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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Stellating polygons</span> </div> </a> <ul id="toc-Stellating_polygons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stellating_polyhedra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stellating_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Stellating polyhedra</span> </div> </a> <button aria-controls="toc-Stellating_polyhedra-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 Stellating polyhedra subsection</span> </button> <ul id="toc-Stellating_polyhedra-sublist" class="vector-toc-list"> <li id="toc-Miller's_rules" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Miller's_rules"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Miller's rules</span> </div> </a> <ul id="toc-Miller's_rules-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_rules_for_stellation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_rules_for_stellation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Other rules for stellation</span> </div> </a> <ul id="toc-Other_rules_for_stellation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Stellating_polytopes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stellating_polytopes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Stellating polytopes</span> </div> </a> <ul id="toc-Stellating_polytopes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Naming_stellations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Naming_stellations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Naming stellations</span> </div> </a> <ul id="toc-Naming_stellations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stellation_to_infinity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stellation_to_infinity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Stellation to infinity</span> </div> </a> <ul id="toc-Stellation_to_infinity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-From_mathematics_to_art" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#From_mathematics_to_art"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>From mathematics to art</span> </div> </a> <ul id="toc-From_mathematics_to_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">8</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">9</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">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> 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href="https://ca.wikipedia.org/wiki/Estelaci%C3%B3" title="Estelació – Catalan" lang="ca" hreflang="ca" data-title="Estelació" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Estelaci%C3%B3n" title="Estelación – Spanish" lang="es" hreflang="es" data-title="Estelación" 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/Steligo" title="Steligo – Esperanto" lang="eo" hreflang="eo" data-title="Steligo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Stellation" title="Stellation – French" lang="fr" hreflang="fr" data-title="Stellation" 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%B3%84%EB%AA%A8%EC%96%91%ED%99%94" title="별모양화 – Korean" lang="ko" hreflang="ko" data-title="별모양화" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Stellazione" title="Stellazione – Italian" lang="it" hreflang="it" data-title="Stellazione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%98%9F%E5%9E%8B%E5%A4%9A%E9%9D%A2%E4%BD%93" 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data-title="Stelare" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%B7%D0%B2%D1%91%D0%B7%D0%B4%D1%87%D0%B0%D1%82%D0%BE%D0%B9_%D1%84%D0%BE%D1%80%D0%BC%D1%8B" title="Образование звёздчатой формы – Russian" lang="ru" hreflang="ru" data-title="Образование звёздчатой формы" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Stelacija" title="Stelacija – Slovenian" lang="sl" hreflang="sl" data-title="Stelacija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Extending the elements of a polytope to form a new figure</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Academ_Stellated_dodecagon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Academ_Stellated_dodecagon.svg/220px-Academ_Stellated_dodecagon.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Academ_Stellated_dodecagon.svg/330px-Academ_Stellated_dodecagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Academ_Stellated_dodecagon.svg/440px-Academ_Stellated_dodecagon.svg.png 2x" data-file-width="750" data-file-height="750" /></a><figcaption>Construction of a stellated <a href="/wiki/Dodecagon" title="Dodecagon">dodecagon</a>: <a href="/wiki/Regular_polygon" title="Regular polygon">a regular</a> polygon with <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> {12/5}.