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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Iterated function system</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <b>iterated function systems</b> (<b>IFSs</b>) are a method of constructing <a href="Fractal" title="Fractal">fractals</a>; the resulting fractals are often <a href="Self-similar" class="mw-redirect" title="Self-similar">self-similar</a>. IFS fractals are more related to <a href="Set_theory" title="Set theory">set theory</a> than fractal geometry.<sup id="cite_ref-picg_1-0" class="reference"><a href="#cite_note-picg-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> They were introduced in 1981.

</p><p><b>IFS</b> fractals, as they are normally called, can be of any number of dimensions, but are commonly computed and drawn in 2D. The fractal is made up of the union of several copies of itself, each copy being transformed by a function (hence "function system"). The canonical example is the <a href="Sierpi%C5%84ski_triangle" title="Sierpiński triangle">Sierpiński triangle</a>. The functions are normally <a href="Contraction_mapping" title="Contraction mapping">contractive</a>, which means they bring points closer together and make shapes smaller. Hence, the shape of an IFS fractal is made up of several possibly-overlapping smaller copies of itself, each of which is also made up of copies of itself, <a href="Ad_infinitum" title="Ad infinitum">ad infinitum</a>. This is the source of its self-similar fractal nature.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Formally, an <a href="Iterated_function" title="Iterated function">iterated function</a> system is a finite set of <a href="Contraction_mapping" title="Contraction mapping">contraction mappings</a> on a <a href="Complete_metric_space" title="Complete metric space">complete metric space</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Symbolically,

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<annotation encoding="application/x-tex">{\displaystyle \{f_{i}:X\to X\mid i=1,2,\dots ,N\},\ N\in \mathbb {N} }</annotation>

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</math></span><img src="./3e1d4359480d183820a0dda5bc78918da77e9ff6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.411ex; height:2.843ex;" alt="{\displaystyle \{f_{i}:X\to X\mid i=1,2,\dots ,N\},\ N\in \mathbb {N} }" loading="lazy"></span></dd></dl>

<p>is an iterated function system if each <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>Hutchinson showed that, for the metric space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math></span><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span>, or more generally, for a complete metric space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, such a system of functions has a unique nonempty <a href="Compact_space" title="Compact space">compact</a> (closed and bounded) fixed set <i>S</i>.<sup id="cite_ref-hutchinson_3-0" class="reference"><a href="#cite_note-hutchinson-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> One way of constructing a fixed set is to start with an initial nonempty closed and bounded set <i>S</i><sub>0</sub> and iterate the actions of the <i>f</i><sub><i>i</i></sub>, taking <i>S</i><sub><i>n</i>+1</sub> to be the union of the images of <i>S</i><sub><i>n</i></sub> under the <i>f</i><sub><i>i</i></sub>; then taking <i>S</i> to be the <a href="Closure_(topology)" title="Closure (topology)">closure</a> of the limit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\rightarrow \infty }S_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }S_{n}}</annotation>

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</math></span><img src="./07180f682cbb6be31aff1242873fc0dbd148d6bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.303ex; height:3.676ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }S_{n}}" loading="lazy"></span>. Symbolically, the unique fixed (nonempty compact) set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}">

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<annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation>

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</math></span><img src="./44aba72977e43f863dd873b095d1dc0bd3f17608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}" loading="lazy"></span> has the property

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<annotation encoding="application/x-tex">{\displaystyle S={\overline {\bigcup _{i=1}^{N}f_{i}(S)}}.}</annotation>

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</math></span><img src="./eee6a117bf4db7da7f054c053b3a7a5a29224a3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.662ex; height:8.009ex;" alt="{\displaystyle S={\overline {\bigcup _{i=1}^{N}f_{i}(S)}}.}" loading="lazy"></span></dd></dl>

<p>The set <i>S</i> is thus the fixed set of the <a href="Hutchinson_operator" title="Hutchinson operator">Hutchinson operator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:2^{X}\to 2^{X}}">

