Fractal

<h2 id="etymology">Etymology</h2>
The term "fractal" was coined by the mathematician <a href="/facts/Beno%25C3%25AEt_Mandelbrot/fa8tKP93">Benoît Mandelbrot</a> in 1975.<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a> Mandelbrot based it on the Latin frāctus, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional <a href="/facts/Fractal_dimension/pJ9tVnwn">dimensions</a> to geometric <a href="/facts/Patterns_in_nature/handzeJh">patterns in nature</a>.<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a><a class="footnote-ref" id="fnref:40" href="#fn:40">40</a><a class="footnote-ref" id="fnref:41" href="#fn:41">41</a>

<h2 id="introduction">Introduction</h2>

The word "fractal" often has different connotations for mathematicians and the general public, where the public is more likely to be familiar with <a href="/facts/Fractal_art/hjnu1JP2">fractal art</a> than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background.
The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the <a href="/facts/Infinite_regress/Y3rk8wQe">infinite regress</a> in parallel mirrors or the <a href="/facts/Homunculus/lxorfR15">homunculus</a>, the little man inside the head of the little man inside the head ...). The difference for fractals is that the pattern reproduced must be detailed.<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a>: 166, 18 <a class="footnote-ref" id="fnref:43" href="#fn:43">43</a><a class="footnote-ref" id="fnref:44" href="#fn:44">44</a>
This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a <a href="/facts/Fractal_dimension/pJ9tVnwn">fractal dimension</a> greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric <a href="/facts/Shapes/nww0x85d">shapes</a> are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is <a href="/facts/Rep-tile/Q5chIBLN">rep-tiled</a> into pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled into pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 32 = 9 pieces.
We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/r, there are a total of rn pieces. Now, consider the <a href="/facts/Koch_curve/JTckxQVK">Koch curve</a>. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the "dimension" of the Koch curve as being the unique real number D that satisfies 3D = 4. This number is called the fractal dimension of the Koch curve; it is not the conventionally perceived dimension of a curve. In general, a key property of fractals is that the fractal dimension differs from the conventionally understood dimension (formally called the topological dimension).

This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a>". In a concrete sense, this means fractals cannot be measured in traditional ways.<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a><a class="footnote-ref" id="fnref:46" href="#fn:46">46</a><a class="footnote-ref" id="fnref:47" href="#fn:47">47</a> To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of <a href="/facts/Rectifiable_curve/QgTrmLxY">measuring</a> with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has an infinite perimeter.<a class="footnote-ref" id="fnref:48" href="#fn:48">48</a>

<h2 id="history">History</h2>

The history of fractals traces a path from chiefly theoretical studies to modern applications in <a href="/facts/Computer_graphics/lapBE8wv">computer graphics</a>, with several notable people contributing canonical fractal forms along the way.<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a><a class="footnote-ref" id="fnref:50" href="#fn:50">50</a> 
A common theme in traditional <a href="/facts/Architecture_of_Africa/PwUaz3hk">African architecture</a> is the use of fractal scaling, whereby small parts of the structure tend to look similar to larger parts, such as a circular village made of circular houses.<a class="footnote-ref" id="fnref:51" href="#fn:51">51</a>
According to <a href="/facts/Clifford_A._Pickover/tsRe4UWE">Pickover</a>, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher <a href="/facts/Gottfried_Leibniz/cTc2er83">Gottfried Leibniz</a> pondered <a href="/facts/Recursion/CxSKxJoW">recursive</a> <a href="/facts/Self-similarity/9Nvo7edY">self-similarity</a> (although he made the mistake of thinking that only the <a href="/facts/Straight_line/gJqsr4Gj">straight line</a> was self-similar in this sense).<a class="footnote-ref" id="fnref:52" href="#fn:52">52</a>
In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them.<a class="footnote-ref" id="fnref:53" href="#fn:53">53</a>: 405  Indeed, according to various historical accounts, after that point few mathematicians tackled the issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters".<a class="footnote-ref" id="fnref:54" href="#fn:54">54</a><a class="footnote-ref" id="fnref:55" href="#fn:55">55</a><a class="footnote-ref" id="fnref:56" href="#fn:56">56</a> Thus, it was not until two centuries had passed that on July 18, 1872 <a href="/facts/Karl_Weierstrass/4U8vdQzL">Karl Weierstrass</a> presented the first definition of a <a href="/facts/Weierstrass_function/m9jU4LE5">function</a> with a <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> that would today be considered a fractal, having the non-<a href="/facts/Intuition_(knowledge)/Mshu0Vxb">intuitive</a> property of being everywhere <a href="/facts/Continuous_function/sKbl02pB">continuous</a> but <a href="/facts/Nowhere_differentiable/jQqTgLk1">nowhere differentiable</a> at the Royal Prussian Academy of Sciences.<a class="footnote-ref" id="fnref:57" href="#fn:57">57</a>: 7 <a class="footnote-ref" id="fnref:58" href="#fn:58">58</a>
In addition, the quotient difference becomes arbitrarily large as the summation index increases.<a class="footnote-ref" id="fnref:59" href="#fn:59">59</a> Not long after that, in 1883, <a href="/facts/Georg_Cantor/k5v1DvW4">Georg Cantor</a>, who attended lectures by Weierstrass,<a class="footnote-ref" id="fnref:60" href="#fn:60">60</a> published examples of <a href="/facts/Subset/SJGERmbA">subsets</a> of the real line known as <a href="/facts/Cantor_set/6NUcMJvN">Cantor sets</a>, which had unusual properties and are now recognized as fractals.<a class="footnote-ref" id="fnref:61" href="#fn:61">61</a>: 11–24  Also in the last part of that century, <a href="/facts/Felix_Klein/XMILhEfy">Felix Klein</a> and <a href="/facts/Henri_Poincar%25C3%25A9/AazL2j4X">Henri Poincaré</a> introduced a category of fractal that has come to be called "self-inverse" fractals.<a class="footnote-ref" id="fnref:62" href="#fn:62">62</a>: 166

One of the next milestones came in 1904, when <a href="/facts/Helge_von_Koch/EXRVVl5a">Helge von Koch</a>, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called the <a href="/facts/Koch_snowflake/JTckxQVK">Koch snowflake</a>.<a class="footnote-ref" id="fnref:63" href="#fn:63">63</a>: 25 <a class="footnote-ref" id="fnref:64" href="#fn:64">64</a> Another milestone came a decade later in 1915, when <a href="/facts/Wac%25C5%2582aw_Sierpi%25C5%2584ski/ywT4mREz">Wacław Sierpiński</a> constructed his famous <a href="/facts/Sierpinski_triangle/gFmLRJxM">triangle</a> then, one year later, his <a href="/facts/Sierpinski_carpet/3ogstvh2">carpet</a>. By 1918, two French mathematicians, <a href="/facts/Pierre_Fatou/mXA5yhVW">Pierre Fatou</a> and <a href="/facts/Gaston_Julia/SALf3AWG">Gaston Julia</a>, though working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mapping <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a> and iterative functions and leading to further ideas about <a href="/facts/Strange_attractors/FAx5Db4w">attractors and repellors</a> (i.e., points that attract or repel other points), which have become very important in the study of fractals.<a class="footnote-ref" id="fnref:65" href="#fn:65">65</a><a class="footnote-ref" id="fnref:66" href="#fn:66">66</a><a class="footnote-ref" id="fnref:67" href="#fn:67">67</a>
Very shortly after that work was submitted, by March 1918, <a href="/facts/Felix_Hausdorff/Rn6DuY2x">Felix Hausdorff</a> expanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have non-integer dimensions.<a class="footnote-ref" id="fnref:68" href="#fn:68">68</a> The idea of self-similar curves was taken further by <a href="/facts/Paul_L%25C3%25A9vy_(mathematician)/RKFMXMkL">Paul Lévy</a>, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, the <a href="/facts/L%25C3%25A9vy_C_curve/znugnMnE">Lévy C curve</a>.<a class="footnote-ref" id="fnref:69" href="#fn:69">69</a>

Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings).<a class="footnote-ref" id="fnref:70" href="#fn:70">70</a>: 179 <a class="footnote-ref" id="fnref:71" href="#fn:71">71</a><a class="footnote-ref" id="fnref:72" href="#fn:72">72</a> That changed, however, in the 1960s, when <a href="/facts/Benoit_Mandelbrot/fa8tKP93">Benoit Mandelbrot</a> started writing about self-similarity in papers such as <a href="/facts/How_Long_Is_the_Coast_of_Britain%253F_Statistical_Self-Similarity_and_Fractional_Dimension/AzckgsE5">How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension</a>,<a class="footnote-ref" id="fnref:73" href="#fn:73">73</a><a class="footnote-ref" id="fnref:74" href="#fn:74">74</a> which built on earlier work by <a href="/facts/Lewis_Fry_Richardson/kKQR6lFu">Lewis Fry Richardson</a>.
In 1975,<a class="footnote-ref" id="fnref:75" href="#fn:75">75</a> Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a>, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".<a class="footnote-ref" id="fnref:76" href="#fn:76">76</a><a class="footnote-ref" id="fnref:77" href="#fn:77">77</a><a class="footnote-ref" id="fnref:78" href="#fn:78">78</a><a class="footnote-ref" id="fnref:79" href="#fn:79">79</a>
In 1980, <a href="/facts/Loren_Carpenter/22scYjGc">Loren Carpenter</a> gave a presentation at the <a href="/facts/SIGGRAPH/gmRKkIlG">SIGGRAPH</a> where he introduced his software for generating and rendering fractally generated landscapes.<a class="footnote-ref" id="fnref:80" href="#fn:80">80</a>

<h2 id="definition-and-characteristics">Definition and characteristics</h2>
One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented <a href="/facts/Shape/NbaGtA8f">geometric shape</a> that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole";<a class="footnote-ref" id="fnref:81" href="#fn:81">81</a> this is generally helpful but limited. Authors disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in.<a class="footnote-ref" id="fnref:82" href="#fn:82">82</a><a class="footnote-ref" id="fnref:83" href="#fn:83">83</a><a class="footnote-ref" id="fnref:84" href="#fn:84">84</a><a class="footnote-ref" id="fnref:85" href="#fn:85">85</a><a class="footnote-ref" id="fnref:86" href="#fn:86">86</a>
One point agreed on is that fractal patterns are characterized by <a href="/facts/Fractal_dimension/pJ9tVnwn">fractal dimensions</a>, but whereas these numbers quantify <a href="/facts/Complexity/yaw0UrRj">complexity</a> (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns.<a class="footnote-ref" id="fnref:87" href="#fn:87">87</a> In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose <a href="/facts/Hausdorff%25E2%2580%2593Besicovitch_dimension/rtX9ryrp">Hausdorff–Besicovitch dimension</a> is greater than its <a href="/facts/Lebesgue_covering_dimension/xdCJLVdw">topological dimension</a>.<a class="footnote-ref" id="fnref:88" href="#fn:88">88</a> However, this requirement is not met by <a href="/facts/Space-filling_curve/b70uGLXh">space-filling curves</a> such as the <a href="/facts/Hilbert_curve/bSEWkPDn">Hilbert curve</a>.<a class="footnote-ref" id="fnref:89" href="#fn:89">89</a>
Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to <a href="/facts/Kenneth_Falconer_(mathematician)/zf1e1BIG">Falconer</a>, fractals should be only generally characterized by a gestalt of the following features;<a class="footnote-ref" id="fnref:90" href="#fn:90">90</a>

<ul><li>Self-similarity, which may include:</li></ul>
<ul><li>Exact self-similarity: identical at all scales, such as the Koch snowflake</li>
<li>Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a>'s satellites are approximations of the entire set, but not exact copies.</li>
<li>Statistical self-similarity: repeats a pattern <a href="/facts/Stochastic/uMbgL94r">stochastically</a> so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals like the well-known example of the <a href="/facts/How_Long_Is_the_Coast_of_Britain%253F_Statistical_Self-Similarity_and_Fractional_Dimension/AzckgsE5">coastline of Britain</a> for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines fractals like the Koch snowflake.<a class="footnote-ref" id="fnref:91" href="#fn:91">91</a></li>
<li>Qualitative self-similarity: as in a time series<a class="footnote-ref" id="fnref:92" href="#fn:92">92</a></li>
<li><a href="/facts/Multifractal/Fly3MCTb">Multifractal</a> scaling: characterized by more than one fractal dimension or scaling rule</li></ul>
<ul><li>Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have <a href="/facts/Emergent_properties/gLAd6wda">emergent properties</a><a class="footnote-ref" id="fnref:93" href="#fn:93">93</a> (related to the next criterion in this list).</li>
<li>Irregularity locally and globally that cannot easily be described in the language of traditional <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a> other than as the limit of a <a href="/facts/Recursion/CxSKxJoW">recursively</a> defined sequence of stages. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls";<a class="footnote-ref" id="fnref:94" href="#fn:94">94</a>see Common techniques for generating fractals.</li></ul>
As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, and is easily described in Euclidean language without a need for recursion.<a class="footnote-ref" id="fnref:95" href="#fn:95">95</a><a class="footnote-ref" id="fnref:96" href="#fn:96">96</a>

<h2 id="common-techniques-for-generating-fractals">Common techniques for generating fractals</h2>
See also: <a href="/facts/Fractal-generating_software/Qgu3C0eI">Fractal-generating software</a>

Images of fractals can be created by <a href="/facts/Fractal-generating_software/Qgu3C0eI">fractal generating programs</a>. Because of the <a href="/facts/Butterfly_effect/Iex6LFKu">butterfly effect</a>, a small change in a single variable can have an <a href="/facts/Predictability/FNSpNGQF">unpredictable</a> outcome.

<ul><li><a href="/facts/Iterated_function_systems/j56g6yEM">Iterated function systems</a> (IFS) – use fixed geometric replacement rules; may be stochastic or deterministic;<a class="footnote-ref" id="fnref:97" href="#fn:97">97</a> e.g., <a href="/facts/Koch_snowflake/JTckxQVK">Koch snowflake</a>, <a href="/facts/Cantor_set/6NUcMJvN">Cantor set</a>, Haferman carpet,<a class="footnote-ref" id="fnref:98" href="#fn:98">98</a> <a href="/facts/Sierpinski_carpet/3ogstvh2">Sierpinski carpet</a>, <a href="/facts/Sierpinski_gasket/gFmLRJxM">Sierpinski gasket</a>, <a href="/facts/Peano_curve/tFe9dQE6">Peano curve</a>, <a href="/facts/Dragon_curve/U3Kl61Bq">Harter-Heighway dragon curve</a>, <a href="/facts/T-square_(fractal)/z4ERel87">T-square</a>, <a href="/facts/Menger_sponge/xRtJcfc9">Menger sponge</a></li>
<li><a href="/facts/Strange_attractor/FAx5Db4w">Strange attractors</a> – use iterations of a map or solutions of a system of initial-value differential or difference equations that exhibit chaos (e.g., see multifractal image, or the <a href="/facts/Logistic_map/wskv7mRQ">logistic map</a>)</li>
<li><a href="/facts/L-system/SGfQ416R">L-systems</a> – use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells<a class="footnote-ref" id="fnref:99" href="#fn:99">99</a>), blood vessels, pulmonary structure,<a class="footnote-ref" id="fnref:100" href="#fn:100">100</a> etc. or <a href="/facts/Turtle_graphics/3UBFgoIE">turtle graphics</a> patterns such as <a href="/facts/Space-filling_curves/b70uGLXh">space-filling curves</a> and tilings</li>
<li>Escape-time fractals – use a <a href="/facts/Formula/BosTcKnv">formula</a> or <a href="/facts/Recurrence_relation/JolXC71M">recurrence relation</a> at each point in a space (such as the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>); usually quasi-self-similar; also known as "orbit" fractals; e.g., the <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a>, <a href="/facts/Julia_set/dmn8xIcG">Julia set</a>, <a href="/facts/Burning_Ship_fractal/zYk9tpa2">Burning Ship fractal</a>, <a href="/facts/Nova_fractal/PRtA7cDg">Nova fractal</a> and <a href="/facts/Lyapunov_fractal/B8HgogxJ">Lyapunov fractal</a>. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.</li>
<li>Random fractals – use stochastic rules; e.g., <a href="/facts/L%25C3%25A9vy_flight/0XrkLdzU">Lévy flight</a>, <a href="/facts/Percolation_theory/Bop4vG7u">percolation clusters</a>, <a href="/facts/Self-avoiding_walk/23chozBY">self avoiding walks</a>, <a href="/facts/Fractal_landscapes/JTkWfeGm">fractal landscapes</a>, trajectories of <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a> and the <a href="/facts/Brownian_tree/5b1CAhKG">Brownian tree</a> (i.e., dendritic fractals generated by modeling <a href="/facts/Diffusion-limited_aggregation/5b1CAhKG">diffusion-limited aggregation</a> or reaction-limited aggregation clusters).<a class="footnote-ref" id="fnref:101" href="#fn:101">101</a></li></ul>

