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href="//web.archive.org/web/20131009133555/http://meta.wikimedia.org/"/></head> <body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-Menger_sponge skin-vector action-view vector-animateLayout"> <div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="mw-js-message" style="display:none;"></div> <div id="siteNotice"></div> <h1 id="firstHeading" class="firstHeading" lang="en"><span dir="auto">Menger sponge</span></h1> <div id="bodyContent"> <div id="siteSub">From Wikipedia, the free encyclopedia</div> <div id="contentSub"></div> <div id="jump-to-nav" class="mw-jump"> Jump to: <a href="#mw-navigation">navigation</a>, <a href="#p-search">search</a> </div> <div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><table class="metadata plainlinks ambox ambox-style ambox-No_footnotes" style="" role="presentation"> <tr> <td class="mbox-image"> <div style="width: 52px;"><img alt="" src="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" width="40" height="40" srcset="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x"/></div> </td> <td class="mbox-text" style=""><span class="mbox-text-span">This article includes a <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, related reading or <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, but <b>its sources remain unclear because it lacks <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>. <span class="hide-when-compact">Please <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by introducing more precise citations.</span> <small><i>(June 2012)</i></small> </span></td> </tr> </table> <div class="thumb tright"> <div class="thumbinner" style="width:312px;"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/File:Menger-Schwamm-farbig.png" class="image"><img alt="" src="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/5/52/Menger-Schwamm-farbig.png/310px-Menger-Schwamm-farbig.png" width="310" height="233" class="thumbimage" srcset="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/5/52/Menger-Schwamm-farbig.png/465px-Menger-Schwamm-farbig.png 1.5x, //web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/5/52/Menger-Schwamm-farbig.png/620px-Menger-Schwamm-farbig.png 2x"/></a> <div class="thumbcaption"> <div class="magnify"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/File:Menger-Schwamm-farbig.png" class="internal" title="Enlarge"><img src="//web.archive.org/web/20131009133555im_/http://bits.wikimedia.org/static-1.22wmf17/skins/common/images/magnify-clip.png" width="15" height="11" alt=""/></a></div> An illustration of <i>M<sub>4</sub></i>, the fourth iteration of the construction process.</div> </div> </div> <p>In <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Menger sponge</b> is a <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Fractal" title="Fractal">fractal</a> curve. It is a <b>universal curve</b>, in that it has <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Topological_dimension" title="Topological dimension" class="mw-redirect">topological dimension</a> one, and any other <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Curve" title="Curve">curve</a> (more precisely: any compact metric space of topological dimension 1) is <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Homeomorphic" title="Homeomorphic" class="mw-redirect">homeomorphic</a> to some subset of it. It is sometimes called the <i>Menger-Sierpinski sponge</i> or the <i>Sierpinski sponge</i>. It is a three-dimensional extension of the <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Cantor_set" title="Cantor set">Cantor set</a> and <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Sierpinski_carpet" title="Sierpinski carpet">Sierpinski carpet</a>. It was first described by <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Karl_Menger" title="Karl Menger">Karl Menger</a> (<a href="#CITEREFMenger1926">1926</a>) while exploring the concept of topological dimension.</p> <p>The Menger sponge simultaneously exhibits an infinite surface area and encloses zero volume.</p> <div id="toc" class="toc"> <div id="toctitle"> <h2>Contents</h2> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Construction"><span class="tocnumber">1</span> <span class="toctext">Construction</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Properties"><span class="tocnumber">2</span> <span class="toctext">Properties</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Formal_definition"><span class="tocnumber">3</span> <span class="toctext">Formal definition</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Other"><span class="tocnumber">4</span> <span class="toctext">Other</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#See_also"><span class="tocnumber">5</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#References"><span class="tocnumber">6</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#External_links"><span class="tocnumber">7</span> <span class="toctext">External links</span></a></li> </ul> </div> <h2><span class="mw-headline" id="Construction">Construction</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Menger_sponge&action=edit&section=1" title="Edit section: Construction">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <p>The construction of a Menger sponge can be described as follows:</p> <ol> <li>Begin with a cube (<i>first image</i>).