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<annotation encoding="application/x-tex">{\displaystyle F:2^{X}\to 2^{X}}</annotation>

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</math></span><img src="./e105901c8036600b8ef9fe008186506ece6656fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.882ex; height:2.676ex;" alt="{\displaystyle F:2^{X}\to 2^{X}}" loading="lazy"></span> defined for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X}">

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</math></span><img src="./1dce86da0107830a9a97287f9486d9b4ff022875.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.822ex; height:2.343ex;" alt="{\displaystyle A\subseteq X}" loading="lazy"></span> via

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<annotation encoding="application/x-tex">{\displaystyle F(A)={\overline {\bigcup _{i=1}^{N}f_{i}(A)}}.}</annotation>

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</math></span><img src="./13e1c15bc05b0e1871dd95ffa7e434ebe81fc26b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.7ex; height:8.009ex;" alt="{\displaystyle F(A)={\overline {\bigcup _{i=1}^{N}f_{i}(A)}}.}" loading="lazy"></span></dd></dl>

<p>The existence and uniqueness of <i>S</i> is a consequence of the <a href="Contraction_mapping_principle" class="mw-redirect" title="Contraction mapping principle">contraction mapping principle</a>, as is the fact that

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }F^{n}(A)=S}</annotation>

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</math></span><img src="./87a906e0854cd7f443b4ba93de18ec68c57dd964.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.843ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }F^{n}(A)=S}" loading="lazy"></span></dd></dl>

<p>for any nonempty compact set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>). Random elements arbitrarily close to <i>S</i> may be obtained by the "chaos game," described below.

</p><p>Recently it was shown that the IFSs of non-contractive type (i.e. composed of maps that are not contractions with respect to any topologically equivalent metric in <i>X</i>) can yield attractors.

These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too.<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>The collection of functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

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</math></span><img src="./65da883ca3d16b461e46c94777b0d9c4aa010e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}" loading="lazy"></span> <a href="Generating_set" class="mw-redirect" title="Generating set">generates</a> a <a href="Monoid" title="Monoid">monoid</a> under <a href="Function_composition" title="Function composition">composition</a>. If there are only two such functions, the monoid can be visualized as a <a href="Binary_tree" title="Binary tree">binary tree</a>, where, at each node of the tree, one may compose with the one or the other function (<i>i.e.</i> take the left or the right branch). In general, if there are <i>k</i> functions, then one may visualize the monoid as a full <a href="K-ary_tree" class="mw-redirect" title="K-ary tree"><i>k</i>-ary tree</a>, also known as a <a href="Cayley_tree" class="mw-redirect" title="Cayley tree">Cayley tree</a>.

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<div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2></div>

<p>Sometimes each function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

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</math></span><img src="./65da883ca3d16b461e46c94777b0d9c4aa010e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}" loading="lazy"></span> is required to be a <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear</a>, or more generally an <a href="Affine_transformation" title="Affine transformation">affine</a>, transformation, and hence represented by a <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. However, IFSs may also be built from non-linear functions, including <a href="Projective_transformation" class="mw-redirect" title="Projective transformation">projective transformations</a> and <a href="M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>. The <a href="Fractal_flame" title="Fractal flame">Fractal flame</a> is an example of an IFS with nonlinear functions.

</p><p>The most common algorithm to compute IFS fractals is called the "<a href="Chaos_game" title="Chaos game">chaos game</a>". It consists of picking a random point in the plane, then iteratively applying one of the functions chosen at random from the function system to transform the point to get a next point. An alternative algorithm is to generate each possible sequence of functions up to a given maximum length, and then to plot the results of applying each of these sequences of functions to an initial point or shape.

</p><p>Each of these algorithms provides a global construction which generates points distributed across the whole fractal. If a small area of the fractal is being drawn, many of these points will fall outside of the screen boundaries. This makes zooming into an IFS construction drawn in this manner impractical.