</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, <b>stellation</b> is the process of extending a <a href="/wiki/Polygon" title="Polygon">polygon</a> in two <a href="/wiki/Dimension" title="Dimension">dimensions</a>, a <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> in three dimensions, or, in general, a <a href="/wiki/Polytope" title="Polytope">polytope</a> in <i>n</i> dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word <i>stellation</i> comes from the Latin <i>stellātus</i>, "starred", which in turn comes from the Latin <i>stella</i>, "star". Stellation is the reciprocal or dual process to <i><a href="/wiki/Faceting" title="Faceting">faceting</a></i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Kepler's_definition"><span id="Kepler.27s_definition"></span>Kepler's definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stellation&action=edit&section=1" title="Edit section: Kepler's definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1619 <a href="/wiki/Kepler" class="mw-redirect" title="Kepler">Kepler</a> defined stellation for polygons and polyhedra as the process of extending edges or faces until they meet to form a new polygon or polyhedron. </p><p>He stellated the regular <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a> to obtain two regular star polyhedra, the <a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">small stellated dodecahedron</a> and the <a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">great stellated dodecahedron</a>. He also stellated the regular <a href="/wiki/Octahedron" title="Octahedron">octahedron</a> to obtain the <a href="/wiki/Stella_octangula" class="mw-redirect" title="Stella octangula">stella octangula</a>, a regular compound of two tetrahedra. </p> <div class="mw-heading mw-heading2"><h2 id="Stellating_polygons">Stellating polygons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stellation&action=edit&section=2" title="Edit section: Stellating polygons"><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:Regular_star_polygons.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Regular_star_polygons.svg/330px-Regular_star_polygons.svg.png" decoding="async" width="330" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Regular_star_polygons.svg/495px-Regular_star_polygons.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Regular_star_polygons.svg/660px-Regular_star_polygons.svg.png 2x" data-file-width="512" data-file-height="320" /></a><figcaption>Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols</figcaption></figure> <p>Stellating a regular polygon symmetrically creates a regular <a href="/wiki/Star_polygon" title="Star polygon">star polygon</a> or <a href="/wiki/Star_polygon#Star_figures" title="Star polygon">polygonal compound</a>. These polygons are characterised by the number of times <i>m</i> that the polygonal boundary winds around the centre of the figure. Like all regular polygons, their vertices lie on a circle. <i>m</i> also corresponds to the number of vertices around the circle to get from one end of a given edge to the other, starting at 1. </p><p>A regular star polygon is represented by its <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> {<i>n</i>/<i>m</i>}, where <i>n</i> is the number of vertices, <i>m</i> is the <i>step</i> used in sequencing the edges around it, and <i>m</i> and <i>n</i> are <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> (have no common <a href="/wiki/Divisor" title="Divisor">factor</a>). The case <i>m</i> = 1 gives the convex polygon {<i>n</i>}. <i>m</i> also must be less than half of <i>n</i>; otherwise the lines will either be parallel or diverge, preventing the figure from ever closing. </p><p>If <i>n</i> and <i>m</i> do have a common factor, then the figure is a regular compound. For example {6/2} is the regular compound of two triangles {3} or <a href="/wiki/Hexagram" title="Hexagram">hexagram</a>, while {10/4} is a compound of two pentagrams {5/2}. </p><p>Some authors use the Schläfli symbol for such regular compounds. Others regard the symbol as indicating a single path which is wound <i>m</i> times around <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den"><i>m</i></span></span>⁠</span> vertex points, such that one edge is superimposed upon another and each vertex point is visited <i>m</i> times. In this case a modified symbol may be used for the compound, for example 2{3} for the hexagram and 2{5/2} for the regular compound of two pentagrams. </p><p>A regular <i>n</i>-gon has <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i> – 4</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> stellations if <i>n</i> is <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a> (assuming compounds of multiple degenerate <a href="/wiki/Digon" title="Digon">digons</a> are not considered), and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i> – 3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> stellations if <i>n</i> is <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a>. </p> <div style="clear:both;" class=""></div> <table width="640" class="wikitable"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Pentagram_green.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Pentagram_green.svg/150px-Pentagram_green.