</li> <li>Divide every face of the cube into 9 squares, like a <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Rubik%27s_Cube" title="Rubik's Cube">Rubik's Cube</a>. This will sub-divide the cube into 27 smaller cubes.</li> <li>Remove the smaller cube in the middle of each face, and remove the smaller cube in the very center of the larger cube, leaving 20 smaller cubes (<i>second image</i>). This is a level-1 Menger sponge (resembling a <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Void_Cube" title="Void Cube">Void Cube</a>).</li> <li>Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue to iterate <i>ad infinitum</i>.</li> </ol> <p>The second iteration will give you a level-2 sponge (<i>third image</i>), the third iteration gives a level-3 sponge (<i>fourth image</i>), and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations.</p> <div class="center"> <div class="thumb tnone"> <div class="thumbinner" style="width:552px;"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/File:Menger_sponge_(Level_0-3).jpg" class="image"><img alt="" src="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/d/de/Menger_sponge_%28Level_0-3%29.jpg/550px-Menger_sponge_%28Level_0-3%29.jpg" width="550" height="157" class="thumbimage" srcset="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/d/de/Menger_sponge_%28Level_0-3%29.jpg 1.5x, //web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/d/de/Menger_sponge_%28Level_0-3%29.jpg 2x"/></a> <div class="thumbcaption"> <div class="magnify"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/File:Menger_sponge_(Level_0-3).jpg" class="internal" title="Enlarge"><img src="//web.archive.org/web/20131009133555im_/http://bits.wikimedia.org/static-1.22wmf17/skins/common/images/magnify-clip.png" width="15" height="11" alt=""/></a></div> An illustration of the iterative construction of a Menger sponge up to <i>M<sub>3</sub></i>, the third iteration.</div> </div> </div> </div> <p>The number of cubes is 20<sup><i>n</i></sup>, with <i>n</i> being the number of iterations performed on the first cube.</p> <div class="thumb tright"> <div class="thumbinner" style="width:222px;"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/File:Mengerova_houba.jpg" class="image"><img alt="" src="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Mengerova_houba.jpg/220px-Mengerova_houba.jpg" width="220" height="293" class="thumbimage" srcset="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Mengerova_houba.jpg/330px-Mengerova_houba.jpg 1.5x, //web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Mengerova_houba.jpg/440px-Mengerova_houba.jpg 2x"/></a> <div class="thumbcaption"> <div class="magnify"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/File:Mengerova_houba.jpg" class="internal" title="Enlarge"><img src="//web.archive.org/web/20131009133555im_/http://bits.wikimedia.org/static-1.22wmf17/skins/common/images/magnify-clip.png" width="15" height="11" alt=""/></a></div> A sculptural representation of the previous illustration.</div> </div> </div> <h2><span class="mw-headline" id="Properties">Properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Menger_sponge&action=edit&section=2" title="Edit section: Properties">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <p>Each face of the Menger sponge is a <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Sierpinski_carpet" title="Sierpinski carpet">Sierpinski carpet</a>; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube <i>M</i><sub>0</sub> is a <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Cantor_set" title="Cantor set">Cantor set</a>.</p> <p>The Menger sponge is a <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Closed_set" title="Closed set">closed set</a>; since it is also bounded, the <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem" title="Heine–Borel theorem">Heine–Borel theorem</a> implies that it is <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Compact_set" title="Compact set" class="mw-redirect">compact</a>. It has <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> 0. It is an <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Uncountable_set" title="Uncountable set">uncountable set</a>.