</p><p>Although the theory of IFS requires each function to be contractive, in practice software that implements IFS only require that the whole system be contractive on average.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Partitioned_iterated_function_systems">Partitioned iterated function systems</h2></div>

<p>PIFS (partitioned iterated function systems), also called local iterated function systems,<sup id="cite_ref-lacroix_6-0" class="reference"><a href="#cite_note-lacroix-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> give surprisingly good image compression, even for photographs that don't seem to have the kinds of self-similar structure shown by simple IFS fractals.<sup id="cite_ref-SIGGRAPH'92_7-0" class="reference"><a href="#cite_note-SIGGRAPH'92-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="The_inverse_problem">The inverse problem</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Fractal_compression" title="Fractal compression">Fractal compression</a></div>

<p>Very fast algorithms exist to generate an image from a set of IFS or PIFS parameters. It is faster and requires much less storage space to store a description of how it was created, transmit that description to a destination device, and regenerate that image anew on the destination device, than to store and transmit the color of each pixel in the image.<sup id="cite_ref-lacroix_6-1" class="reference"><a href="#cite_note-lacroix-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>The <a href="Inverse_problem" title="Inverse problem">inverse problem</a> is more difficult: given some original arbitrary digital image such as a digital photograph, try to find a set of IFS parameters which, when evaluated by iteration, produces another image visually similar to the original.

In 1989, Arnaud Jacquin presented a solution to a restricted form of the inverse problem using only PIFS; the general form of the inverse problem remains unsolved.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kominek_9-0" class="reference"><a href="#cite_note-kominek-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lacroix_6-2" class="reference"><a href="#cite_note-lacroix-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>As of 1995, all <a href="Fractal_compression" title="Fractal compression">fractal compression</a> software is based on Jacquin's approach.<sup id="cite_ref-kominek_9-1" class="reference"><a href="#cite_note-kominek-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>The diagram shows the construction on an IFS from two affine functions. The functions are represented by their effect on the bi-unit square (the function transforms the outlined square into the shaded square). The combination of the two functions forms the <a href="Hutchinson_operator" title="Hutchinson operator">Hutchinson operator</a>. Three iterations of the operator are shown, and then the final image is of the fixed point, the final fractal.

</p><p>Early examples of fractals which may be generated by an IFS include the <a href="Cantor_set" title="Cantor set">Cantor set</a>, first described in 1884; and <a href="De_Rham_curve" title="De Rham curve">de Rham curves</a>, a type of self-similar curve described by <a href="Georges_de_Rham" title="Georges de Rham">Georges de Rham</a> in 1957.

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>IFSs were conceived in their present form by John E. Hutchinson in 1981<sup id="cite_ref-hutchinson_3-1" class="reference"><a href="#cite_note-hutchinson-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and popularized by <a href="Michael_Barnsley" title="Michael Barnsley">Michael Barnsley</a>'s book <i>Fractals Everywhere</i>.

</p>

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</style><blockquote class="templatequote"><p>IFSs provide models for certain plants, leaves, and ferns, by virtue of the self-similarity which often occurs in branching structures in nature.</p></blockquote><div class="templatequotecite"><p style="display: inline; padding-left: 2.3em;">— Michael Barnsley <i>et al.</i><sup id="cite_ref-V-variable_10-0" class="reference"><a href="#cite_note-V-variable-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></p></div>

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<ul><li><a href="Complex-base_system#Base_.E2.88.921.C2.B1i" title="Complex-base system">Complex-base system</a></li>

<li><a href="Collage_theorem" title="Collage theorem">Collage theorem</a></li>

<li><a href="Infinite_compositions_of_analytic_functions" title="Infinite compositions of analytic functions">Infinite compositions of analytic functions</a></li>

<li><a href="L-system" title="L-system">L-system</a></li>

<li><a href="Fractal_compression" title="Fractal compression">Fractal compression</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFZobristChaman_Sabharwal1992" class="citation book cs1">Zobrist, George Winston; Chaman Sabharwal (1992). <a rel="nofollow" class="external text" href="https://play.google.com/store/books/details?id=Ai6Qo0qoE9EC"><i>Progress in Computer Graphics: Volume 1</i></a>. Intellect Books. p. 135. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780893916510</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">7 May</span> 2017</span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Michael Barnsley (1988). <i>Fractals Everywhere</i>, p.82. Academic Press, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780120790623</bdi>.</span>