<ul><li><a href="/facts/Finite_subdivision_rule/M40dDJUm">Finite subdivision rules</a> – use a recursive <a href="/facts/Topological/2EqW47cU">topological</a> algorithm for refining tilings<a class="footnote-ref" id="fnref:102" href="#fn:102">102</a> and they are similar to the process of <a href="/facts/Cell_division/n5MrunWj">cell division</a>.<a class="footnote-ref" id="fnref:103" href="#fn:103">103</a> The iterative processes used in creating the <a href="/facts/Cantor_set/6NUcMJvN">Cantor set</a> and the <a href="/facts/Sierpinski_carpet/3ogstvh2">Sierpinski carpet</a> are examples of finite subdivision rules, as is <a href="/facts/Barycentric_subdivision/hsk1bw6b">barycentric subdivision</a>.</li></ul>
<h2 id="applications">Applications</h2>
<h3>Simulated fractals</h3>

Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of physical time and space. Models may simulate theoretical fractals or natural phenomena with fractal features. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for <a href="/facts/Fractal_analysis/UlKkdapU">fractal analysis</a>. Some specific applications of fractals to technology are listed elsewhere. Images and other outputs of modelling are normally referred to as being "fractals" even if they do not have strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or display <a href="/facts/Artifact_(error)/et9onNmx">artifacts</a> which are not characteristics of true fractals.
Modeled fractals may be sounds,<a class="footnote-ref" id="fnref:104" href="#fn:104">104</a> digital images, electrochemical patterns, <a href="/facts/Circadian_rhythm/rwQh0ClJ">circadian rhythms</a>,<a class="footnote-ref" id="fnref:105" href="#fn:105">105</a> etc.
Fractal patterns have been reconstructed in physical 3-dimensional space<a class="footnote-ref" id="fnref:106" href="#fn:106">106</a>: 10  and virtually, often called "<a href="/facts/In_silico/g2Cz0ejg">in silico</a>" modeling.<a class="footnote-ref" id="fnref:107" href="#fn:107">107</a> Models of fractals are generally created using <a href="/facts/Fractal-generating_software/Qgu3C0eI">fractal-generating software</a> that implements techniques such as those outlined above.<a class="footnote-ref" id="fnref:108" href="#fn:108">108</a><a class="footnote-ref" id="fnref:109" href="#fn:109">109</a><a class="footnote-ref" id="fnref:110" href="#fn:110">110</a> As one illustration, trees, ferns, cells of the nervous system,<a class="footnote-ref" id="fnref:111" href="#fn:111">111</a> blood and lung vasculature,<a class="footnote-ref" id="fnref:112" href="#fn:112">112</a> and other branching <a href="/facts/Patterns_in_nature/handzeJh">patterns in nature</a> can be modeled on a computer by using recursive <a href="/facts/Algorithm/fnl5NmRt">algorithms</a> and <a href="/facts/L-systems/SGfQ416R">L-systems</a> techniques.<a class="footnote-ref" id="fnref:113" href="#fn:113">113</a>
The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a <a href="/facts/Frond/msza4qEa">frond</a> from a <a href="/facts/Fern/DKlstRPL">fern</a> is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects, such as coastlines and mountains. A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms.

<h3>Natural phenomena with fractal features</h3>
Further information: <a href="/facts/Patterns_in_nature/handzeJh">Patterns in nature</a>
Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees.<a class="footnote-ref" id="fnref:114" href="#fn:114">114</a> Phenomena known to have fractal features include:

<ul><li><a href="/facts/Actin_cytoskeleton/TR7qU9Jv">Actin cytoskeleton</a><a class="footnote-ref" id="fnref:115" href="#fn:115">115</a></li>
<li><a href="/facts/Algae/U1ftVKU3">Algae</a></li>
<li><a href="/facts/Animal_coloration/ULNm6v70">Animal coloration</a> patterns</li>
<li><a href="/facts/Blood_vessel/FG3GjPNW">Blood vessels</a> and <a href="/facts/Pulmonary_vessels/mBM1zMzc">pulmonary vessels</a><a class="footnote-ref" id="fnref:116" href="#fn:116">116</a></li>
<li>Brownian motion (generated by a one-dimensional <a href="/facts/Wiener_process/KPh7vYd7">Wiener process</a>).<a class="footnote-ref" id="fnref:117" href="#fn:117">117</a></li>
<li>Clouds and rainfall areas<a class="footnote-ref" id="fnref:118" href="#fn:118">118</a></li>
<li>Coastlines <a class="footnote-ref" id="fnref:119" href="#fn:119">119</a></li>
<li><a href="/facts/Impact_crater/nWntooh4">Craters</a></li>
<li>Crystals<a class="footnote-ref" id="fnref:120" href="#fn:120">120</a></li>
<li><a href="/facts/DNA/nB89Iauz">DNA</a></li>
<li>Dust grains<a class="footnote-ref" id="fnref:121" href="#fn:121">121</a></li>
<li><a href="/facts/Earthquakes/NGjIzagw">Earthquakes</a><a class="footnote-ref" id="fnref:122" href="#fn:122">122</a><a class="footnote-ref" id="fnref:123" href="#fn:123">123</a></li>
<li><a href="/facts/Fault_line/0Fw2ladx">Fault lines</a></li>
<li>Geometrical optics<a class="footnote-ref" id="fnref:124" href="#fn:124">124</a></li>
<li>Heart rates<a class="footnote-ref" id="fnref:125" href="#fn:125">125</a></li>
<li><a href="/facts/Heart_sounds/8DbSR4oq">Heart sounds</a></li>
<li><a href="/facts/Lake/YP0xxn0E">Lake</a> shorelines and areas<a class="footnote-ref" id="fnref:126" href="#fn:126">126</a><a class="footnote-ref" id="fnref:127" href="#fn:127">127</a><a class="footnote-ref" id="fnref:128" href="#fn:128">128</a></li>
<li><a href="/facts/Lightning/dLTuxz8n">Lightning</a> bolts</li>
<li>Mountain-goat horns</li>
<li><a href="/facts/Neuron/G7CyTVdH">Neurons</a></li>
<li>Polymers</li>
<li>Percolation</li>
<li><a href="/facts/Mountain/wboT5GJc">Mountain ranges</a></li>
<li><a href="/facts/Wind_wave/7IsEqwJp">Ocean waves</a><a class="footnote-ref" id="fnref:129" href="#fn:129">129</a></li>
<li>Pineapple</li>
<li><a href="/facts/Proteins/Jxmagu3E">Proteins</a><a class="footnote-ref" id="fnref:130" href="#fn:130">130</a></li>
<li><a href="/facts/Psychedelic_experience/8uJ4ce7r">Psychedelic experience</a><a class="footnote-ref" id="fnref:131" href="#fn:131">131</a></li>
<li><a href="/facts/Purkinje_cell/QjFsk7Qn">Purkinje cells</a><a class="footnote-ref" id="fnref:132" href="#fn:132">132</a></li>
<li><a href="/facts/Rings_of_Saturn/fFaoFV4T">Rings of Saturn</a><a class="footnote-ref" id="fnref:133" href="#fn:133">133</a><a class="footnote-ref" id="fnref:134" href="#fn:134">134</a></li>
<li><a href="/facts/River/GP7MSur6">River networks</a><a class="footnote-ref" id="fnref:135" href="#fn:135">135</a></li>
<li><a href="/facts/Romanesco_broccoli/01ybQnlR">Romanesco broccoli</a></li>
<li>Snowflakes<a class="footnote-ref" id="fnref:136" href="#fn:136">136</a></li>
<li>Soil pores<a class="footnote-ref" id="fnref:137" href="#fn:137">137</a></li>
<li>Surfaces in <a href="/facts/Turbulence/NpgFUhG5">turbulent</a> flows<a class="footnote-ref" id="fnref:138" href="#fn:138">138</a><a class="footnote-ref" id="fnref:139" href="#fn:139">139</a></li>
<li>Trees <a class="footnote-ref" id="fnref:140" href="#fn:140">140</a></li></ul>