</p> <p>The <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Lebesgue_covering_dimension" title="Lebesgue covering dimension">Lebesgue covering dimension</a> of the Menger sponge is one, the same as any <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Curve" title="Curve">curve</a>. Menger showed, in the 1926 construction, that the sponge is a <b>universal curve</b>, in that any possible one-dimensional curve is <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Homeomorphic" title="Homeomorphic" class="mw-redirect">homeomorphic</a> to a subset of the Menger sponge, where here a curve means any <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Compact_space" title="Compact space">compact</a> <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Metric_space" title="Metric space">metric space</a> of Lebesgue covering dimension one; this includes <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Tree_(graph_theory)" title="Tree (graph theory)">trees</a> and <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Graph_theory" title="Graph theory">graphs</a> with an arbitrary <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Countable" title="Countable" class="mw-redirect">countable</a> number of edges, vertices and closed loops, connected in arbitrary ways. In a similar way, the <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Sierpinski_carpet" title="Sierpinski carpet">Sierpinski carpet</a> is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Planar_graph" title="Planar graph">planar</a>, and might be embedded in any number of dimensions.</p> <p>The Menger sponge simultaneously exhibits an infinite surface area and encloses zero volume. In spite of this, there exists a homeomorphism of the cube having finite <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Distortion_(mathematics)" title="Distortion (mathematics)">distortion</a> that "squeezes the sponge" in the sense that the holes in the sponge go to a Cantor set of zero measure (<a href="#CITEREFIwaniecMartin2001">Iwaniec & Martin 2001</a>, §6.5.6).</p> <p>The sponge has a <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a> of (log 20) / (log 3) (approximately 2.726833).<br clear="all"/></p> <h2><span class="mw-headline" id="Formal_definition">Formal definition</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Menger_sponge&action=edit&section=3" title="Edit section: Formal definition">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <p>Formally, a Menger sponge can be defined as follows:</p> <dl> <dd><img class="tex" alt="M := \bigcap_{n\in\mathbb{N}} M_n" src="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/math/8/3/7/8376fe62e86ff527200f62a6aa7230e1.png"/></dd> </dl> <p>where <i>M<sub>0</sub></i> is the <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Unit_cube" title="Unit cube">unit cube</a> and</p> <dl> <dd><img class="tex" alt="M_{n+1} := \left\{\begin{matrix} (x,y,z)\in\mathbb{R}^3: & \begin{matrix}\exists i,j,k\in\{0,1,2\}: (3x-i,3y-j,3z-k)\in M_n \\ \mbox{and at most one of }i,j,k\mbox{ is equal to 1}\end{matrix} \end{matrix}\right\}." src="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/math/d/0/6/d0667b54927ae87f0fb006f958fdaa56.png"/></dd> </dl> <h2><span class="mw-headline" id="Other">Other</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Menger_sponge&action=edit&section=4" title="Edit section: Other">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <p>The phrase 'Menger Sponge' is seen in the game Minecraft as one of the many splashes that appear on the main menu screen; this is most likely due to the fact that the game is all about blocks, and the Menger Sponge is based around the standard cube.</p> <p>The character Funny Valentine from the manga Jojo's Bizarre Adventure Part 7 Steel Ball Run uses Menger Sponges in one of his attacks. Funny has the ability to pull other versions of people from other dimensions. If two versions of the same person come into contact with each other, they explode into Menger Sponges. This causes the death of both versions of the character.</p> <h2><span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Menger_sponge&action=edit&section=5" title="Edit section: See also">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <ul> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Apollonian_gasket" title="Apollonian gasket">Apollonian gasket</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Koch_snowflake" title="Koch snowflake">Koch snowflake</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension" title="List of fractals by Hausdorff dimension">List of fractals by Hausdorff dimension</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Sierpinski_triangle#Analogues_in_higher_dimensions" title="Sierpinski triangle">Sierpiński tetrahedron</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Sierpinski_triangle" title="Sierpinski triangle">Sierpiński triangle</a></li> </ul> <h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Menger_sponge&action=edit&section=6" title="Edit section: References">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <ul> <li><span id="CITEREFIwaniecMartin2001" class="citation"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Tadeusz_Iwaniec" title="Tadeusz Iwaniec">Iwaniec, Tadeusz</a>; Martin, Gaven (2001), <i>Geometric function theory and non-linear analysis</i>, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, <a href="/web/20131009133555/http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Special:BookSources/978-0-19-850929-5" title="Special:BookSources/978-0-19-850929-5">978-0-19-850929-5</a>, <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.ams.org/mathscinet-getitem?mr=1859913">1859913</a></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMenger+sponge&rft.aufirst=Tadeusz&rft.au=Iwaniec%2C+Tadeusz&rft.aulast=Iwaniec&rft.au=Martin%2C+Gaven&rft.btitle=Geometric+function+theory+and+non-linear+analysis&rft.date=2001&rft.genre=book&rft.isbn=978-0-19-850929-5&rft.pub=The+Clarendon+Press+Oxford+University+Press&rft.series=Oxford+Mathematical+Monographs&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITEREFMenger1926" class="citation">Menger, Karl (1926), "Allgemeine Räume und Cartesische Räume. I.", <i>Communications to the Amsterdam Academy of Sciences</i></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMenger+sponge&rft.atitle=Allgemeine+R%C3%A4ume+und+Cartesische+R%C3%A4ume.