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Pentagram_green.svg/225px-Pentagram_green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Pentagram_green.svg/300px-Pentagram_green.svg.png 2x" data-file-width="500" data-file-height="500" /></a></span><br />The <a href="/wiki/Pentagram" title="Pentagram">pentagram</a>, {5/2}, is the only stellation of a <a href="/wiki/Pentagon" title="Pentagon">pentagon</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Regular_star_figure_2(3,1).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Regular_star_figure_2%283%2C1%29.svg/150px-Regular_star_figure_2%283%2C1%29.svg.png" decoding="async" width="150" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Regular_star_figure_2%283%2C1%29.svg/225px-Regular_star_figure_2%283%2C1%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Regular_star_figure_2%283%2C1%29.svg/300px-Regular_star_figure_2%283%2C1%29.svg.png 2x" data-file-width="1000" data-file-height="866" /></a></span><br />The <a href="/wiki/Hexagram" title="Hexagram">hexagram</a>, {6/2}, the stellation of a <a href="/wiki/Hexagon" title="Hexagon">hexagon</a> and a compound of two triangles. </td> <td rowspan="2"><span typeof="mw:File"><a href="/wiki/File:Enneagon_stellations.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Enneagon_stellations.svg/320px-Enneagon_stellations.svg.png" decoding="async" width="320" height="375" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Enneagon_stellations.svg/480px-Enneagon_stellations.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Enneagon_stellations.svg/640px-Enneagon_stellations.svg.png 2x" data-file-width="600" data-file-height="703" /></a></span><br />The <a href="/wiki/Enneagon" class="mw-redirect" title="Enneagon">enneagon</a> (nonagon) {9} has 3 <a href="/wiki/Enneagram_(geometry)" title="Enneagram (geometry)">enneagrammic</a> forms:<br />{9/2}, {9/3}, {9/4}, with {9/3} being a compound of 3 triangles. </td></tr> <tr> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Obtuse_heptagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Obtuse_heptagram.svg/150px-Obtuse_heptagram.svg.png" decoding="async" width="150" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Obtuse_heptagram.svg/225px-Obtuse_heptagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Obtuse_heptagram.svg/300px-Obtuse_heptagram.svg.png 2x" data-file-width="818" data-file-height="798" /></a></span><span typeof="mw:File"><a href="/wiki/File:Acute_heptagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Acute_heptagram.svg/150px-Acute_heptagram.svg.png" decoding="async" width="150" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Acute_heptagram.svg/225px-Acute_heptagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Acute_heptagram.svg/300px-Acute_heptagram.svg.png 2x" data-file-width="818" data-file-height="798" /></a></span> <p><br />The <a href="/wiki/Heptagon" title="Heptagon">heptagon</a> has two <a href="/wiki/Heptagram" title="Heptagram">heptagrammic</a> forms:<br />{7/2}, {7/3} </p> </td></tr></tbody></table> <p>Like the <a href="/wiki/Heptagon" title="Heptagon">heptagon</a>, the <a href="/wiki/Octagon" title="Octagon">octagon</a> also has two <a href="/wiki/Octagram" title="Octagram">octagrammic</a> stellations, one, {8/3} being a <a href="/wiki/Star_polygon" title="Star polygon">star polygon</a>, and the other, {8/2}, being the compound of two <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">squares</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Stellating_polyhedra">Stellating polyhedra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stellation&action=edit&section=3" title="Edit section: Stellating polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:First_stellation_of_octahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/First_stellation_of_octahedron.svg/70px-First_stellation_of_octahedron.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/First_stellation_of_octahedron.svg/105px-First_stellation_of_octahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/First_stellation_of_octahedron.svg/140px-First_stellation_of_octahedron.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:First_stellation_of_dodecahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/First_stellation_of_dodecahedron.svg/70px-First_stellation_of_dodecahedron.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/First_stellation_of_dodecahedron.svg/105px-First_stellation_of_dodecahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/First_stellation_of_dodecahedron.svg/140px-First_stellation_of_dodecahedron.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Second_stellation_of_dodecahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Second_stellation_of_dodecahedron.svg/70px-Second_stellation_of_dodecahedron.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Second_stellation_of_dodecahedron.svg/105px-Second_stellation_of_dodecahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Second_stellation_of_dodecahedron.svg/140px-Second_stellation_of_dodecahedron.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Third_stellation_of_dodecahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Third_stellation_of_dodecahedron.svg/70px-Third_stellation_of_dodecahedron.