<h3>Fractals in cell biology</h3>
Fractals often appear in the realm of living organisms where they arise through branching processes and other complex pattern formation. Ian Wong and co-workers have shown that migrating cells can form fractals by clustering and <a href="/facts/Branching_process/JzdGdWrQ">branching</a>.<a class="footnote-ref" id="fnref:141" href="#fn:141">141</a> <a href="/facts/Neuron/G7CyTVdH">Nerve cells</a> function through processes at the cell surface, with phenomena that are enhanced by largely increasing the surface to volume ratio. As a consequence nerve cells often are found to form into fractal patterns.<a class="footnote-ref" id="fnref:142" href="#fn:142">142</a> These processes are crucial in cell <a href="/facts/Physiology/fh25y2pQ">physiology</a> and different <a href="/facts/Pathology/U9b18LJG">pathologies</a>.<a class="footnote-ref" id="fnref:143" href="#fn:143">143</a>
Multiple subcellular structures also are found to assemble into fractals. <a href="/facts/Diego_Krapf/g1H7hN9y">Diego Krapf</a> has shown that through branching processes the <a href="/facts/Actin/Hu3qIUL9">actin</a> filaments in human cells assemble into fractal patterns.<a class="footnote-ref" id="fnref:144" href="#fn:144">144</a> Similarly Matthias Weiss showed that the <a href="/facts/Endoplasmic_reticulum/DVbLA7GE">endoplasmic reticulum</a> displays fractal features.<a class="footnote-ref" id="fnref:145" href="#fn:145">145</a> The current understanding is that fractals are ubiquitous in cell biology, from <a href="/facts/Protein/Jxmagu3E">proteins</a>, to <a href="/facts/Organelle/vhDFnGbs">organelles</a>, to whole cells.

<h3>In creative works</h3>
Further information: <a href="/facts/Fractal_art/hjnu1JP2">Fractal art</a> and <a href="/facts/Mathematics_and_art/kT01G7la">Mathematics and art</a>
Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by <a href="/facts/Jackson_Pollock/WQ6WvcYX">Jackson Pollock</a> by pouring paint directly onto horizontal canvasses.<a class="footnote-ref" id="fnref:146" href="#fn:146">146</a><a class="footnote-ref" id="fnref:147" href="#fn:147">147</a><a class="footnote-ref" id="fnref:148" href="#fn:148">148</a>
Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks.<a class="footnote-ref" id="fnref:149" href="#fn:149">149</a> Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals.<a class="footnote-ref" id="fnref:150" href="#fn:150">150</a>
<a href="/facts/Decalcomania/SXAuDKdl">Decalcomania</a>, a technique used by artists such as <a href="/facts/Max_Ernst/lwXCga3M">Max Ernst</a>, can produce fractal-like patterns.<a class="footnote-ref" id="fnref:151" href="#fn:151">151</a> It involves pressing paint between two surfaces and pulling them apart.
Cyberneticist <a href="/facts/Ron_Eglash/JL9KLA37">Ron Eglash</a> has suggested that fractal geometry and mathematics are prevalent in <a href="/facts/African_art/FhTC0IaV">African art</a>, games, <a href="/facts/Divination/b8gLkFfv">divination</a>, trade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.<a class="footnote-ref" id="fnref:152" href="#fn:152">152</a><a class="footnote-ref" id="fnref:153" href="#fn:153">153</a> <a href="/facts/Hokky_Situngkir/bytfU3xJ">Hokky Situngkir</a> also suggested the similar properties in Indonesian traditional art, <a href="/facts/Batik/gzNjRaE3">batik</a>, and <a href="/facts/Ornament_(art)/byhAkelV">ornaments</a> found in traditional houses.<a class="footnote-ref" id="fnref:154" href="#fn:154">154</a><a class="footnote-ref" id="fnref:155" href="#fn:155">155</a>
Ethnomathematician Ron Eglash has discussed the planned layout of <a href="/facts/Benin_city/rvm1QCh0">Benin city</a> using fractals as the basis, not only in the city itself and the villages but even in the rooms of houses. He commented that "When Europeans first came to Africa, they considered the architecture very disorganised and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn't even discovered yet."<a class="footnote-ref" id="fnref:156" href="#fn:156">156</a>
In a 1996 interview with <a href="/facts/Michael_Silverblatt/Mp4TN3LX">Michael Silverblatt</a>, <a href="/facts/David_Foster_Wallace/5qNGP8J7">David Foster Wallace</a> explained that the structure of the first draft of <a href="/facts/Infinite_Jest/tquM5CBg">Infinite Jest</a> he gave to his editor Michael Pietsch was inspired by fractals, specifically the <a href="/facts/Sierpinski_triangle/gFmLRJxM">Sierpinski triangle</a> (a.k.a. Sierpinski gasket), but that the edited novel is "more like a lopsided Sierpinsky Gasket".<a class="footnote-ref" id="fnref:157" href="#fn:157">157</a>
Some works by the Dutch artist <a href="/facts/M._C._Escher/6uuAGlDx">M. C. Escher</a>, such as <a href="/facts/Circle_Limit_III/LaWu4Xze">Circle Limit III</a>, contain shapes repeated to infinity that become smaller and smaller as they get near to the edges, in a pattern that would always look the same if zoomed in.
Aesthetics and Psychological Effects of Fractal Based Design:<a class="footnote-ref" id="fnref:158" href="#fn:158">158</a> Highly prevalent in nature, fractal patterns possess self-similar components that repeat at varying size scales. The perceptual experience of human-made environments can be impacted with inclusion of these natural patterns. Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on the impact of other visual judgments. Here we examine the aesthetic and perceptual experience of fractal ‘global-forest’ designs already installed in humanmade spaces and demonstrate how fractal pattern components are associated with positive psychological experiences that can be utilized to promote occupant well-being. These designs are composite fractal patterns consisting of individual fractal ‘tree-seeds’ which combine to create a ‘global fractal forest.’ The local ‘tree-seed’ patterns, global configuration of tree-seed locations, and overall resulting ‘global-forest’ patterns have fractal qualities. These designs span multiple mediums yet are all intended to lower occupant stress without detracting from the function and overall design of the space. In this series of studies, we first establish divergent relationships between various visual attributes, with pattern complexity, preference, and engagement ratings increasing with fractal complexity compared to ratings of refreshment and relaxation which stay the same or decrease with complexity. Subsequently, we determine that the local constituent fractal (‘tree-seed’) patterns contribute to the perception of the overall fractal design, and address how to balance aesthetic and psychological effects (such as individual experiences of perceived engagement and relaxation) in fractal design installations. This set of studies demonstrates that fractal preference is driven by a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment). Installations of these composite mid-high complexity ‘global-forest’ patterns consisting of ‘tree-seed’ components balance these contrasting needs, and can serve as a practical implementation of biophilic patterns in human-made environments to promote occupant well-being.