+I.&rft.aufirst=Karl&rft.aulast=Menger&rft.au=Menger%2C+Karl&rft.date=1926&rft.genre=article&rft.jtitle=Communications+to+the+Amsterdam+Academy+of+Sciences&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span> English translation reprinted in <span id="CITEREFEdgar2004" class="citation">Edgar, Gerald A., ed. (2004), <i>Classics on fractals</i>, Studies in Nonlinearity, Westview Press. Advanced Book Program, Boulder, CO, <a href="/web/20131009133555/http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Special:BookSources/978-0-8133-4153-8" title="Special:BookSources/978-0-8133-4153-8">978-0-8133-4153-8</a>, <a href="/web/20131009133555/http://en.wikipedia.org/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.ams.org/mathscinet-getitem?mr=2049443">2049443</a></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMenger+sponge&rft.btitle=Classics+on+fractals&rft.date=2004&rft.genre=book&rft.isbn=978-0-8133-4153-8&rft.pub=Westview+Press.+Advanced+Book+Program%2C+Boulder%2C+CO&rft.series=Studies+in+Nonlinearity&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li>Karl Menger, <i>Dimensionstheorie</i>, (1928) B.G Teubner Publishers, Leipzig.</li> <li><span id="CITEREFZhou2007" class="citation">Zhou, Li (2007), "Problem 11208: Chromatic numbers of the Menger sponges", <i><a href="/web/20131009133555/http://en.wikipedia.org/wiki/American_Mathematical_Monthly" title="American Mathematical Monthly">American Mathematical Monthly</a></i> <b>114</b> (9): 842</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMenger+sponge&rft.atitle=Problem+11208%3A+Chromatic+numbers+of+the+Menger+sponges&rft.aufirst=Li&rft.aulast=Zhou&rft.au=Zhou%2C+Li&rft.date=2007&rft.genre=article&rft.issue=9&rft.jtitle=American+Mathematical+Monthly&rft.pages=842&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.volume=114" class="Z3988"><span style="display:none;"> </span></span></li> </ul> <h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Menger_sponge&action=edit&section=7" title="Edit section: External links">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <table class="metadata mbox-small plainlinks" style="border:1px solid #aaa; background-color:#f9f9f9;"> <tr> <td class="mbox-image"><img alt="" src="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" width="30" height="40" srcset="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x"/></td> <td class="mbox-text plainlist" style="">Wikimedia Commons has media related to: <i><b><a href="//web.archive.org/web/20131009133555/http://commons.wikimedia.org/wiki/Menger_sponge" class="extiw" title="commons:Menger sponge">Menger sponge</a></b></i></td> </tr> </table> <ul> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://mathworld.wolfram.com/MengerSponge.html">Menger sponge at Wolfram MathWorld</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://theiff.org/oexhibits/menger01.html">The 'Business Card Menger Sponge' by Dr. Jeannine Mosely - an online exhibit about this giant origami fractal at the Institute For Figuring</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.mathematik.com/Menger/Menger2.html">An interactive Menger sponge</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://ibiblio.org/e-notes/3Dapp/Sponge.htm">Interactive Java models</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://santisan.free.fr/coco/extras2.htm">Puzzle Hunt</a> — Video explaining Zeno's paradoxes using Menger-Sierpinski sponge</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.flickr.com/photos/84445194@N00/sets/72157594256801666/">Menger Sponge Assembly</a> Construction of a Level-3 Menger Sponge from business cards</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.pure-mirage.com/html/Optimized%20Menger%20Sponges.htm">Menger Sponge Animations</a> — Menger sponge animations up to level 9, discussion of optimization for 3d.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.cornellcollege.edu/mathematics/">L3 Menger Sponge with business cards 2006</a> – An L3 Menger sponge by students at Cornell College built in 2006</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.msstate.edu/web/phototemplate.php?Id=1737">Level 3 Menger Sponge made of Business Cards</a> – A level-3 Menger sponge built by students at Mississippi State University out of 48,000 folded business cards.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.flickr.com/photos/fpsunflower/337024546/">Menger sphere</a>, rendered in <a href="/web/20131009133555/http://en.wikipedia.org/wiki/SunFlow" title="SunFlow" class="mw-redirect">SunFlow</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.flickr.com/photos/rougeux/sets/72157621702780335/">Post-It Menger Sponge</a> – a level 3 Menger sponge being built from Post-its</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.nytimes.com/2011/06/28/science/28math-menger.html?