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Third_stellation_of_dodecahedron.svg/105px-Third_stellation_of_dodecahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Third_stellation_of_dodecahedron.svg/140px-Third_stellation_of_dodecahedron.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Sixteenth_stellation_of_icosahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Sixteenth_stellation_of_icosahedron.svg/70px-Sixteenth_stellation_of_icosahedron.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Sixteenth_stellation_of_icosahedron.svg/105px-Sixteenth_stellation_of_icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Sixteenth_stellation_of_icosahedron.svg/140px-Sixteenth_stellation_of_icosahedron.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:First_stellation_of_icosahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/First_stellation_of_icosahedron.svg/70px-First_stellation_of_icosahedron.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/First_stellation_of_icosahedron.svg/105px-First_stellation_of_icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/First_stellation_of_icosahedron.svg/140px-First_stellation_of_icosahedron.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Seventeenth_stellation_of_icosahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Seventeenth_stellation_of_icosahedron.svg/70px-Seventeenth_stellation_of_icosahedron.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Seventeenth_stellation_of_icosahedron.svg/105px-Seventeenth_stellation_of_icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Seventeenth_stellation_of_icosahedron.svg/140px-Seventeenth_stellation_of_icosahedron.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span> </td></tr></tbody></table> <p>A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound. The interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, and as the stellation process continues then more of these cells will be enclosed. For a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells – we say that the cells in such a congruent set are of the same type. A common method of finding stellations involves selecting one or more cell types. </p><p>This can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way. </p><p>A set of cells forming a closed layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types. </p><p>Based on such ideas, several restrictive categories of interest have been identified. </p> <ul><li><b>Main-line stellations.</b> Adding successive shells to the core polyhedron leads to the set of main-line stellations.</li> <li><b>Fully supported stellations.</b> The underside faces of a cell can appear externally as an "overhang." In a fully supported stellation there are no such overhangs, and all visible parts of a face are seen from the same side.</li> <li><b>Monoacral stellations.</b> Literally "single-peaked." Where there is only one kind of peak, or vertex, in a stellation (i.e. all vertices are congruent within a single symmetry orbit), the stellation is monoacral. All such stellations are fully supported.</li> <li><b>Primary stellations.</b> Where a polyhedron has planes of mirror symmetry, edges falling in these planes are said to lie in primary lines. If all edges lie in primary lines, the stellation is primary. All primary stellations are fully supported.</li> <li><b>Miller stellations.</b> In "The Fifty-Nine Icosahedra" <a href="/wiki/H.S.M._Coxeter" class="mw-redirect" title="H.S.M. Coxeter">Coxeter</a>, Du Val, Flather and Petrie record five rules suggested by <a href="/wiki/J._C._P._Miller" title="J. C. P. Miller">Miller</a>. Although these rules refer specifically to the icosahedron's geometry, they have been adapted to work for arbitrary polyhedra. They ensure, among other things, that the rotational symmetry of the original polyhedron is preserved, and that each stellation is different in outward appearance. The four kinds of stellation just defined are all subsets of the Miller stellations.</li></ul> <p>We can also identify some other categories: </p> <ul><li>A <b>partial stellation</b> is one where not all elements of a given dimensionality are extended.</li> <li>A <b>sub-symmetric stellation</b> is one where not all elements are extended symmetrically.</li></ul> <p>The <a href="/wiki/Archimedean_solids" class="mw-redirect" title="Archimedean solids">Archimedean solids</a> and their duals can also be stellated. Here we usually add the rule that all of the original face planes must be present in the stellation, i.e. we do not consider partial stellations. For example the <a href="/wiki/Cube" title="Cube">cube</a> is not usually considered a stellation of the <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a>. </p><p>Generalising Miller's rules there are: </p> <ul><li>4 stellations of the <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a></li> <li>187 stellations of the <a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedron</a></li> <li>358,833,097 stellations of the <a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a></li> <li>17 stellations of the <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a> (4 are shown in <a href="/wiki/List_of_Wenninger_polyhedron_models#Stellations:_models_W19_to_W66" title="List