<h3>Physiological responses</h3>
Humans appear to be especially well-adapted to processing fractal patterns with <a href="/facts/Fractal_dimension/pJ9tVnwn">fractal dimension</a> between 1.3 and 1.5.<a class="footnote-ref" id="fnref:159" href="#fn:159">159</a> When humans view fractal patterns with fractal dimension between 1.3 and 1.5, this tends to reduce physiological stress.<a class="footnote-ref" id="fnref:160" href="#fn:160">160</a><a class="footnote-ref" id="fnref:161" href="#fn:161">161</a>

<h3>Applications in technology</h3>
Main article: <a href="/facts/Fractal_analysis/UlKkdapU">Fractal analysis</a>

<ul><li><a href="/facts/Fractal_antenna/7UKkLXnA">Fractal antennas</a><a class="footnote-ref" id="fnref:162" href="#fn:162">162</a></li>
<li>Fractal transistor<a class="footnote-ref" id="fnref:163" href="#fn:163">163</a></li>
<li>Fractal heat exchangers<a class="footnote-ref" id="fnref:164" href="#fn:164">164</a></li>
<li>Digital imaging</li>
<li>Architecture<a class="footnote-ref" id="fnref:165" href="#fn:165">165</a></li>
<li>Urban growth<a class="footnote-ref" id="fnref:166" href="#fn:166">166</a><a class="footnote-ref" id="fnref:167" href="#fn:167">167</a></li>
<li><a href="/facts/Categorisation/R6rWYttP">Classification</a> of <a href="/facts/Histopathology/9Eggjrg4">histopathology</a> slides</li>
<li><a href="/facts/Fractal_landscape/JTkWfeGm">Fractal landscape</a> or <a href="/facts/Coast/RBD0wxn3">Coastline</a> <a href="/facts/Complexity/yaw0UrRj">complexity</a></li>
<li>Detecting 'life as we don't know it' by fractal analysis<a class="footnote-ref" id="fnref:168" href="#fn:168">168</a></li>
<li>Enzymes (<a href="/facts/Michaelis%25E2%2580%2593Menten_kinetics/lYFykW3p">Michaelis–Menten kinetics</a>)</li>
<li><a href="/facts/Algorithmic_composition/dGeVh6GP">Generation of new music</a></li>
<li><a href="/facts/Signal_(information_theory)/41YqUJ8c">Signal</a> and <a href="/facts/Fractal_compression/d2qUg7sc">image compression</a></li>
<li>Creation of digital photographic enlargements</li>
<li><a href="/facts/Fractal_in_soil_mechanics/hALDG5FV">Fractal in soil mechanics</a></li>
<li><a href="/facts/Game_Design/VayxBcbN">Computer and video game design</a></li>
<li><a href="/facts/Computer_Graphics/lapBE8wv">Computer Graphics</a></li>
<li><a href="/facts/Life/DNw7NUId">Organic</a> environments</li>
<li><a href="/facts/Procedural_generation/I94vq0AQ">Procedural generation</a></li>
<li><a href="/facts/Fractography/oo2sEQxH">Fractography</a> and <a href="/facts/Fracture_mechanics/f9QogJzF">fracture mechanics</a></li>
<li><a href="/facts/SAXS/Q4tvaRn7">Small angle scattering theory of fractally rough systems</a></li>
<li><a href="/facts/T-shirt/qj59PYsF">T-shirts</a> and other fashion</li>
<li>Generation of patterns for camouflage, such as <a href="/facts/MARPAT/2F77nryD">MARPAT</a></li>
<li><a href="/facts/Digital_sundial/gr2AddNf">Digital sundial</a></li>
<li>Technical analysis of price series</li>
<li><a href="/facts/Fractal_dimension_on_networks/krTWIm0h">Fractals in networks</a></li>
<li>Medicine<a class="footnote-ref" id="fnref:169" href="#fn:169">169</a></li>
<li><a href="/facts/Neuroscience/o5RV8Cas">Neuroscience</a><a class="footnote-ref" id="fnref:170" href="#fn:170">170</a><a class="footnote-ref" id="fnref:171" href="#fn:171">171</a></li>
<li><a href="/facts/Diagnostic_Imaging/pL6JyJEp">Diagnostic Imaging</a><a class="footnote-ref" id="fnref:172" href="#fn:172">172</a></li>
<li><a href="/facts/Pathology/U9b18LJG">Pathology</a><a class="footnote-ref" id="fnref:173" href="#fn:173">173</a><a class="footnote-ref" id="fnref:174" href="#fn:174">174</a></li>
<li>Geology<a class="footnote-ref" id="fnref:175" href="#fn:175">175</a></li>
<li><a href="/facts/Geography/TRd1Toyo">Geography</a><a class="footnote-ref" id="fnref:176" href="#fn:176">176</a></li>
<li><a href="/facts/Archaeology/K3Q5P38R">Archaeology</a><a class="footnote-ref" id="fnref:177" href="#fn:177">177</a><a class="footnote-ref" id="fnref:178" href="#fn:178">178</a></li>
<li><a href="/facts/Soil_mechanics/WfVDFMAW">Soil mechanics</a><a class="footnote-ref" id="fnref:179" href="#fn:179">179</a></li>
<li><a href="/facts/Seismology/Lony1LYX">Seismology</a><a class="footnote-ref" id="fnref:180" href="#fn:180">180</a></li>
<li><a href="/facts/Search_and_rescue/pdLTUPgl">Search and rescue</a><a class="footnote-ref" id="fnref:181" href="#fn:181">181</a></li>
<li><a href="/facts/Morton_order/0QMjw8nF">Morton order</a> space filling curves for <a href="/facts/GPU/PTK1RQVp">GPU</a> <a href="/facts/Cache_coherency/MhUD576q">cache coherency</a> in <a href="/facts/Texture_mapping/9gDTsuXN">texture mapping</a>,<a class="footnote-ref" id="fnref:182" href="#fn:182">182</a><a class="footnote-ref" id="fnref:183" href="#fn:183">183</a><a class="footnote-ref" id="fnref:184" href="#fn:184">184</a> <a href="/facts/Rasterisation/QXfz5t7E">rasterisation</a><a class="footnote-ref" id="fnref:185" href="#fn:185">185</a><a class="footnote-ref" id="fnref:186" href="#fn:186">186</a> and indexing of turbulence data.<a class="footnote-ref" id="fnref:187" href="#fn:187">187</a><a class="footnote-ref" id="fnref:188" href="#fn:188">188</a></li></ul>

<h2 id="see-also">See also</h2>
<ul>
<li>Mathematics portal</li><li>Systems science portal</li></ul>