_r=1">The Mystery of the Menger Sponge.</a> Sliced diagonally to reveal stars.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://oeis.org/A212596">Number of cards required to build a Menger sponge of level n in origami.</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131009133555/http://www.minecraftwiki.net/wiki/Splashes">Minecraft Menger Sponge</a> A list of the splashes in Minecraft including "Menger Sponge!"</li> </ul> <table cellspacing="0" class="navbox" style="border-spacing:0;"> <tr> <td style="padding:2px;"> <table cellspacing="0" class="nowraplinks collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit;"> <tr> <th scope="col" class="navbox-title" colspan="3"> <div class="noprint plainlinks hlist navbar mini"> <ul> <li class="nv-view"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Template:Fractals" title="Template:Fractals"><span title="View this template" style=";;background:none transparent;border:none;;">v</span></a></li> <li class="nv-talk"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Template_talk:Fractals" title="Template talk:Fractals"><span title="Discuss this template" style=";;background:none transparent;border:none;;">t</span></a></li> <li class="nv-edit"><a class="external text" href="//web.archive.org/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Template:Fractals&action=edit"><span title="Edit this template" style=";;background:none transparent;border:none;;">e</span></a></li> </ul> </div> <div style="font-size:110%;"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Fractal" title="Fractal">Fractals</a></div> </th> </tr> <tr style="height:2px;"> <td></td> </tr> <tr> <th scope="row" class="navbox-group">Characteristics</th> <td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px;"> <div style="padding:0em 0.25em;"> <ul> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Fractal_dimension" title="Fractal dimension">Fractal dimension</a> <ul> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Lebesgue_covering_dimension" title="Lebesgue covering dimension">Topological dimension</a></li> </ul> </li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Self-similarity" title="Self-similarity">Self-similarity</a></li> </ul> </div> </td> <td class="navbox-image" rowspan="15" style="width:0%;padding:0px 0px 0px 2px;"> <div><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Barnsley_fern" title="Barnsley fern"><img alt="An image of a fern which exhibits affine self-similarity." src="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Fractal_fern_explained.png/80px-Fractal_fern_explained.png" width="80" height="112" srcset="//web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Fractal_fern_explained.png/120px-Fractal_fern_explained.png 1.5x, //web.archive.org/web/20131009133555im_/http://upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Fractal_fern_explained.png/160px-Fractal_fern_explained.png 2x"/></a></div> </td> </tr> <tr style="height:2px;"> <td></td> </tr> <tr> <th scope="row" class="navbox-group"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Iterated_function_system" title="Iterated function system">Iterated function system</a></th> <td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px;"> <div style="padding:0em 0.25em;"> <ul> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Barnsley_fern" title="Barnsley fern">Barnsley fern</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Cantor_set" title="Cantor set">Cantor set</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Dragon_curve" title="Dragon curve">Dragon curve</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Koch_snowflake" title="Koch snowflake">Koch snowflake</a></li> <li><strong class="selflink">Menger sponge</strong></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Sierpinski_carpet" title="Sierpinski carpet">Sierpinski carpet</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Sierpinski_triangle" title="Sierpinski triangle">Sierpinski triangle</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Space-filling_curve" title="Space-filling curve">Space-filling curve</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/T-square_(fractal)" title="T-square (fractal)">T-square</a></li> </ul> </div> </td> </tr> <tr style="height:2px;"> <td></td> </tr> <tr> <th scope="row" class="navbox-group"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Strange_attractor" title="Strange attractor" class="mw-redirect">Strange attractor</a></th> <td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px;"> <div style="padding:0em 0.25em;"> <ul> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Multifractal_system" title="Multifractal system">Multifractal system</a></li> </ul> </div> </td> </tr> <tr style="height:2px;"> <td></td> </tr> <tr> <th scope="row" class="navbox-group"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/L-system" title="L-system">L-system</a></th> <td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px;"> <div