of Wenninger polyhedron models">Wenninger</a>'s <i>Polyhedron Models</i>)</li> <li>An unknown number of stellations of the <a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a>; there are 7,071,671 non-<a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">chiral</a> stellations, but the number of chiral stellations is unknown. (20 are shown in <a href="/wiki/List_of_Wenninger_polyhedron_models#Stellations_of_icosidodecahedron" title="List of Wenninger polyhedron models">Wenninger</a>'s <i>Polyhedron Models</i>)</li></ul> <p>Seventeen of the nonconvex uniform polyhedra are stellations of Archimedean solids. </p> <div class="mw-heading mw-heading3"><h3 id="Miller's_rules"><span id="Miller.27s_rules"></span>Miller's rules</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stellation&action=edit&section=4" title="Edit section: Miller's rules"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the book <i><a href="/wiki/The_fifty_nine_icosahedra" class="mw-redirect" title="The fifty nine icosahedra">The Fifty-Nine Icosahedra</a></i>, J.C.P. Miller proposed a <a href="/wiki/The_fifty_nine_icosahedra#Miller's_rules" class="mw-redirect" title="The fifty nine icosahedra">set of rules</a> for defining which stellation forms should be considered "properly significant and distinct". </p><p>These rules have been adapted for use with stellations of many other polyhedra. Under Miller's rules we find: </p> <ul><li>There are no stellations of the <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>, because all faces are adjacent</li> <li>There are no stellations of the <a href="/wiki/Cube" title="Cube">cube</a>, because non-adjacent faces are parallel and thus cannot be extended to meet in new edges</li> <li>There is 1 stellation of the <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>, the <a href="/wiki/Stella_octangula" class="mw-redirect" title="Stella octangula">stella octangula</a></li> <li>There are 3 stellations of the <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a>: the <a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">small stellated dodecahedron</a>, the <a href="/wiki/Great_dodecahedron" title="Great dodecahedron">great dodecahedron</a> and the <a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">great stellated dodecahedron</a>, all of which are Kepler–Poinsot polyhedra.</li> <li>There are 58 stellations of the <a href="/wiki/Icosahedron" title="Icosahedron">icosahedron</a>, including the <a href="/wiki/Great_icosahedron" title="Great icosahedron">great icosahedron</a> (one of the Kepler–Poinsot polyhedra), and the <a href="/wiki/Second_stellation_of_icosahedron" class="mw-redirect" title="Second stellation of icosahedron">second</a> and <a href="/wiki/Final_stellation_of_the_icosahedron" title="Final stellation of the icosahedron">final</a> stellations of the icosahedron. The 59th model in <i>The fifty nine icosahedra</i> is the original icosahedron itself.</li></ul> <p>Many "Miller stellations" cannot be obtained directly by using Kepler's method. For example many have hollow centres where the original faces and edges of the core polyhedron are entirely missing: there is nothing left to be stellated. On the other hand, Kepler's method also yields stellations which are forbidden by Miller's rules since their cells are edge- or vertex-connected, even though their faces are single polygons. This discrepancy received no real attention until Inchbald (2002). </p> <div class="mw-heading mw-heading3"><h3 id="Other_rules_for_stellation">Other rules for stellation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stellation&action=edit&section=5" title="Edit section: Other rules for stellation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Miller's rules by no means represent the "correct" way to enumerate stellations. They are based on combining parts within the <a href="/wiki/Stellation_diagram" title="Stellation diagram">stellation diagram</a> in certain ways, and don't take into account the topology of the resulting faces. As such there are some quite reasonable stellations of the icosahedron that are not part of their list – one was identified by James Bridge in 1974, while some "Miller stellations" are questionable as to whether they should be regarded as stellations at all – one of the icosahedral set comprises several quite disconnected cells floating symmetrically in space. </p><p>As yet an alternative set of rules that takes this into account has not been fully developed. Most progress has been made based on the notion that stellation is the reciprocal or dual process to <a href="/wiki/Facetting" class="mw-redirect" title="Facetting">facetting</a>, whereby parts are removed from a polyhedron without creating any new vertices. For every stellation of some polyhedron, there is a <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">dual</a> facetting of the <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a>, and vice versa. By studying facettings of the dual, we gain insights into the stellations of the original. Bridge found his new stellation of