7%A0%D7%92%D7%98%D7%95%D7%9F" title="הנמלה של לנגטון – Hebrew" lang="he" hreflang="he" data-title="הנמלה של לנגטון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%B3%E3%82%B0%E3%83%88%E3%83%B3%E3%81%AE%E3%82%A2%E3%83%AA" title="ラングトンのアリ – Japanese" lang="ja" hreflang="ja" data-title="ラングトンのアリ" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Mr%C3%B3wka_Langtona" title="Mrówka Langtona – Polish" lang="pl" hreflang="pl" data-title="Mrówka Langtona" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Formiga_de_Langton" title="Formiga de Langton – Portuguese" lang="pt" hreflang="pt" data-title="Formiga de Langton" 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%9C%D1%83%D1%80%D0%B0%D0%B2%D0%B5%D0%B9_%D0%9B%D1%8D%D0%BD%D0%B3%D1%82%D0%BE%D0%BD%D0%B0" title="Муравей Лэнгтона – Russian" lang="ru" hreflang="ru" data-title="Муравей Лэнгтона" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Langtons_myra" title="Langtons myra – Swedish" lang="sv" hreflang="sv" data-title="Langtons myra" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D1%83%D1%80%D0%B0%D1%85%D0%B0_%D0%9B%D0%B5%D0%BD%D0%B3%D1%82%D0%BE%D0%BD%D0%B0" title="Мураха Ленгтона – Ukrainian" lang="uk" hreflang="uk" data-title="Мураха Ленгтона" 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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">Two-dimensional Turing machine with emergent behavior</div> <p class="mw-empty-elt"> </p> <div id="LangtonsAnt" style="float:right;"></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:LangtonsAnt.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/LangtonsAnt.svg/184px-LangtonsAnt.svg.png" decoding="async" width="184" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/LangtonsAnt.svg/276px-LangtonsAnt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/LangtonsAnt.svg/368px-LangtonsAnt.svg.png 2x" data-file-width="184" data-file-height="184" /></a><figcaption>Langton's ant after 11,000 steps. A red pixel shows the ant's location.</figcaption></figure> <p><b>Langton's ant</b> is a two-dimensional <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> with a very simple set of rules but complex <a href="/wiki/Emergence" title="Emergence">emergent</a> behavior. It was invented by <a href="/wiki/Chris_Langton" class="mw-redirect" title="Chris Langton">Chris Langton</a> in 1986 and runs on a <a href="/wiki/Square_tiling" title="Square tiling">square lattice</a> of black and white cells.<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> The idea has been generalized in several different ways, such as <a href="/wiki/Turmite" title="Turmite">turmites</a> which add more colors and more states. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Rules">Rules</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Langton%27s_ant&action=edit&section=1" title="Edit section: Rules"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:LangtonsAntAnimated.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/09/LangtonsAntAnimated.gif" decoding="async" width="353" height="353" class="mw-file-element" data-file-width="353" data-file-height="353" /></a><figcaption>Animation of first 200 steps of Langton's ant</figcaption></figure> <p>Squares on a plane are colored variously either black or white. We arbitrarily identify one square as the "ant". The ant can travel in any of the four cardinal directions at each step it takes. The "ant" moves according to the rules below: </p> <ul><li>At a white square, turn 90° clockwise, flip the color of the square, move forward one unit</li> <li>At a black square, turn 90° counter-clockwise, flip the color of the square, move forward one unit</li></ul> <p>Langton's ant can also be described as a <a href="/wiki/Cellular_automaton" title="Cellular automaton">cellular automaton</a>, where the grid is colored black or white and the "ant" square has one of eight different colors assigned to encode the combination of black/white state and the current direction of motion of the ant.<sup id="cite_ref-Gajardo2000_2-0" class="reference"><a href="#cite_note-Gajardo2000-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Modes_of_behavior">Modes of behavior</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Langton%27s_ant&action=edit&section=2" title="Edit section: Modes of behavior"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These simple rules lead to complex behavior. Three distinct modes of behavior are apparent,<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> when starting on a completely white grid. </p> <ol><li>Simplicity. During the first few hundred moves it creates very simple patterns which are often <a href="/wiki/Symmetry" title="Symmetry">symmetric</a>.