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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Chaos_theory" title="Chaos theory">chaos theory</a>, the <b>correlation dimension</b> (denoted by <i>ν</i>) is a measure of the <a href="/wiki/Dimensionality" class="mw-redirect" title="Dimensionality">dimensionality</a> of the space occupied by a set of random points, often referred to as a type of <a href="/wiki/Fractal_dimension" title="Fractal dimension">fractal dimension</a>.<sup id="cite_ref-grassberger_1-0" class="reference"><a href="#cite_note-grassberger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-grassberger2_2-0" class="reference"><a href="#cite_note-grassberger2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-grassberger3_3-0" class="reference"><a href="#cite_note-grassberger3-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>For example, if we have a set of random points on the <a href="/wiki/Real_number" title="Real number">real number</a> line between 0 and 1, the correlation dimension will be <i>ν</i> = 1, while if they are distributed on say, a triangle embedded in three-dimensional space (or <i>m</i>-dimensional space), the correlation dimension will be <i>ν</i> = 2. This is what we would intuitively expect from a measure of dimension. The real utility of the correlation dimension is in determining the (possibly fractional) dimensions of fractal objects. There are other methods of measuring dimension (e.g. the <a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a>, the <a href="/wiki/Box-counting_dimension" class="mw-redirect" title="Box-counting dimension">box-counting dimension</a>, and the <a href="/wiki/Information_dimension" title="Information dimension">information dimension</a>) but the correlation dimension has the advantage of being straightforwardly and quickly calculated, of being less noisy when only a small number of points is available, and is often in agreement with other calculations of dimension. </p><p>For any set of <i>N</i> points in an <i>m</i>-dimensional space </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}(i)=[x_{1}(i),x_{2}(i),\ldots ,x_{m}(i)],\qquad i=1,2,\ldots N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="2em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(i)=[x_{1}(i),x_{2}(i),\ldots ,x_{m}(i)],\qquad i=1,2,\ldots N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea71396198569c41d0c097d60a04269a1a1f61b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.3ex; height:2.843ex;" alt="{\displaystyle {\vec {x}}(i)=[x_{1}(i),x_{2}(i),\ldots ,x_{m}(i)],\qquad i=1,2,\ldots N}"></span></dd></dl> <p>then the <a href="/wiki/Correlation_integral" title="Correlation integral">correlation integral</a> <i>C</i>(<i>ε</i>) is calculated by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\varepsilon )=\lim _{N\rightarrow \infty }{\frac {g}{N^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\varepsilon )=\lim _{N\rightarrow \infty }{\frac {g}{N^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cda6d54e96085c8c594480fb9a1dfdc4cb7afec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.903ex; height:5.009ex;" alt="{\displaystyle C(\varepsilon )=\lim _{N\rightarrow \infty }{\frac {g}{N^{2}}}}"></span></dd></dl> <p>where <i>g</i> is the total number of pairs of points which have a distance between them that is less than distance <i>ε</i> (a graphical representation of such close pairs is the <a href="/wiki/Recurrence_plot" title="Recurrence plot">recurrence plot</a>). As the number of points tends to infinity, and the distance between them tends to zero, the correlation integral, for small values of <i>ε</i>, will take the form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\varepsilon )\sim \varepsilon ^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\varepsilon )\sim \varepsilon ^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81df6dfbb705c033290859cfb7ddf0e10b1e0262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.945ex; height:2.843ex;" alt="{\displaystyle C(\varepsilon )\sim \varepsilon ^{\nu }}"></span></dd></dl> <p>If the number of points is sufficiently large, and evenly distributed, a <a href="/wiki/Log-log_graph" class="mw-redirect" title="Log-log graph">log-log graph</a> of the correlation integral versus <i>ε</i> will yield an estimate of <i>ν</i>. This idea can be qualitatively understood by realizing that for higher-dimensional objects, there will be more ways for points to be close to each other, and so the number of pairs close to each other will rise more rapidly for higher dimensions. </p><p><a href="/wiki/Peter_Grassberger" title="Peter Grassberger">Grassberger</a> and <a href="/wiki/Itamar_Procaccia" title="Itamar Procaccia">Procaccia</a> introduced the technique in 1983;<sup id="cite_ref-grassberger_1-1" class="reference"><a href="#cite_note-grassberger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the article gives the results of such estimates for a number of fractal objects, as well as comparing the values to other measures of fractal dimension. The technique can be used to distinguish between (deterministic) chaotic and truly random behavior, although it may not be good at detecting deterministic behavior if the deterministic generating mechanism is very complex.