the icosahedron by studying the facettings of its dual, the dodecahedron. </p><p>Some polyhedronists take the view that stellation is a two-way process, such that any two polyhedra sharing the same face planes are stellations of each other. This is understandable if one is devising a general algorithm suitable for use in a computer program, but is otherwise not particularly helpful. </p><p>Many examples of stellations can be found in the <a href="/wiki/List_of_Wenninger_polyhedron_models#Stellations:_models_W19_to_W66" title="List of Wenninger polyhedron models">list of Wenninger's stellation models</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Stellating_polytopes">Stellating polytopes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stellation&action=edit&section=6" title="Edit section: Stellating polytopes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The stellation process can be applied to higher dimensional polytopes as well. A <a href="/wiki/Stellation_diagram" title="Stellation diagram">stellation diagram</a> of an <i>n</i>-polytope exists in an (<i>n</i> − 1)-dimensional <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> of a given <a href="/wiki/Facet_(geometry)" title="Facet (geometry)">facet</a>. </p><p>For example, in 4-space, the <a href="/wiki/Great_grand_stellated_120-cell" title="Great grand stellated 120-cell">great grand stellated 120-cell</a> is the final stellation of the <a href="/wiki/Regular_4-polytope" title="Regular 4-polytope">regular 4-polytope</a> <a href="/wiki/120-cell" title="120-cell">120-cell</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Naming_stellations">Naming stellations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stellation&action=edit&section=7" title="Edit section: Naming stellations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first systematic naming of stellated polyhedra was <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley</a>'s naming of the regular star polyhedra (nowadays known as the <a href="/wiki/Kepler%E2%80%93Poinsot_polyhedra" class="mw-redirect" title="Kepler–Poinsot polyhedra">Kepler–Poinsot polyhedra</a>). This system was widely, but not always systematically, adopted for other polyhedra and higher polytopes. </p><p><a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Conway</a> devised a terminology for stellated <a href="/wiki/Polygon" title="Polygon">polygons</a>, <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a> and <a href="/wiki/Polychoron" class="mw-redirect" title="Polychoron">polychora</a> (Coxeter 1974). In this system the process of extending edges to create a new figure is called <i>stellation</i>, that of extending faces is called <i>greatening</i> and that of extending cells is called <i>aggrandizement</i> (this last does not apply to polyhedra). This allows a systematic use of words such as 'stellated', 'great', and 'grand' in devising names for the resulting figures. For example Conway proposed some minor variations to the names of the <a href="/wiki/Kepler%E2%80%93Poinsot_polyhedra#History" class="mw-redirect" title="Kepler–Poinsot polyhedra">Kepler–Poinsot polyhedra</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Stellation_to_infinity">Stellation to infinity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stellation&action=edit&section=8" title="Edit section: Stellation to infinity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wenninger noticed that some polyhedra, such as the cube, do not have any finite stellations. However stellation cells can be constructed as prisms which extend to infinity. The figure comprising these prisms may be called a <b>stellation to infinity</b>. By most definitions of a polyhedron, however, these stellations are not strictly polyhedra. </p><p>Wenninger's figures occurred as <a href="/wiki/Hemipolyhedron#Duals_of_the_hemipolyhedra" title="Hemipolyhedron">duals of the uniform hemipolyhedra</a>, where the faces that pass through the center are sent to vertices "at infinity". </p> <div class="mw-heading mw-heading2"><h2 id="From_mathematics_to_art">From mathematics to art</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stellation&action=edit&section=9" title="Edit section: From mathematics to art"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Magnus_Wenninger_polyhedral_models.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Magnus_Wenninger_polyhedral_models.jpg/220px-Magnus_Wenninger_polyhedral_models.jpg" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Magnus_Wenninger_polyhedral_models.jpg/330px-Magnus_Wenninger_polyhedral_models.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Magnus_Wenninger_polyhedral_models.jpg/440px-Magnus_Wenninger_polyhedral_models.jpg 2x" data-file-width="4288" data-file-height="2848" /></a><figcaption>Magnus Wenninger with some of his models of stellated polyhedra in 2009</figcaption></figure> <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">Further information: <a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></div> <p>Alongside from his contributions to mathematics, <a href="/wiki/Magnus_Wenninger" title="Magnus Wenninger">Magnus Wenninger</a> is described in the context of the relationship of <a href="/wiki/Mathematics_and_art" title="Mathematics and art">mathematics and art</a> as making "especially beautiful" models of complex stellated polyhedra.