</li> <li>Chaos. After a few hundred moves, a large, irregular pattern of black and white squares appears. The ant traces a <a href="/wiki/Pseudorandomness" title="Pseudorandomness">pseudo-random</a> path until around 10,000 steps.</li> <li>Emergent order. Finally the ant starts building a recurrent "highway" pattern of 104 steps that repeats indefinitely.</li></ol> <p>All <a href="/wiki/Finite-state_machine" title="Finite-state machine">finite</a> initial configurations tested eventually converge to the same repetitive pattern, suggesting that the "highway" is an <a href="/wiki/Attractor" title="Attractor">attractor</a> of Langton's ant, but no one has been able to prove that this is true for all such initial configurations. It is only known that the ant's trajectory is always unbounded regardless of the initial configuration<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> – this is known as the <a href="/wiki/E.G.D._Cohen" class="mw-redirect" title="E.G.D. Cohen">Cohen</a>-Kong theorem.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Computational_properties">Computational properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Langton%27s_ant&action=edit&section=3" title="Edit section: Computational properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2000, Gajardo et al. showed a construction that calculates any <a href="/wiki/Boolean_circuit" title="Boolean circuit">boolean circuit</a> using the trajectory of a single instance of Langton's ant.<sup id="cite_ref-Gajardo2000_2-1" class="reference"><a href="#cite_note-Gajardo2000-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Extension_to_multiple_colors">Extension to multiple colors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Langton%27s_ant&action=edit&section=4" title="Edit section: Extension to multiple colors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Greg_Turk" title="Greg Turk">Greg Turk</a> and <a href="/wiki/Jim_Propp" title="Jim Propp">Jim Propp</a> considered a simple extension to Langton's ant where instead of just two colors, more colors are used.<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> The colors are modified in a cyclic fashion. A simple naming scheme is used: for each of the successive colors, a letter "L" or "R" is used to indicate whether a left or right turn should be taken. Langton's ant has the name "RL" in this naming scheme. </p><p>Some of these extended Langton's ants produce patterns that become <a href="/wiki/Symmetric" class="mw-redirect" title="Symmetric">symmetric</a> over and over again. One of the simplest examples is the ant "RLLR". One sufficient condition for this to happen is that the ant's name, seen as a cyclic list, consists of consecutive pairs of identical letters "LL" or "RR". (the term "cyclic list" indicates that the last letter may pair with the first one) The proof involves <a href="/wiki/Truchet_tiles" title="Truchet tiles">Truchet tiles</a>. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption">Some example patterns in the multiple-color extension of Langton's ants:</li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:LangtonsAnt-nColor_RLR_13937.png" class="mw-file-description" title="RLR: Grows chaotically. It is not known whether this ant ever produces a highway."><img alt="RLR: Grows chaotically. It is not known whether this ant ever produces a highway." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/LangtonsAnt-nColor_RLR_13937.png/120px-LangtonsAnt-nColor_RLR_13937.png" decoding="async" width="120" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/LangtonsAnt-nColor_RLR_13937.png/180px-LangtonsAnt-nColor_RLR_13937.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/LangtonsAnt-nColor_RLR_13937.png/240px-LangtonsAnt-nColor_RLR_13937.png 2x" data-file-width="496" data-file-height="472" /></a></span></div> <div class="gallerytext">RLR: Grows chaotically. It is not known whether this ant ever produces a highway.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:LangtonsAnt-nColor_LLRR_123157.png" class="mw-file-description" title="LLRR: Grows symmetrically."><img alt="LLRR: Grows symmetrically." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/LangtonsAnt-nColor_LLRR_123157.png/120px-LangtonsAnt-nColor_LLRR_123157.png" decoding="async" width="120" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/LangtonsAnt-nColor_LLRR_123157.png/180px-LangtonsAnt-nColor_LLRR_123157.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/LangtonsAnt-nColor_LLRR_123157.png/240px-LangtonsAnt-nColor_LLRR_123157.png 2x" data-file-width="496" data-file-height="472" /></a></span></div> <div class="gallerytext">LLRR: Grows symmetrically.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:LangtonsAnt-nColor_LRRRRRLLR_70273.png" class="mw-file-description" title="LRRRRRLLR: Fills space in a square around itself."><img alt="LRRRRRLLR: Fills space in a square around itself." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/LangtonsAnt-nColor_LRRRRRLLR_70273.png/120px-LangtonsAnt-nColor_LRRRRRLLR_70273.png" decoding="async" width="120" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/LangtonsAnt-nColor_LRRRRRLLR_70273.png/180px-LangtonsAnt-nColor_LRRRRRLLR_70273.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/LangtonsAnt-nColor_LRRRRRLLR_70273.png/240px-LangtonsAnt-nColor_LRRRRRLLR_70273.png 2x" data-file-width="496" data-file-height="472" /></a></span></div> <div class="gallerytext">LRRRRRLLR: Fills space in a square around itself.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:LangtonsAnt-nColor_LLRRRLRLRLLR_36437.png" class="mw-file-description" title="LLRRRLRLRLLR: Creates a convoluted highway."><img alt="LLRRRLRLRLLR: Creates a convoluted highway." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/LangtonsAnt-nColor_LLRRRLRLRLLR_36437.png/120px-LangtonsAnt-nColor_LLRRRLRLRLLR_36437.png" decoding="async" width="120" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/LangtonsAnt-nColor_LLRRRLRLRLLR_36437.png/180px-LangtonsAnt-nColor_LLRRRLRLRLLR_36437.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/LangtonsAnt-nColor_LLRRRLRLRLLR_36437.png/240px-LangtonsAnt-nColor_LLRRRLRLRLLR_36437.png 2x" data-file-width="496" data-file-height="472" /></a></span></div> <div class="gallerytext">LLRRRLRLRLLR: Creates a convoluted highway.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:LangtonsAnt-nColor_RRLLLRLLLRRR_32734.png" class="mw-file-description" title="RRLLLRLLLRRR: Creates a filled triangle shape that grows and moves after 15900~ iterations."><img alt="RRLLLRLLLRRR: Creates a filled triangle shape that grows and moves after 15900~ iterations." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/LangtonsAnt-nColor_RRLLLRLLLRRR_32734.png/120px-LangtonsAnt-nColor_RRLLLRLLLRRR_32734.png" decoding="async" width="120" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/LangtonsAnt-nColor_RRLLLRLLLRRR_32734.png/180px-LangtonsAnt-nColor_RRLLLRLLLRRR_32734.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/LangtonsAnt-nColor_RRLLLRLLLRRR_32734.png/240px-LangtonsAnt-nColor_RRLLLRLLLRRR_32734.png 2x" data-file-width="496" data-file-height="472" /></a></span></div> <div class="gallerytext">RRLLLRLLLRRR: Creates a filled triangle shape that grows and moves after 15900~ iterations.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:CA3061-81k7.png" class="mw-file-description" title="L2NNL1L2L1: Hexagonal grid, grows circularly."><img alt="L2NNL1L2L1: Hexagonal grid, grows circularly." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/CA3061-81k7.png/120px-CA3061-81k7.png" decoding="async" width="120" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/CA3061-81k7.png/180px-CA3061-81k7.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/CA3061-81k7.png/240px-CA3061-81k7.png 2x" data-file-width="2560" data-file-height="2420" /></a></span></div> <div class="gallerytext">L<sub>2</sub>NNL<sub>1</sub>L<sub>2</sub>L<sub>1</sub>: Hexagonal grid, grows circularly.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:CA174906.png" class="mw-file-description" title="L1L2NUL2L1R2: Hexagonal grid, spiral growth."><img alt="L1L2NUL2L1R2: Hexagonal grid, spiral growth." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/CA174906.png/120px-CA174906.png" decoding="async" width="120" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/CA174906.png/180px-CA174906.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/CA174906.png/240px-CA174906.png 2x" data-file-width="3590" data-file-height="3260" /></a></span></div> <div class="gallerytext">L<sub>1</sub>L<sub>2</sub>NUL<sub>2</sub>L<sub>1</sub>R<sub>2</sub>: Hexagonal grid, spiral growth.