</li>

<li id="cite_note-hutchinson-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-hutchinson_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-hutchinson_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFHutchinson1981" class="citation journal cs1">Hutchinson, John E. (1981). <a rel="nofollow" class="external text" href="https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf">"Fractals and self similarity"</a> <span class="cs1-format">(PDF)</span>. <i>Indiana Univ. Math. J</i>. <b>30</b> (5): <span class="nowrap">713–</span>747. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1512%2Fiumj.1981.30.30055">10.1512/iumj.1981.30.30055</a></span>.</cite></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">M. Barnsley, A. Vince, The Chaos Game on a General Iterated Function System</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFDravesErik_Reckase2007" class="citation web cs1"><a href="Scott_Draves" title="Scott Draves">Draves, Scott</a>; Erik Reckase (July 2007). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080509073421/http://flam3.com/flame.pdf">"The Fractal Flame Algorithm"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://flam3.com/flame.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2008-05-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-07-17</span></span>.</cite></span>

</li>

<li id="cite_note-lacroix-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-lacroix_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-lacroix_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-lacroix_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Bruno Lacroix. <a rel="nofollow" class="external text" href="http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp01/MQ36939.pdf">"Fractal Image Compression"</a>. 1998.</span>

</li>

<li id="cite_note-SIGGRAPH'92-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-SIGGRAPH'92_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFFischer1992" class="citation conference cs1 cs1-prop-long-vol">Fischer, Yuval (1992-08-12). Przemyslaw Prusinkiewicz (ed.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170912012035/https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf"><i>SIGGRAPH'92 course notes - Fractal Image Compression</i></a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="http://www.siggraph.org/">SIGGRAPH</a>. Vol. Fractals - From Folk Art to Hyperreality. <a href="ACM_SIGGRAPH" title="ACM SIGGRAPH">ACM SIGGRAPH</a>. Archived from <a rel="nofollow" class="external text" href="https://karczmarczuk.users.greyc.fr/matrs/Dess/RADI/Refs/fractal_paper.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-09-12<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-06-30</span></span>.</cite></span>

</li>

Dietmar Saupe, Raouf Hamzaoui.

<a rel="nofollow" class="external text" href="https://www.uni-konstanz.de/mmsp/pubsys/publishedFiles/SaHa94.pdf">"A Review of the Fractal Image Compression Literature"</a>.</span>

</li>

John Kominek.

<a rel="nofollow" class="external text" href="https://web.archive.org/web/20181123231446/https://pdfs.semanticscholar.org/d77b/ffac2560c92771d3756c0e29863a44919d6f.pdf">"Algorithm for Fast Fractal Image Compression"</a>.

</li>

<li id="cite_note-V-variable-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-V-variable_10-0">^</a></b></span> <span class="reference-text"><a href="Michael_Barnsley" title="Michael Barnsley">Michael Barnsley</a>, <i>et al.</i>,<cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.maths.anu.edu.au/~barnsley/pdfs/V-var_super_fractals.pdf">"V-variable fractals and superfractals"</a> <span class="cs1-format">(PDF)</span>.</cite> <span style="font-size: 85%;">(2.22 MB)</span></span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFDravesErik_Reckase2007" class="citation web cs1"><a href="Scott_Draves" title="Scott Draves">Draves, Scott</a>; Erik Reckase (July 2007). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080509073421/http://flam3.com/flame.pdf">"The Fractal Flame Algorithm"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://flam3.com/flame.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2008-05-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-07-17</span></span>.</cite></li>

<li><cite id="CITEREFFalconer1990" class="citation book cs1"><a href="Kenneth_Falconer_(mathematician)" title="Kenneth Falconer (mathematician)">Falconer, Kenneth</a> (1990). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fractalgeometrym0000falc/page/113"><i>Fractal geometry: Mathematical foundations and applications</i></a></span>. John Wiley and Sons. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/fractalgeometrym0000falc/page/113">113–117, 136</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-92287-0</bdi>.</cite></li>