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg/170px-Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg" decoding="async" width="170" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg/255px-Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg/340px-Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg 2x" data-file-width="543" data-file-height="566" /></a><figcaption>Marble floor <a href="/wiki/Mosaic" title="Mosaic">mosaic</a> by <a href="/wiki/Paolo_Uccello" title="Paolo Uccello">Paolo Uccello</a>, <a href="/wiki/Basilica_of_St_Mark,_Venice" class="mw-redirect" title="Basilica of St Mark, Venice">Basilica of St Mark, Venice</a>, c. 1430</figcaption></figure> <p>The <a href="/wiki/Italian_Renaissance" title="Italian Renaissance">Italian Renaissance</a> artist <a href="/wiki/Paolo_Uccello" title="Paolo Uccello">Paolo Uccello</a> created a floor mosaic showing a small stellated dodecahedron in the <a href="/wiki/Basilica_of_St_Mark,_Venice" class="mw-redirect" title="Basilica of St Mark, Venice">Basilica of St Mark, Venice</a>, c. 1430. Uccello's depiction was used as the symbol for the <a href="/wiki/Venice_Biennale" title="Venice Biennale">Venice Biennale</a> in 1986 on the topic of "Art and Science".<sup id="cite_ref-Emmer2003_2-0" class="reference"><a href="#cite_note-Emmer2003-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The same stellation is central to two <a href="/wiki/Lithograph" class="mw-redirect" title="Lithograph">lithographs</a> by <a href="/wiki/M._C._Escher" title="M. C. Escher">M. C. Escher</a>: <i>Contrast (Order and Chaos)</i>, 1950, and <i><a href="/wiki/Gravitation_(M._C._Escher)" title="Gravitation (M. C. Escher)">Gravitation</a></i>, 1952.<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> </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=Stellation&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i><a href="/wiki/The_Fifty-Nine_Icosahedra" title="The Fifty-Nine Icosahedra">The Fifty-Nine Icosahedra</a></i></li> <li><a href="/wiki/List_of_Wenninger_polyhedron_models" title="List of Wenninger polyhedron models">List of Wenninger polyhedron models</a> includes 44 stellated forms of the octahedron, dodecahedron, icosahedron, and icosidodecahedron, enumerated in the book <i>Polyhedron Models</i> (1974) by <a href="/wiki/Magnus_Wenninger" title="Magnus Wenninger">Magnus Wenninger</a></li> <li><a href="/wiki/Polyhedral_compound" class="mw-redirect" title="Polyhedral compound">Polyhedral compound</a> Includes 5 regular compounds and 4 dual regular compounds.</li> <li><a href="/wiki/List_of_polyhedral_stellations" title="List of polyhedral stellations">List of polyhedral stellations</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=Stellation&action=edit&section=11" 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 a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMalkevitch" class="citation web cs1">Malkevitch, Joseph. <a rel="nofollow" class="external text" href="http://www.ams.org/samplings/feature-column/fcarc-art5">"Mathematics and Art. 5. Polyhedra, tilings, and dissections"</a>. American Mathematical Society<span class="reference-accessdate">. Retrieved <span class="nowrap">1 September</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mathematics+and+Art.+5.+Polyhedra%2C+tilings%2C+and+dissections&rft.pub=American+Mathematical+Society&rft.aulast=Malkevitch&rft.aufirst=Joseph&rft_id=http%3A%2F%2Fwww.ams.org%2Fsamplings%2Ffeature-column%2Ffcarc-art5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStellation" class="Z3988"></span></span> </li> <li id="cite_note-Emmer2003-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Emmer2003_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmmer2003" class="citation book cs1">Emmer, Michele (2 December 2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EHRDnU29PO8C&pg=PA269"><i>Mathematics and Culture I</i></a>. Springer Science & Business Media. p. 269. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-01770-7" title="Special:BookSources/978-3-540-01770-7"><bdi>978-3-540-01770-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Culture+I&rft.pages=269&rft.pub=Springer+Science+%26+Business+Media&rft.date=2003-12-02&rft.isbn=978-3-540-01770-7&rft.aulast=Emmer&rft.aufirst=Michele&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEHRDnU29PO8C%26pg%3DPA269&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStellation" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLocher,_J._L.2000" class="citation book cs1">Locher, J. L. (2000). <i>The Magic of M. C. Escher</i>. Harry N. Abrams, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-810-96720-0" title="Special:BookSources/0-810-96720-0"><bdi>0-810-96720-0</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+Magic+of+M.+C.+Escher&rft.pub=Harry+N.+Abrams%2C+Inc.&rft.date=2000&rft.isbn=0-810-96720-0&rft.au=Locher%2C+J.