<ul><li><a href="/facts/Banach_fixed_point_theorem/TRygeoCP">Banach fixed point theorem</a> – Theorem about metric spacesPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Bifurcation_theory/sLsN54Rn">Bifurcation theory</a> – Study of sudden qualitative behavior changes caused by small parameter changes</li>
<li><a href="/facts/Box_counting/GtNjNeBe">Box counting</a> – Fractal analysis technique</li>
<li><a href="/facts/Cymatics/XnTFdX8U">Cymatics</a> – Creation of visible patterns on a vibrated plate</li>
<li><a href="/facts/Determinism/NhvNlAbe">Determinism</a> – Philosophical view that events are determined by prior events</li>
<li><a href="/facts/Diamond-square_algorithm/k3hhV2Vm">Diamond-square algorithm</a> – Method for generating heightmaps for computer graphics</li>
<li><a href="/facts/Droste_effect/I4rgh6ls">Droste effect</a> – Recursive visual effect</li>
<li><a href="/facts/Feigenbaum_function/FrKDAMvj">Feigenbaum function</a> – Concept in dynamical systems</li>
<li><a href="/facts/Form_constant/PANwj7g4">Form constant</a> – Recurringly observed geometric pattern</li>
<li><a href="/facts/Fractal_cosmology/xwqkrfjp">Fractal cosmology</a> – Absolute Infinite Everything</li>
<li><a href="/facts/Fractal_derivative/eF6LbrrQ">Fractal derivative</a> – Generalization of derivative to fractals</li>
<li><a href="/facts/Fractalgrid/c7CalmPW">Fractalgrid</a> – Concept in electric power distribution</li>
<li><a href="/facts/Fractal_sequence/60fgawsN">Fractal sequence</a> – Sequence that contains itself as a subsequence</li>
<li><a href="/facts/Fractal_string/tmd99a7A">Fractal string</a> – Open subset of the real–number line</li>
<li><a href="/facts/Fracton/8wWmeIMX">Fracton</a> – Synonym of phonon</li>
<li><a href="/facts/Graftal/SGfQ416R">Graftal</a> – Rewriting system and type of formal grammarPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Greeble/adqplVtc">Greeble</a> – Detailing added to make models appear more complex</li>
<li><a href="/facts/H_tree/HcHp6u0X">H tree</a> – Right-angled fractal canopy</li>
<li><a href="/facts/Infinite_regress/Y3rk8wQe">Infinite regress</a> – Philosophical problem</li>
<li><a href="/facts/Lacunarity/2xYyCCM3">Lacunarity</a> – Term in geometry and fractal analysis</li>
<li><a href="/facts/List_of_fractals_by_Hausdorff_dimension/THnqUPeF">List of fractals by Hausdorff dimension</a></li>
<li><a href="/facts/Mandelbulb/pBK47wKj">Mandelbulb</a> – Three-dimensional fractal</li>
<li><a href="/facts/Mandelbox/1yFdFXen">Mandelbox</a> – Fractal with a boxlike shape</li>
<li><a href="/facts/Macrocosm_and_microcosm/vzdlMsfW">Macrocosm and microcosm</a> – Analogy between man and cosmosPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Matryoshka_doll/ASCqmLEC">Matryoshka doll</a> – Russian nested wooden toy created in 1890</li>
<li><a href="/facts/Menger_Sponge/xRtJcfc9">Menger Sponge</a> – Three-dimensional fractalPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Multifractal_system/Fly3MCTb">Multifractal system</a> – System with multiple fractal dimensions</li>
<li><a href="/facts/Newton_fractal/PRtA7cDg">Newton fractal</a> – Boundary set in the complex plane</li>
<li><a href="/facts/Percolation/CRP2RCt6">Percolation</a> – Filtration of fluids through porous materials</li>
<li><a href="/facts/Power_law/DhAjYwwz">Power law</a> – Functional relationship between two quantities</li>
<li><a href="/facts/List_of_important_publications_in_mathematics/P9LM1RYR">Publications in fractal geometry</a></li>
<li><a href="/facts/Random_walk/08v0jmfv">Random walk</a> – Process forming a path from many random steps</li>
<li><a href="/facts/Self-reference/C6suy6u1">Self-reference</a> – Sentence, idea or formula that refers to itself</li>
<li><a href="/facts/Self-similarity/9Nvo7edY">Self-similarity</a> – Whole of an object being mathematically similar to part of itself</li>
<li><a href="/facts/Systems_theory/C56Ck7DQ">Systems theory</a> – Interdisciplinary study of systems</li>
<li><a href="/facts/Strange_loop/RhbBHH3X">Strange loop</a> – Cyclic structure that goes through several levels in a hierarchical system</li>
<li><a href="/facts/Turbulence/NpgFUhG5">Turbulence</a> – Motion characterized by chaotic changes in pressure and flow velocity</li>
<li><a href="/facts/Wiener_process/KPh7vYd7">Wiener process</a> – Stochastic process generalizing Brownian motion</li></ul>

<h2 id="notes">Notes</h2>

<h2 id="further-reading">Further reading</h2>
<ul><li>Stanley, Eugene H, Ostrowsky, N. (editors); On Growth and Fractal Form Fractal and Non-Fractal Patterns in Physics, Martinus Nijhoff Publisher, 1986. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-89838-850-3</li>
<li>Barnsley, Michael F.; and Rising, Hawley; Fractals Everywhere. Boston: Academic Press Professional, 1993. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-079061-0</li>
<li>Duarte, German A.; Fractal Narrative. About the Relationship Between Geometries and Technology and Its Impact on Narrative Spaces. Bielefeld: Transcript, 2014. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-8376-2829-6</li>
<li>Falconer, Kenneth; Techniques in Fractal Geometry. John Wiley and Sons, 1997. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-92287-0</li>
<li>Jürgens, Hartmut; <a href="/facts/Heinz-Otto_Peitgen/5NkEd1Gp">Peitgen, Heinz-Otto</a>; and Saupe, Dietmar; Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-97903-4</li>
<li><a href="/facts/Benoit_Mandelbrot/fa8tKP93">Mandelbrot, Benoit B.</a>; <a href="/facts/The_Fractal_Geometry_of_Nature/FKlWQtdw">The Fractal Geometry of Nature</a>. New York: W. H. Freeman and Co., 1982. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7167-1186-9</li>
<li>Peitgen, Heinz-Otto; and Saupe, Dietmar; eds.; The Science of Fractal Images. New York: Springer-Verlag, 1988. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-96608-0</li>
<li><a href="/facts/Clifford_A._Pickover/tsRe4UWE">Pickover, Clifford A.</a>; ed.; Chaos and Fractals: A Computer Graphical Journey – A 10 Year Compilation of Advanced Research. Elsevier, 1998. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-444-50002-2</li>
<li>Jones, Jesse; Fractals for the Macintosh, Waite Group Press, Corte Madera, CA, 1993. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-878739-46-8.</li>
<li>Lauwerier, Hans; Fractals: Endlessly Repeated Geometrical Figures, Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-08551-X, cloth. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-02445-6 paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.</li>
<li>Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-850839-7.</li>
<li>Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric; <a href="http://www.fractalexplorer.com">Exploring Fractals on the Macintosh</a>, Addison Wesley, 1995. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-62630-6</li>
<li>Lesmoir-Gordon, Nigel; The Colours of Infinity: The Beauty, The Power and the Sense of Fractals. 2004. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-904555-05-5 (The book comes with a related DVD of the <a href="/facts/Arthur_C._Clarke/2XJ6UGDB">Arthur C. Clarke</a> documentary introduction to the fractal concept and the <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a>.)</li>
<li>Liu, Huajie; Fractal Art, Changsha: Hunan Science and Technology Press, 1997, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9787535722348.</li>
<li>Gouyet, Jean-François; Physics and Fractal Structures (Foreword by B. Mandelbrot); Masson, 1996. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 2-225-85130-1, and New York: Springer-Verlag, 1996. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94153-0. Out-of-print. Available in PDF version at.<a href="http://www.jfgouyet.fr/fractal/fractauk.html">"Physics and Fractal Structures"</a> (in French). Jfgouyet.fr. Retrieved October 17, 2010.</li>
<li>Falconer, Kenneth (2013). Fractals, A Very Short Introduction. Oxford University Press.</li></ul>
<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Fractal.