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:CA50338_animation.gif" class="mw-file-description" title="R1R2NUR2R1L2: Animation."><img alt="R1R2NUR2R1L2: Animation." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/CA50338_animation.gif/114px-CA50338_animation.gif" decoding="async" width="114" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/CA50338_animation.gif/171px-CA50338_animation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/CA50338_animation.gif/229px-CA50338_animation.gif 2x" data-file-width="464" data-file-height="487" /></a></span></div> <div class="gallerytext">R<sub>1</sub>R<sub>2</sub>NUR<sub>2</sub>R<sub>1</sub>L<sub>2</sub>: Animation.</div> </li> </ul> <p>The hexagonal grid permits up to six different rotations, which are notated here as N (no change), R<sub>1</sub> (60° clockwise), R<sub>2</sub> (120° clockwise), U (180°), L<sub>2</sub> (120° counter-clockwise), L<sub>1</sub> (60° counter-clockwise). </p> <div class="mw-heading mw-heading2"><h2 id="Extension_to_multiple_states">Extension to multiple states</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Langton%27s_ant&action=edit&section=5" title="Edit section: Extension to multiple states"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Turmite" title="Turmite">Turmite</a></div> <p>A further extension of Langton's ants is to consider multiple states of the Turing machine – as if the ant itself has a color that can change. These ants are called <a href="/wiki/Turmite" title="Turmite">turmites</a>, a contraction of "Turing machine <a href="/wiki/Termite" title="Termite">termites</a>". Common behaviours include the production of highways, chaotic growth and spiral growth.<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> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption">Some example turmites:</li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Turmite-111180121010-12536.svg" class="mw-file-description" title="Spiral growth."><img alt="Spiral growth." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Turmite-111180121010-12536.svg/105px-Turmite-111180121010-12536.svg.png" decoding="async" width="105" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Turmite-111180121010-12536.svg/158px-Turmite-111180121010-12536.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Turmite-111180121010-12536.svg/211px-Turmite-111180121010-12536.svg.png 2x" data-file-width="204" data-file-height="232" /></a></span></div> <div class="gallerytext">Spiral growth.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Turmite-120121010011-8342.svg" class="mw-file-description" title="Semi-chaotic growth."><img alt="Semi-chaotic growth." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Turmite-120121010011-8342.svg/106px-Turmite-120121010011-8342.svg.png" decoding="async" width="106" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Turmite-120121010011-8342.svg/158px-Turmite-120121010011-8342.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Turmite-120121010011-8342.svg/211px-Turmite-120121010011-8342.svg.png 2x" data-file-width="512" data-file-height="582" /></a></span></div> <div class="gallerytext">Semi-chaotic growth.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Turmite-121021110111-27731.svg" class="mw-file-description" title="Production of a highway after a period of chaotic growth."><img alt="Production of a highway after a period of chaotic growth." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Turmite-121021110111-27731.svg/105px-Turmite-121021110111-27731.svg.png" decoding="async" width="105" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Turmite-121021110111-27731.svg/158px-Turmite-121021110111-27731.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Turmite-121021110111-27731.svg/211px-Turmite-121021110111-27731.svg.png 2x" data-file-width="204" data-file-height="232" /></a></span></div> <div class="gallerytext">Production of a highway after a period of chaotic growth.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Turmite-121181121020-65932.svg" class="mw-file-description" title="Chaotic growth with a distinctive texture."><img alt="Chaotic growth with a distinctive texture." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Turmite-121181121020-65932.svg/105px-Turmite-121181121020-65932.svg.png" decoding="async" width="105" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Turmite-121181121020-65932.svg/158px-Turmite-121181121020-65932.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Turmite-121181121020-65932.svg/211px-Turmite-121181121020-65932.svg.png 2x" data-file-width="204" data-file-height="232" /></a></span></div> <div class="gallerytext">Chaotic growth with a distinctive texture.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Turmite-180121020081-223577.svg" class="mw-file-description" title="Growth with a distinctive texture inside an expanding frame."><img alt="Growth with a distinctive texture inside an expanding frame." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Turmite-180121020081-223577.svg/105px-Turmite-180121020081-223577.svg.png" decoding="async" width="105" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Turmite-180121020081-223577.svg/158px-Turmite-180121020081-223577.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Turmite-180121020081-223577.svg/211px-Turmite-180121020081-223577.svg.png 2x" data-file-width="204" data-file-height="232" /></a></span></div> <div class="gallerytext">Growth with a distinctive texture inside an expanding frame.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Turmite-181181121010-10211.svg" class="mw-file-description" title="Constructing a Fibonacci spiral."><img alt="Constructing a Fibonacci spiral." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Turmite-181181121010-10211.svg/105px-Turmite-181181121010-10211.svg.png" decoding="async" width="105" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Turmite-181181121010-10211.svg/158px-Turmite-181181121010-10211.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Turmite-181181121010-10211.svg/211px-Turmite-181181121010-10211.svg.png 2x" data-file-width="204" data-file-height="232" /></a></span></div> <div class="gallerytext">Constructing a <a href="/wiki/Fibonacci_spiral" class="mw-redirect" title="Fibonacci spiral">Fibonacci spiral</a>.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Turmite_creating_a_growing_diamond.png" class="mw-file-description" title="Constructing a growing diamond"><img alt="Constructing a growing diamond" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Turmite_creating_a_growing_diamond.png/120px-Turmite_creating_a_growing_diamond.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Turmite_creating_a_growing_diamond.png/180px-Turmite_creating_a_growing_diamond.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d5/Turmite_creating_a_growing_diamond.png 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">Constructing a growing diamond</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Extension_to_multiple_ants">Extension to multiple ants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Langton%27s_ant&action=edit&section=6" title="Edit section: Extension to multiple ants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Langton%27s_Ant_colony.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/82/Langton%27s_Ant_colony.gif" decoding="async" width="402" height="265" class="mw-file-element" data-file-width="402" data-file-height="265" /></a><figcaption>A colony (as an absolute oscillator) builds a triangle</figcaption></figure> <p>Multiple Langton's ants can co-exist on the 2D plane, and their interactions give rise to complex, higher-order automata that collectively build a wide variety of organized structures. </p><p>There are different ways of modelling their interaction and the results of the simulation may strongly depend on the choices made.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Multiple turmites can co-exist on the 2D plane as long as there is a rule that defines what happens when they meet. <a href="/wiki/Ed_Pegg,_Jr." class="mw-redirect" title="Ed Pegg, Jr.">Ed Pegg, Jr.</a> considered ants that can turn for example <i>both</i> left and right, splitting in two and annihilating each other when they meet.