<li><cite id="CITEREFBarnsleyAndrew_Vince2011" class="citation journal cs1"><a href="Michael_Barnsley" title="Michael Barnsley">Barnsley, Michael</a>; Andrew Vince (2011). "The Chaos Game on a General Iterated Function System". <i>Ergodic Theory Dynam. Systems</i>. <b>31</b> (4): <span class="nowrap">1073–</span>1079. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1005.0322">1005.0322</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010arXiv1005.0322B">2010arXiv1005.0322B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0143385710000428">10.1017/S0143385710000428</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122674315">122674315</a>.</cite></li>

<li><b>For an historical overview, and the generalization :</b> <cite id="CITEREFDavid2019" class="citation journal cs1">David, Claire (2019). <a rel="nofollow" class="external text" href="https://journals.onaft.edu.ua/index.php/geometry/article/view/1485">"fractal properties of Weierstrass-type functions"</a>. <i>Proceedings of the International Geometry Center</i>. <b>12</b> (2): <span class="nowrap">43–</span>61. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.15673%2Ftmgc.v12i2.1485">10.15673/tmgc.v12i2.1485</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:209964068">209964068</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/362010262_A_Primer_on_the_Elementary_Theory_of_Infinite_Compositions_of_Complex_Functions_Images">A Primer on the Elementary Theory of Infinite Compositions of Complex Functions</a></li></ul>

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</style><div id="Fractal_software217" style="font-size:114%;margin:0 4em"><a href="Fractal-generating_software" title="Fractal-generating software">Fractal software</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Digital_art" title="Digital art">Digital art</a></li>

<li><a href="Graphics_software" title="Graphics software">Graphics software</a></li>

<li><a href="Fractal_art" title="Fractal art">Fractal art</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open-source</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Apophysis_(software)" title="Apophysis (software)">Apophysis</a></li>

<li><a href="Blender_(software)" title="Blender (software)">Blender</a></li>

<li><a href="Fractint" title="Fractint">Fractint</a></li>

<li>Kalles Fraktaler</li>

<li><a href="MilkDrop" title="MilkDrop">MilkDrop</a></li>

<li><a href="Sterling_(program)" title="Sterling (program)">Sterling</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">GNU</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Electric_Sheep" title="Electric Sheep">Electric Sheep</a></li>

<li><a href="OpenPlaG" title="OpenPlaG">openPlaG</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Freeware</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>IFStile</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Retail</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Cross-platform</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bryce_(software)" title="Bryce (software)">Bryce</a></li>

<li><a href="Maple_(software)" title="Maple (software)">Maple</a></li>

<li><a href="Ultra_Fractal" title="Ultra Fractal">Ultra Fractal</a></li>

<li><a href="Wolfram_Mathematica" class="mw-redirect" title="Wolfram Mathematica">Wolfram Mathematica</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Windows only</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="VisSim" title="VisSim">VisSim</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Scenery_generator" title="Scenery generator">Scenery generator</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="MojoWorld_Generator" title="MojoWorld Generator">MojoWorld Generator</a></li>

<li><a href="Picogen" title="Picogen">Picogen</a></li>

<li><a href="Terragen" title="Terragen">Terragen</a></li>

<li><a href="VistaPro" title="VistaPro">VistaPro</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Found objects</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Burning_Ship_fractal" title="Burning Ship fractal">Burning Ship fractal</a></li>

<li><a href="Jerusalem_cube" class="mw-redirect" title="Jerusalem cube">Jerusalem cube</a></li>

<li><a href="Julia_set" title="Julia set">Julia set</a></li>

<li><a href="Mandelbox" title="Mandelbox">Mandelbox</a></li>

<li><a href="Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a></li>

<li><a href="Mandelbulb" title="Mandelbulb">Mandelbulb</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Computer-generated_imagery" title="Computer-generated imagery">Computer-generated imagery</a></li>