Wikibooks has a book on the topic of: Fractals

<ul><li><a href="http://webarchive.loc.gov/all/20011116035117/http://dmoz.org/science/math/chaos_and_fractals/">Fractals</a> at the <a href="/facts/Library_of_Congress/TfB6dEN5">Library of Congress</a> Web Archives (archived November 16, 2001)</li>
<li>"<a href="https://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html">Hunting the Hidden Dimension</a>", <a href="/facts/PBS/E6q9udH8">PBS</a> <a href="/facts/Nova_(American_TV_series)/yRj6GvQB">NOVA</a>, first aired August 24, 2011</li>
<li><a href="https://www.ted.com/talks/benoit_mandelbrot_fractals_and_the_art_of_roughness">Benoit Mandelbrot: Fractals and the Art of Roughness</a> (<a href="https://web.archive.org/web/20140217132541/http://www.ted.com/talks/benoit_mandelbrot_fractals_the_art_of_roughness.html">Archived</a> February 17, 2014, at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>), <a href="/facts/TED_(conference)/Sc3JxnA1">TED</a>, February 2010</li>
<li><a href="http://adsabs.harvard.edu/abs/2007PrGeo..22..451Y">Equations of self-similar fractal measure based on the fractional-order calculus</a>（2007）</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons. xxv. ISBN 978-0-470-84862-3. <a href="978-0-470-84862-3" target="_blank">978-0-470-84862-3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Briggs, John (1992). Fractals:The Patterns of Chaos. London: Thames and Hudson. p. 148. ISBN 978-0-500-27693-8. <a href="978-0-500-27693-8" target="_blank">978-0-500-27693-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Vicsek, Tamás (1992). Fractal growth phenomena. Singapore/New Jersey: World Scientific. pp. 31, 139–146. ISBN 978-981-02-0668-0. <a href="978-981-02-0668-0" target="_blank">978-981-02-0668-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Gouyet, Jean-François (1996). Physics and fractal structures. Paris/New York: Masson Springer. ISBN 978-0-387-94153-0. <a href="978-0-387-94153-0" target="_blank">978-0-387-94153-0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Mandelbrot, Benoît B. (2004). Fractals and Chaos. Berlin: Springer. p. 38. ISBN 978-0-387-20158-0. A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension <a href="978-0-387-20158-0" target="_blank">978-0-387-20158-0</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Vicsek, Tamás (1992). Fractal growth phenomena. Singapore/New Jersey: World Scientific. pp. 31, 139–146. ISBN 978-981-02-0668-0. <a href="978-981-02-0668-0" target="_blank">978-981-02-0668-0</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Mandelbrot, Benoît B. (2004). Fractals and Chaos. Berlin: Springer. p. 38. ISBN 978-0-387-20158-0. A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension <a href="978-0-387-20158-0" target="_blank">978-0-387-20158-0</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Segal, S. L. (June 1978). "Riemann's example of a continuous 'nondifferentiable' function continued". The Mathematical Intelligencer. 1 (2): 81–82. doi:10.1007/BF03023065. S2CID 120037858. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8. <a href="978-0-8133-4153-8" target="_blank">978-0-8133-4153-8</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012. <a href="https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html" target="_blank">https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Mandelbrot, Benoit (July 8, 2013). "24/7 Lecture on Fractals". 2006 Ig Nobel Awards. Improbable Research. Archived from the original on December 11, 2021. <a href="https://www.youtube.com/watch?v=5e7HB5Oze4g#t=70" target="_blank">https://www.youtube.com/watch?v=5e7HB5Oze4g#t=70</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Mandelbrot, B. B.: The Fractal Geometry of Nature. W. H. Freeman and Company, New York (1982); p. 15. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Edgar, Gerald (2007). Measure, Topology, and Fractal Geometry. Springer Science & Business Media. p. 7. ISBN 978-0-387-74749-1. <a href="978-0-387-74749-1" target="_blank">978-0-387-74749-1</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons. xxv. ISBN 978-0-470-84862-3. <a href="978-0-470-84862-3" target="_blank">978-0-470-84862-3</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Briggs, John (1992). Fractals:The Patterns of Chaos. London: Thames and Hudson. p. 148. ISBN 978-0-500-27693-8. <a href="978-0-500-27693-8" target="_blank">978-0-500-27693-8</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Gouyet, Jean-François (1996). Physics and fractal structures. Paris/New York: Masson Springer. ISBN 978-0-387-94153-0. <a href="978-0-387-94153-0" target="_blank">978-0-387-94153-0</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Vicsek, Tamás (1992). Fractal growth phenomena. Singapore/New Jersey: World Scientific. pp. 31, 139–146. ISBN 978-981-02-0668-0. <a href="978-981-02-0668-0" target="_blank">978-981-02-0668-0</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Peters, Edgar (1996). Chaos and order in the capital markets : a new view of cycles, prices, and market volatility. New York: Wiley. ISBN 978-0-471-13938-6. <a href="978-0-471-13938-6" target="_blank">978-0-471-13938-6</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Brothers, Harlan J. (2007). "Structural Scaling in Bach's Cello Suite No. 3". Fractals. 15 (1): 89–95. doi:10.1142/S0218348X0700337X. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Tan, Can Ozan; Cohen, Michael A.; Eckberg, Dwain L.; Taylor, J. Andrew (2009). "Fractal properties of human heart period variability: Physiological and methodological implications". The Journal of Physiology. 587 (15): 3929–41. doi:10.1113/jphysiol.2009.169219. PMC 2746620. PMID 19528254. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2746620" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2746620</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Liu, Jing Z.; Zhang, Lu D.; Yue, Guang H. (2003). "Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging". Biophysical Journal. 85 (6): 4041–4046. Bibcode:2003BpJ....85.4041L. doi:10.1016/S0006-3495(03)74817-6. PMC 1303704. PMID 14645092. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1303704" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1303704</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Karperien, Audrey L.; Jelinek, Herbert F.; Buchan, Alastair M. (2008). "Box-Counting Analysis of Microglia Form in Schizophrenia, Alzheimer's Disease and Affective Disorder". Fractals. 16 (2): 103. doi:10.1142/S0218348X08003880. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
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<li id="fn:30">Hu, Shougeng; Cheng, Qiuming; Wang, Le; Xie, Shuyun (2012). "Multifractal characterization of urban residential land price in space and time". Applied Geography. 34: 161–170. Bibcode:2012AppGe..34..161H. doi:10.1016/j.apgeog.2011.10.016. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Karperien, Audrey; Jelinek, Herbert F.; Leandro, Jorge de Jesus Gomes; Soares, João V. B.; Cesar Jr, Roberto M.; Luckie, Alan (2008). "Automated detection of proliferative retinopathy in clinical practice". Clinical Ophthalmology. 2 (1): 109–122. doi:10.2147/OPTH.S1579. PMC 2698675. PMID 19668394. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2698675" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2698675</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Losa, Gabriele A.; Nonnenmacher, Theo F. (2005). Fractals in biology and medicine. Springer. ISBN 978-3-7643-7172-2. <a href="978-3-7643-7172-2" target="_blank">978-3-7643-7172-2</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