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStellation" class="Z3988"></span></span> </li> </ol></div> <ul><li>Bridge, N. J.; Facetting the dodecahedron, <i>Acta Crystallographica</i> <b>A30</b> (1974), pp. 548–552.</li> <li><a href="/wiki/Coxeter" class="mw-redirect" title="Coxeter">Coxeter</a>, H.S.M.; <i>Regular complex polytopes</i> (1974).</li> <li><a href="/wiki/Coxeter" class="mw-redirect" title="Coxeter">Coxeter</a>, H.S.M.; Du Val, P.; Flather, H. T.; and Petrie, J. F. <i>The Fifty-Nine Icosahedra</i>, 3rd Edition. Stradbroke, England: Tarquin Publications (1999).</li> <li>Inchbald, G.; In search of the lost icosahedra, <i>The Mathematical Gazette</i> <b>86</b> (2002), pp. 208-215.</li> <li>Messer, P.; Stellations of the rhombic triacontahedron and beyond, <i>Symmetry: culture and science</i>, 11 (2000), pp. 201–230.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWenninger1974" class="citation book cs1"><a href="/wiki/Magnus_Wenninger" title="Magnus Wenninger">Wenninger, Magnus</a> (1974). <i>Polyhedron Models</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-09859-9" title="Special:BookSources/0-521-09859-9"><bdi>0-521-09859-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polyhedron+Models&rft.pub=Cambridge+University+Press&rft.date=1974&rft.isbn=0-521-09859-9&rft.aulast=Wenninger&rft.aufirst=Magnus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStellation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWenninger1983" class="citation book cs1"><a href="/wiki/Magnus_Wenninger" title="Magnus Wenninger">Wenninger, Magnus</a> (1983). <i>Dual Models</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-24524-9" title="Special:BookSources/0-521-24524-9"><bdi>0-521-24524-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dual+Models&rft.pub=Cambridge+University+Press&rft.date=1983&rft.isbn=0-521-24524-9&rft.aulast=Wenninger&rft.aufirst=Magnus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStellation" class="Z3988"></span></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=Stellation&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Stellation"><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/Stellation.html">"Stellation"</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=Stellation&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FStellation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStellation" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.steelpillow.com/polyhedra/icosa/">Stellating the Icosahedron and Facetting the Dodecahedron</a></li> <li><a rel="nofollow" class="external text" href="http://www.software3d.com/Stella.php">Stella: Polyhedron Navigator</a> – Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.</li> <li><a rel="nofollow" class="external text" href="http://www.software3d.com/Enumerate.php">Enumeration of stellations</a></li> <li><a rel="nofollow" class="external text" href="http://bulatov.org/polyhedra/stellation/">Vladimir Bulatov <i>Polyhedra Stellation.</i></a></li> <li><a rel="nofollow" class="external text" href="http://davidcool.com/Stellation.zip">Vladimir Bulatov's Polyhedra Stellations Applet packaged as an OS X application</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304025232/http://davidcool.com/Stellation.zip">Archived</a> 2016-03-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://bulatov.org/polyhedra/stellation_applet/">Stellation Applet</a></li> <li><a rel="nofollow" class="external text" href="http://bulatov.org/polyhedra/StellationWithVariousSymmetries/">An Interactive Creation of Polyhedra Stellations with Various Symmetries</a></li> <li><a rel="nofollow" class="external text" href="http://members.ozemail.com.au/~llan/i59.html">The Fifty-Nine Icosahedra – Applet</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191228054914/http://members.ozemail.com.au/~llan/i59.html">Archived</a> 2019-12-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/stellations-icosahedron-index.html">59 Stellations of the Icosahedron, George Hart</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20061006212927/http://www.cacr.caltech.edu/~roy/Stellate/explain.html">Stellation: Beautiful Math</a></li> <li><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00283-009-9061-y">Further Stellations of the Uniform Polyhedra, John Lawrence Hudson</a> The Mathematical Intelligencer, Volume 31, Number 4, 2009</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=Stellation&oldid=1245424269">https://en.wikipedia.org/w/index.php?title=Stellation&oldid=1245424269</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:Polygons" title="Category:Polygons">Polygons</a></li><li><a href="/wiki/Category:Polyhedra" title="Category:Polyhedra">Polyhedra</a></li><li><a href="/wiki/Category:Polytopes" title="Category:Polytopes">Polytopes</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_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</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 12 September 2024, at 22:41<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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