<li><a href="Fractal_compression" title="Fractal compression">Fractal compression</a></li>

<li><a href="Fractal_landscape" title="Fractal landscape">Fractal landscape</a></li>

<li><a href="Fractal_flame" title="Fractal flame">Fractal flame</a></li>

<li><a href="Mathematical_visualization" title="Mathematical visualization">Mathematical visualization</a></li>

<li><a href="Orbit_trap" title="Orbit trap">Orbit trap</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li>Category</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Fractals328" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Fractals328" style="font-size:114%;margin:0 4em"><a href="Fractal" title="Fractal">Fractals</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Characteristics</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="Fractal_dimension" title="Fractal dimension">Fractal dimensions</a>

<ul><li><a href="Assouad_dimension" title="Assouad dimension">Assouad</a></li>

<li><a href="Minkowski%E2%80%93Bouligand_dimension" title="Minkowski–Bouligand dimension">Box-counting</a>

<ul><li><a href="Higuchi_dimension" title="Higuchi dimension">Higuchi</a></li></ul></li>

<li><a href="Correlation_dimension" title="Correlation dimension">Correlation</a></li>

<li><a href="Hausdorff_dimension" title="Hausdorff dimension">Hausdorff</a></li>

<li><a href="Packing_dimension" title="Packing dimension">Packing</a></li>

<li><a href="Lebesgue_covering_dimension" title="Lebesgue covering dimension">Topological</a></li></ul></li>

<li><a href="Recursion" title="Recursion">Recursion</a></li>

<li><a href="Self-similarity" title="Self-similarity">Self-similarity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"></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="Barnsley_fern" title="Barnsley fern">Barnsley fern</a></li>

<li><a href="Cantor_set" title="Cantor set">Cantor set</a></li>

<li><a href="Koch_snowflake" title="Koch snowflake">Koch snowflake</a></li>

<li><a href="Menger_sponge" title="Menger sponge">Menger sponge</a></li>

<li><a href="Sierpi%C5%84ski_carpet" title="Sierpiński carpet">Sierpiński carpet</a></li>

<li><a href="Sierpi%C5%84ski_triangle" title="Sierpiński triangle">Sierpiński triangle</a></li>

<li><a href="Apollonian_gasket" title="Apollonian gasket">Apollonian gasket</a></li>

<li><a href="Fibonacci_word_fractal" title="Fibonacci word fractal">Fibonacci word</a></li>

<li><a href="Space-filling_curve" title="Space-filling curve">Space-filling curve</a>

<ul><li><a href="Blancmange_curve" title="Blancmange curve">Blancmange curve</a></li>

<li><a href="De_Rham_curve" title="De Rham curve">De Rham curve</a>

<ul><li><a href="Minkowski_sausage" title="Minkowski sausage">Minkowski</a></li></ul></li>

<li><a href="Dragon_curve" title="Dragon curve">Dragon curve</a></li>

<li><a href="Hilbert_curve" title="Hilbert curve">Hilbert curve</a></li>

<li><a href="Koch_snowflake" title="Koch snowflake">Koch curve</a></li>

<li><a href="L%C3%A9vy_C_curve" title="Lévy C curve">Lévy C curve</a></li>

<li><a href="Moore_curve" title="Moore curve">Moore curve</a></li>

<li><a href="Peano_curve" title="Peano curve">Peano curve</a></li>

<li><a href="Sierpi%C5%84ski_curve" title="Sierpiński curve">Sierpiński curve</a></li>

<li><a href="Z-order_curve" title="Z-order curve">Z-order curve</a></li></ul></li>

<li><a href="Fractal_string" title="Fractal string">String</a></li>

<li><a href="T-square_(fractal)" title="T-square (fractal)">T-square</a></li>

<li><a href="N-flake" title="N-flake">n-flake</a></li>

<li><a href="Vicsek_fractal" title="Vicsek fractal">Vicsek fractal</a></li>

<li><a href="Gosper_curve" title="Gosper curve">Gosper curve</a></li>

<li><a href="Pythagoras_tree_(fractal)" title="Pythagoras tree (fractal)">Pythagoras tree</a></li>