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Langton%27s_ant&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Conway%27s_Game_of_Life" title="Conway's Game of Life">Conway's Game of Life</a> – Two-dimensional cellular automaton</li> <li><a href="/wiki/Langton%27s_loops" title="Langton's loops">Langton's loops</a> – Self-reproducing cellular automaton patterns</li> <li><a href="/wiki/Paterson%27s_worms" title="Paterson's worms">Paterson's worms</a> – Family of cellular automata to model feeding behaviour</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=Langton%27s_ant&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free 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(2012). <a rel="nofollow" class="external text" href="https://hal.inria.fr/inria-00462438/file/AsynchRobustnessTurmites.pdf">"Robustness of Multi-agent Models: The Example of Collaboration between Turmites with Synchronous and Asynchronous Updating"</a> <span class="cs1-format">(PDF)</span>. <i>Complex Systems</i>. <b>21</b> (3): 165–182. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.25088%2FComplexSystems.21.3.165">10.25088/ComplexSystems.21.3.165</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Complex+Systems&rft.atitle=Robustness+of+Multi-agent+Models%3A+The+Example+of+Collaboration+between+Turmites+with+Synchronous+and+Asynchronous+Updating&rft.volume=21&rft.issue=3&rft.pages=165-182&rft.date=2012&rft_id=info%3Adoi%2F10.25088%2FComplexSystems.21.3.165&rft.aulast=Belgacem&rft.aufirst=S.&rft.au=Fat%C3%A8s%2C+N.&rft_id=https%3A%2F%2Fhal.inria.fr%2Finria-00462438%2Ffile%2FAsynchRobustnessTurmites.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALangton%27s+ant" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPegg,_Jr." class="citation web cs1">Pegg, Jr., Ed. <a rel="nofollow" class="external text" href="http://www.mathpuzzle.com/26Mar03.html">"Math Puzzle"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">15 October</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Math+Puzzle&rft.aulast=Pegg%2C+Jr.&rft.aufirst=Ed&rft_id=http%3A%2F%2Fwww.mathpuzzle.com%2F26Mar03.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALangton%27s+ant" class="Z3988"></span>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Langton%27s_ant&action=edit&section=9" 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 <wbr /><i><b><a href="https://commons.wikimedia.org/wiki/Category:Langton%27s_ant" class="extiw" title="commons:Category:Langton's ant">Langton's ant</a></b></i> and <wbr /><i><b><a href="https://commons.wikimedia.org/wiki/Category:Turmite" class="extiw" title="commons:Category:Turmite">Turmite</a></b></i>.</div></div> </div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Langton's_ant"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LangtonsAnt.html">"Langton's ant"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Langton%27s+ant&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FLangtonsAnt.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALangton%27s+ant" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=w6XQQhCgq5c">Chris Langton demonstrating multiple ants interacting in a "colony"</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303211426/http://dev.whydomath.org/Reading_Room_Material/ian_stewart/AntyParticles.pdf">Mathematical Recreations column</a> by <a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Ian Stewart</a> using Langton's ant as a metaphor for a <a href="/wiki/Theory_of_everything" title="Theory of everything">theory of everything</a>. Contains the proof that Langton's ant is unbounded.</li> <li><a rel="nofollow" class="external text" href="https://gollygang.github.io/ruletablerepository/downloads/Langtons-Ant-nColor.zip">Golly script for generating rules in the multiple color extension of Langton's ant</a></li> <li><a rel="nofollow" class="external text" href="http://datagenetics.com/blog/september22015/index.html">DataGenetics, Langton's Ant (and Life)</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Langton%27s_ant&oldid=1239482896">https://en.wikipedia.org/w/index.php?title=Langton%27s_ant&oldid=1239482896</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Artificial_life" title="Category:Artificial life">Artificial life</a></li><li><a href="/wiki/Category:Cellular_automaton_rules" title="Category:Cellular automaton rules">Cellular automaton rules</a></li><li><a href="/wiki/Category:Turing_machine" title="Category:Turing machine">Turing machine</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_April_2018" title="Category:Use dmy dates from April 2018">Use dmy dates from April 2018</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 9 August 2024, at 15:14<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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