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>As an example, in the "Sun in Time" article,<sup id="cite_ref-sit_5-0" class="reference"><a href="#cite_note-sit-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> the method was used to show that the number of <a href="/wiki/Sunspot" title="Sunspot">sunspots</a> on the <a href="/wiki/Sun" title="Sun">sun</a>, after accounting for the known cycles such as the daily and 11-year cycles, is very likely not random noise, but rather chaotic noise, with a low-dimensional fractal attractor. </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=Correlation_dimension&action=edit&section=1" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Takens%27_theorem" class="mw-redirect" title="Takens' theorem">Takens' theorem</a></li> <li><a href="/wiki/Correlation_integral" title="Correlation integral">Correlation integral</a></li> <li><a href="/wiki/Recurrence_quantification_analysis" title="Recurrence quantification analysis">Recurrence quantification analysis</a></li> <li><a href="/wiki/Approximate_entropy" title="Approximate entropy">Approximate entropy</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Correlation_dimension&action=edit&section=2" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-grassberger-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-grassberger_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-grassberger_1-1"><sup><i><b>b</b></i></sup></a></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="CITEREFPeter_Grassberger_and_Itamar_Procaccia1983" class="citation journal cs1"><a href="/wiki/Peter_Grassberger" title="Peter Grassberger">Peter Grassberger</a> and <a href="/wiki/Itamar_Procaccia" title="Itamar Procaccia">Itamar Procaccia</a> (1983). "Measuring the Strangeness of Strange Attractors". <i>Physica D: Nonlinear Phenomena</i>. <b>9</b> (1‒2): 189‒208. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1983PhyD....9..189G">1983PhyD....9..189G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0167-2789%2883%2990298-1">10.1016/0167-2789(83)90298-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+D%3A+Nonlinear+Phenomena&rft.atitle=Measuring+the+Strangeness+of+Strange+Attractors&rft.volume=9&rft.issue=1%E2%80%922&rft.pages=189%E2%80%92208&rft.date=1983&rft_id=info%3Adoi%2F10.1016%2F0167-2789%2883%2990298-1&rft_id=info%3Abibcode%2F1983PhyD....9..189G&rft.au=Peter+Grassberger+and+Itamar+Procaccia&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACorrelation+dimension" class="Z3988"></span></span> </li> <li id="cite_note-grassberger2-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-grassberger2_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeter_Grassberger_and_Itamar_Procaccia1983" class="citation journal cs1"><a href="/wiki/Peter_Grassberger" title="Peter Grassberger">Peter Grassberger</a> and Itamar Procaccia (1983). "Characterization of Strange Attractors". <i>Physical Review Letters</i>. <b>50</b> (5): 346‒349. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1983PhRvL..50..346G">1983PhRvL..50..346G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.50.346">10.1103/PhysRevLett.50.346</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Characterization+of+Strange+Attractors&rft.volume=50&rft.issue=5&rft.pages=346%E2%80%92349&rft.date=1983&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.50.346&rft_id=info%3Abibcode%2F1983PhRvL..50..346G&rft.au=Peter+Grassberger+and+Itamar+Procaccia&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACorrelation+dimension" class="Z3988"></span></span> </li> <li id="cite_note-grassberger3-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-grassberger3_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeter_Grassberger1983" class="citation journal cs1"><a href="/wiki/Peter_Grassberger" title="Peter Grassberger">Peter Grassberger</a> (1983). "Generalized Dimensions of Strange Attractors". <i>Physics Letters A</i>. <b>97</b> (6): 227‒230. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1983PhLA...97..227G">1983PhLA...97..227G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0375-9601%2883%2990753-3">10.1016/0375-9601(83)90753-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+A&rft.atitle=Generalized+Dimensions+of+Strange+Attractors&rft.volume=97&rft.issue=6&rft.pages=227%E2%80%92230&rft.date=1983&rft_id=info%3Adoi%2F10.1016%2F0375-9601%2883%2990753-3&rft_id=info%3Abibcode%2F1983PhLA...97..227G&rft.au=Peter+Grassberger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACorrelation+dimension" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeCosterMitchell1991" class="citation journal cs1">DeCoster, Gregory P.; Mitchell, Douglas W. (1991). "The efficacy of the correlation dimension technique in detecting determinism in small samples". <i>Journal of Statistical Computation and Simulation</i>. <b>39</b> (4): 221–229. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00949659108811357">10.1080/00949659108811357</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Computation+and+Simulation&rft.atitle=The+efficacy+of+the+correlation+dimension+technique+in+detecting+determinism+in+small+samples&rft.volume=39&rft.issue=4&rft.pages=221-229&rft.date=1991&rft_id=info%3Adoi%2F10.1080%2F00949659108811357&rft.aulast=DeCoster&rft.aufirst=Gregory+P.&rft.au=Mitchell%2C+Douglas+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACorrelation+dimension" class="Z3988"></span></span> </li> <li id="cite_note-sit-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-sit_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSonett,_C.,_Giampapa,_M.,_and_Matthews,_M._(Eds.)1992" class="citation book cs1">Sonett, C., Giampapa, M., and Matthews, M. (Eds.) (1992). <i>The Sun in Time</i>. <a href="/wiki/University_of_Arizona_Press" title="University of Arizona Press">University of Arizona Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8165-1297-3" title="Special:BookSources/0-8165-1297-3"><bdi>0-8165-1297-3</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+Sun+in+Time&rft.pub=University+of+Arizona+Press&rft.date=1992&rft.isbn=0-8165-1297-3&rft.au=Sonett%2C+C.%2C+Giampapa%2C+M.%2C+and+Matthews%2C+M.+%28Eds.%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACorrelation+dimension" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: 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navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fractal_dimension" title="Fractal dimension">Fractal dimensions</a> <ul><li><a href="/wiki/Assouad_dimension" title="Assouad dimension">Assouad</a></li> <li><a href="/wiki/Minkowski%E2%80%93Bouligand_dimension" title="Minkowski–Bouligand dimension">Box-counting</a> <ul><li><a href="/wiki/Higuchi_dimension" title="Higuchi dimension">Higuchi</a></li></ul></li> <li><a class="mw-selflink selflink">Correlation</a></li> <li><a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff</a></li> <li><a href="/wiki/Packing_dimension" title="Packing dimension">Packing</a></li> <li><a href="/wiki/Lebesgue_covering_dimension" title="Lebesgue covering dimension">Topological</a></li></ul></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Self-similarity" title="Self-similarity">Self-similarity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Iterated_function_system" title="Iterated function system">Iterated function <br />system</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barnsley_fern" title="Barnsley fern">Barnsley fern</a></li> <li><a href="/wiki/Cantor_set" title="Cantor set">Cantor set</a></li> <li><a href="/wiki/Koch_snowflake" title="Koch snowflake">Koch snowflake</a></li> <li><a href="/wiki/Menger_sponge" title="Menger sponge">Menger sponge</a></li> <li><a href="/wiki/Sierpi%C5%84ski_carpet" title="Sierpiński carpet">Sierpiński carpet</a></li> <li><a href="/wiki/Sierpi%C5%84ski_triangle" title="Sierpiński triangle">Sierpiński triangle</a></li> <li><a href="/wiki/Apollonian_gasket" title="Apollonian gasket">Apollonian gasket</a></li> <li><a href="/wiki/Fibonacci_word_fractal" title="Fibonacci word fractal">Fibonacci word</a></li> <li><a href="/wiki/Space-filling_curve" title="Space-filling curve">Space-filling curve</a> <ul><li><a href="/wiki/Blancmange_curve" title="Blancmange curve">Blancmange curve</a></li> <li><a href="/wiki/De_Rham_curve" title="De Rham curve">De Rham curve</a> <ul><li><a href="/wiki/Minkowski_sausage" title="Minkowski sausage">Minkowski</a></li></ul></li> <li><a href="/wiki/Dragon_curve" title="Dragon curve">Dragon curve</a></li> <li><a href="/wiki/Hilbert_curve" title="Hilbert curve">Hilbert curve</a></li> <li><a href="/wiki/Koch_snowflake" title="Koch snowflake">Koch curve</a></li> <li><a href="/wiki/L%C3%A9vy_C_curve" title="Lévy C curve">Lévy C curve</a></li> <li><a href="/wiki/Moore_curve" title="Moore curve">Moore curve</a></li> <li><a href="/wiki/Peano_curve" title="Peano curve">Peano curve</a></li> <li><a href="/wiki/Sierpi%C5%84ski_curve" title="Sierpiński curve">Sierpiński curve</a></li> <li><a href="/wiki/Z-order_curve" title="Z-order curve">Z-order curve</a></li></ul></li> <li><a href="/wiki/Fractal_string" title="Fractal string">String</a></li> <li><a href="/wiki/T-square_(fractal)" title="T-square (fractal)">T-square</a></li> <li><a href="/wiki/N-flake" title="N-flake">n-flake</a></li> <li><a href="/wiki/Vicsek_fractal" title="Vicsek fractal">Vicsek fractal</a></li> <li><a href="/wiki/Gosper_curve" title="Gosper curve">Gosper curve</a></li> <li><a