style="padding:0em 0.25em;"> <ul> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Space-filling_curve" title="Space-filling curve">Space-filling curve</a></li> </ul> </div> </td> </tr> <tr style="height:2px;"> <td></td> </tr> <tr> <th scope="row" class="navbox-group">Escape-time fractals</th> <td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px;"> <div style="padding:0em 0.25em;"> <ul> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Burning_Ship_fractal" title="Burning Ship fractal">Burning Ship fractal</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Julia_set" title="Julia set">Julia set</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Lyapunov_fractal" title="Lyapunov fractal">Lyapunov fractal</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Nova_fractal" title="Nova fractal">Nova fractal</a></li> </ul> </div> </td> </tr> <tr style="height:2px;"> <td></td> </tr> <tr> <th scope="row" class="navbox-group">Random fractals</th> <td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px;"> <div style="padding:0em 0.25em;"> <ul> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Brownian_tree" title="Brownian tree">Brownian tree</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Diffusion-limited_aggregation" title="Diffusion-limited aggregation">Diffusion-limited aggregation</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Fractal_landscape" title="Fractal landscape">Fractal landscape</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/L%C3%A9vy_flight" title="Lévy flight">Lévy flight</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Percolation_theory" title="Percolation theory">Percolation theory</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding walk</a></li> </ul> </div> </td> </tr> <tr style="height:2px;"> <td></td> </tr> <tr> <th scope="row" class="navbox-group">People</th> <td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px;"> <div style="padding:0em 0.25em;"> <ul> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Gaston_Julia" title="Gaston Julia">Gaston Julia</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Helge_von_Koch" title="Helge von Koch">Helge von Koch</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Paul Lévy</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Lewis_Fry_Richardson" title="Lewis Fry Richardson">Lewis Fry Richardson</a></li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Wacław Sierpiński</a></li> </ul> </div> </td> </tr> <tr style="height:2px;"> <td></td> </tr> <tr> <th scope="row" class="navbox-group">Other</th> <td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px;"> <div style="padding:0em 0.25em;"> <ul> <li>"<a href="/web/20131009133555/http://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Self-Similarity_and_Fractional_Dimension" title="How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension">How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension</a>"</li> <li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension" title="List of fractals by Hausdorff dimension">List of fractals by Hausdorff dimension</a></li> </ul> </div> </td> </tr> </table> </td> </tr> </table>   <noscript><img src="//web.archive.org/web/20131009133555im_/http://en.wikipedia.org/w/index.php?title=Special:CentralAutoLogin/start&type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;"/></noscript></div> <div class="printfooter"> Retrieved from "<a href="https://web.archive.org/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Menger_sponge&oldid=571099207">http://en.wikipedia.org/w/index.php?title=Menger_sponge&oldid=571099207</a>" </div> <div id="catlinks" class="catlinks"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Category:Fractals" title="Category:Fractals">Fractals</a></li><li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Category:Curves" title="Category:Curves">Curves</a></li><li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Category:Topological_spaces" title="Category:Topological spaces">Topological spaces</a></li><li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Category:Cubes" title="Category:Cubes">Cubes</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Category:Articles_lacking_in-text_citations_from_June_2012" title="Category:Articles lacking in-text citations from June 2012">Articles lacking in-text citations from June 2012</a></li><li><a href="/web/20131009133555/http://en.wikipedia.org/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li></ul></div></div> <div class="visualClear"></div> </div> </div> <div id="mw-navigation"> <h2>Navigation menu</h2> <div id="mw-head"> <div id="p-personal" role="navigation" class="" aria-labelledby="p-personal-label"> <h3 id="p-personal-label">Personal tools</h3> <ul> <li id="pt-createaccount"><a href="/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Menger+sponge&type=signup">Create account</a></li><li id="pt-login"><a href="/web/20131009133555/http://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Menger+sponge" title="You're encouraged to log in; however, it's not mandatory. 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