<li><a href="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="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="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="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="Fractal_canopy" title="Fractal canopy">Fractal canopy</a></li>

<li><a href="Space-filling_curve" title="Space-filling curve">Space-filling curve</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="Burning_Ship_fractal" title="Burning Ship fractal">Burning Ship fractal</a></li>

<li><a href="Julia_set" title="Julia set">Julia set</a>

<ul><li><a href="Filled_Julia_set" title="Filled Julia set">Filled</a></li>

<li><a href="Newton_fractal" title="Newton fractal">Newton fractal</a></li>

<li><a href="Douady_rabbit" title="Douady rabbit">Douady rabbit</a></li></ul></li>

<li><a href="Lyapunov_fractal" title="Lyapunov fractal">Lyapunov fractal</a></li>

<li><a href="Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a>

<ul><li><a href="Misiurewicz_point" title="Misiurewicz point">Misiurewicz point</a></li></ul></li>

<li><a href="Multibrot_set" title="Multibrot set">Multibrot set</a></li>

<li><a href="Newton_fractal" title="Newton fractal">Newton fractal</a></li>

<li><a href="Tricorn_(mathematics)" title="Tricorn (mathematics)">Tricorn</a></li>

<li><a href="Mandelbox" title="Mandelbox">Mandelbox</a></li>

<li><a href="Mandelbulb" title="Mandelbulb">Mandelbulb</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="Buddhabrot" title="Buddhabrot">Buddhabrot</a></li>

<li><a href="Orbit_trap" title="Orbit trap">Orbit trap</a></li>

<li><a href="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="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="Brownian_motion" title="Brownian motion">Brownian motion</a>

<ul><li><a href="Diffusion-limited_aggregation" title="Diffusion-limited aggregation">Brownian tree</a></li>

<li><a href="Brownian_motor" title="Brownian motor">Brownian motor</a></li></ul></li>

<li><a href="Fractal_landscape" title="Fractal landscape">Fractal landscape</a></li>

<li><a href="L%C3%A9vy_flight" title="Lévy flight">Lévy flight</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation theory</a></li>

<li><a href="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="Michael_Barnsley" title="Michael Barnsley">Michael Barnsley</a></li>

<li><a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li>

<li><a href="Bill_Gosper" title="Bill Gosper">Bill Gosper</a></li>

<li><a href="Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a></li>

<li><a href="Desmond_Paul_Henry" title="Desmond Paul Henry">Desmond Paul Henry</a></li>

<li><a href="Gaston_Julia" title="Gaston Julia">Gaston Julia</a></li>

<li><a href="Niels_Fabian_Helge_von_Koch" title="Niels Fabian Helge von Koch">Niels Fabian Helge von Koch</a></li>

<li><a href="Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a></li>

<li><a href="Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a></li>

<li><a href="Hamid_Naderi_Yeganeh" title="Hamid Naderi Yeganeh">Hamid Naderi Yeganeh</a></li>

<li><a href="Lewis_Fry_Richardson" title="Lewis Fry Richardson">Lewis Fry Richardson</a></li>

<li><a href="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="Coastline_paradox" title="Coastline paradox">Coastline paradox</a></li>

<li><a href="Fractal_art" title="Fractal art">Fractal art</a></li>

<li><a href="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="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="The_Beauty_of_Fractals" title="The Beauty of Fractals">The Beauty of Fractals</a></i> (1986 book)</li>

<li><i><a href="Chaos%3A_Making_a_New_Science" title="Chaos: Making a New Science">Chaos: Making a New Science</a></i> (1987 book)</li>

<li><a href="Kaleidoscope" title="Kaleidoscope">Kaleidoscope</a></li>

<li><a href="Chaos_theory" title="Chaos theory">Chaos theory</a></li></ul>

</div></td></tr></tbody></table></div>

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