href="/wiki/Pythagoras_tree_(fractal)" title="Pythagoras tree (fractal)">Pythagoras tree</a></li> <li><a href="/wiki/Weierstrass_function" title="Weierstrass function">Weierstrass function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Attractor#Strange_attractor" title="Attractor">Strange attractor</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multifractal_system" title="Multifractal system">Multifractal system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L-system" title="L-system">L-system</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fractal_canopy" title="Fractal canopy">Fractal canopy</a></li> <li><a href="/wiki/Space-filling_curve" title="Space-filling curve">Space-filling curve</a> <ul><li><a href="/wiki/H_tree" title="H tree">H tree</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fractal#Common_techniques_for_generating_fractals" title="Fractal">Escape-time <br />fractals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burning_Ship_fractal" title="Burning Ship fractal">Burning Ship fractal</a></li> <li><a href="/wiki/Julia_set" title="Julia set">Julia set</a> <ul><li><a href="/wiki/Filled_Julia_set" title="Filled Julia set">Filled</a></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Douady_rabbit" title="Douady rabbit">Douady rabbit</a></li></ul></li> <li><a href="/wiki/Lyapunov_fractal" title="Lyapunov fractal">Lyapunov fractal</a></li> <li><a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a> <ul><li><a href="/wiki/Misiurewicz_point" title="Misiurewicz point">Misiurewicz point</a></li></ul></li> <li><a href="/wiki/Multibrot_set" title="Multibrot set">Multibrot set</a></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Tricorn_(mathematics)" title="Tricorn (mathematics)">Tricorn</a></li> <li><a href="/wiki/Mandelbox" title="Mandelbox">Mandelbox</a></li> <li><a href="/wiki/Mandelbulb" title="Mandelbulb">Mandelbulb</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a> techniques</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Buddhabrot" title="Buddhabrot">Buddhabrot</a></li> <li><a href="/wiki/Orbit_trap" title="Orbit trap">Orbit trap</a></li> <li><a href="/wiki/Pickover_stalk" title="Pickover stalk">Pickover stalk</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Chaos_game" title="Chaos game">Random</a> fractals</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> <ul><li><a href="/wiki/Diffusion-limited_aggregation" title="Diffusion-limited aggregation">Brownian tree</a></li> <li><a href="/wiki/Brownian_motor" title="Brownian motor">Brownian motor</a></li></ul></li> <li><a href="/wiki/Fractal_landscape" title="Fractal landscape">Fractal landscape</a></li> <li><a href="/wiki/L%C3%A9vy_flight" title="Lévy flight">Lévy flight</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation theory</a></li> <li><a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding walk</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Michael_Barnsley" title="Michael Barnsley">Michael Barnsley</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Bill_Gosper" title="Bill Gosper">Bill Gosper</a></li> <li><a href="/wiki/Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a></li> <li><a href="/wiki/Desmond_Paul_Henry" title="Desmond Paul Henry">Desmond Paul Henry</a></li> <li><a href="/wiki/Gaston_Julia" title="Gaston Julia">Gaston Julia</a></li> <li><a href="/wiki/Niels_Fabian_Helge_von_Koch" title="Niels Fabian Helge von Koch">Niels Fabian Helge von Koch</a></li> <li><a href="/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Paul Lévy</a></li> <li><a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a></li> <li><a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a></li> <li><a href="/wiki/Hamid_Naderi_Yeganeh" title="Hamid Naderi Yeganeh">Hamid Naderi Yeganeh</a></li> <li><a href="/wiki/Lewis_Fry_Richardson" title="Lewis Fry Richardson">Lewis Fry Richardson</a></li> <li><a href="/wiki/Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Wacław Sierpiński</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coastline_paradox" title="Coastline paradox">Coastline paradox</a></li> <li><a href="/wiki/Fractal_art" title="Fractal art">Fractal art</a></li> <li><a href="/wiki/List_of_fractals_by_Hausdorff_dimension" title="List of fractals by Hausdorff dimension">List of fractals by Hausdorff dimension</a></li> <li><i><a href="/wiki/The_Fractal_Geometry_of_Nature" title="The Fractal Geometry of Nature">The Fractal Geometry of Nature</a></i> (1982 book)</li> <li><i><a href="/wiki/The_Beauty_of_Fractals" title="The Beauty of Fractals">The Beauty of Fractals</a></i> (1986 book)</li> <li><i><a href="/wiki/Chaos:_Making_a_New_Science" title="Chaos: Making a New Science">Chaos: Making a New Science</a></i> (1987 book)</li> <li><a href="/wiki/Kaleidoscope" title="Kaleidoscope">Kaleidoscope</a></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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