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<li id="fn:34">Wallace, David Foster (August 4, 2006). "Bookworm on KCRW". Kcrw.com. Archived from the original on November 11, 2010. Retrieved October 17, 2010. <a href="https://web.archive.org/web/20101111033857/http://www.kcrw.com/etc/programs/bw/bw960411david_foster_wallace" target="_blank">https://web.archive.org/web/20101111033857/http://www.kcrw.com/etc/programs/bw/bw960411david_foster_wallace</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Eglash, Ron (1999). "African Fractals: Modern Computing and Indigenous Design". New Brunswick: Rutgers University Press. Archived from the original on January 3, 2018. Retrieved October 17, 2010. <a href="https://web.archive.org/web/20180103005701/http://homepages.rpi.edu/~eglash/eglash.dir/afractal/afbook.htm" target="_blank">https://web.archive.org/web/20180103005701/http://homepages.rpi.edu/~eglash/eglash.dir/afractal/afbook.htm</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">Ostwald, Michael J., and Vaughan, Josephine (2016) The Fractal Dimension of Architecture Birhauser, Basel. doi:10.1007/978-3-319-32426-5. <a href="/wiki/The_Fractal_Dimension_of_Architecture" target="_blank">/wiki/The_Fractal_Dimension_of_Architecture</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Baranger, Michael. "Chaos, Complexity, and Entropy: A physics talk for non-physicists" (PDF). <a href="http://necsi.edu/projects/baranger/cce.pdf" target="_blank">http://necsi.edu/projects/baranger/cce.pdf</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Benoît Mandelbrot, Objets fractals, 1975, p. 4 <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Albers, Donald J.; Alexanderson, Gerald L. (2008). "Benoît Mandelbrot: In his own words". Mathematical people : profiles and interviews. Wellesley, MA: AK Peters. p. 214. ISBN 978-1-56881-340-0. <a href="978-1-56881-340-0" target="_blank">978-1-56881-340-0</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">"fractal". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.) <a href="https://www.oed.com/view/Entry/74094" target="_blank">https://www.oed.com/view/Entry/74094</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons. xxv. ISBN 978-0-470-84862-3. <a href="978-0-470-84862-3" target="_blank">978-0-470-84862-3</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
<li id="fn:44">Albers, Donald J.; Alexanderson, Gerald L. (2008). "Benoît Mandelbrot: In his own words". Mathematical people : profiles and interviews. Wellesley, MA: AK Peters. p. 214. ISBN 978-1-56881-340-0. <a href="978-1-56881-340-0" target="_blank">978-1-56881-340-0</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">Vicsek, Tamás (1992). Fractal growth phenomena. Singapore/New Jersey: World Scientific. pp. 31, 139–146. ISBN 978-981-02-0668-0. <a href="978-981-02-0668-0" target="_blank">978-981-02-0668-0</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
<li id="fn:47">Gordon, Nigel (2000). Introducing fractal geometry. Duxford: Icon. p. 71. ISBN 978-1-84046-123-7. <a href="978-1-84046-123-7" target="_blank">978-1-84046-123-7</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></li>
<li id="fn:48">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></li>
<li id="fn:49">Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8. <a href="978-0-8133-4153-8" target="_blank">978-0-8133-4153-8</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></li>
<li id="fn:50">Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012. <a href="https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html" target="_blank">https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></li>
<li id="fn:51">Eglash, Ron (1999). African Fractals Modern Computing and Indigenous Design. Rutgers University Press. ISBN 978-0-8135-2613-3. <a href="978-0-8135-2613-3" target="_blank">978-0-8135-2613-3</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></li>
<li id="fn:52">Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling. p. 310. ISBN 978-1-4027-5796-9. <a href="978-1-4027-5796-9" target="_blank">978-1-4027-5796-9</a> <a href="#fnref:52" class="footnote-back-ref">↩</a></li>
<li id="fn:53">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:53" class="footnote-back-ref">↩</a></li>
<li id="fn:54">Gordon, Nigel (2000). Introducing fractal geometry. Duxford: Icon. p. 71. ISBN 978-1-84046-123-7. <a href="978-1-84046-123-7" target="_blank">978-1-84046-123-7</a> <a href="#fnref:54" class="footnote-back-ref">↩</a></li>
<li id="fn:55">Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8. <a href="978-0-8133-4153-8" target="_blank">978-0-8133-4153-8</a> <a href="#fnref:55" class="footnote-back-ref">↩</a></li>
<li id="fn:56">Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012. <a href="https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html" target="_blank">https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html</a> <a href="#fnref:56" class="footnote-back-ref">↩</a></li>
<li id="fn:57">Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8. <a href="978-0-8133-4153-8" target="_blank">978-0-8133-4153-8</a> <a href="#fnref:57" class="footnote-back-ref">↩</a></li>
<li id="fn:58">Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012. <a href="https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html" target="_blank">https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html</a> <a href="#fnref:58" class="footnote-back-ref">↩</a></li>
<li id="fn:59">"Fractal Geometry". www-history.mcs.st-and.ac.uk. Retrieved April 11, 2017. <a href="http://www-history.mcs.st-and.ac.uk/HistTopics/fractals.html" target="_blank">http://www-history.mcs.st-and.ac.uk/HistTopics/fractals.html</a> <a href="#fnref:59" class="footnote-back-ref">↩</a></li>
<li id="fn:60">Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012. <a href="https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html" target="_blank">https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html</a> <a href="#fnref:60" class="footnote-back-ref">↩</a></li>
<li id="fn:61">Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8. <a href="978-0-8133-4153-8" target="_blank">978-0-8133-4153-8</a> <a href="#fnref:61" class="footnote-back-ref">↩</a></li>
<li id="fn:62">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:62" class="footnote-back-ref">↩</a></li>
<li id="fn:63">Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8. <a href="978-0-8133-4153-8" target="_blank">978-0-8133-4153-8</a> <a href="#fnref:63" class="footnote-back-ref">↩</a></li>
<li id="fn:64">Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012. <a href="https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html" target="_blank">https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html</a> <a href="#fnref:64" class="footnote-back-ref">↩</a></li>
<li id="fn:65">Vicsek, Tamás (1992). Fractal growth phenomena. Singapore/New Jersey: World Scientific. pp. 31, 139–146. ISBN 978-981-02-0668-0. <a href="978-981-02-0668-0" target="_blank">978-981-02-0668-0</a> <a href="#fnref:65" class="footnote-back-ref">↩</a></li>
<li id="fn:66">Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8. <a href="978-0-8133-4153-8" target="_blank">978-0-8133-4153-8</a> <a href="#fnref:66" class="footnote-back-ref">↩</a></li>
<li id="fn:67">Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012. <a href="https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html" target="_blank">https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html</a> <a href="#fnref:67" class="footnote-back-ref">↩</a></li>
<li id="fn:68">Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012. <a href="https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html" target="_blank">https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html</a> <a href="#fnref:68" class="footnote-back-ref">↩</a></li>
<li id="fn:69">The original paper, Lévy, Paul (1938). "Les Courbes planes ou gauches et les surfaces composées de parties semblables au tout". Journal de l'École Polytechnique: 227–247, 249–291., is translated in Edgar, pages 181–239. <a href="#fnref:69" class="footnote-back-ref">↩</a></li>
<li id="fn:70">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:70" class="footnote-back-ref">↩</a></li>
<li id="fn:71">Gordon, Nigel (2000). Introducing fractal geometry. Duxford: Icon. p. 71. ISBN 978-1-84046-123-7. <a href="978-1-84046-123-7" target="_blank">978-1-84046-123-7</a> <a href="#fnref:71" class="footnote-back-ref">↩</a></li>
<li id="fn:72">Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012. <a href="https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html" target="_blank">https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html</a> <a href="#fnref:72" class="footnote-back-ref">↩</a></li>
<li id="fn:73">Mandelbrot, B. (1967). "How Long Is the Coast of Britain?". Science. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158. S2CID 15662830. Archived from the original on October 19, 2021. Retrieved October 31, 2020. <a href="https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473" target="_blank">https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473</a> <a href="#fnref:73" class="footnote-back-ref">↩</a></li>
<li id="fn:74">Batty, Michael (April 4, 1985). "Fractals – Geometry Between Dimensions". New Scientist. 105 (1450): 31. <a href="#fnref:74" class="footnote-back-ref">↩</a></li>
<li id="fn:75">Albers, Donald J.; Alexanderson, Gerald L. (2008). "Benoît Mandelbrot: In his own words". Mathematical people : profiles and interviews. Wellesley, MA: AK Peters. p. 214. ISBN 978-1-56881-340-0. <a href="978-1-56881-340-0" target="_blank">978-1-56881-340-0</a> <a href="#fnref:75" class="footnote-back-ref">↩</a></li>
<li id="fn:76">Russ, John C. (1994). Fractal surfaces. Vol. 1. Springer. p. 1. ISBN 978-0-306-44702-0. Retrieved February 5, 2011. <a href="978-0-306-44702-0" target="_blank">978-0-306-44702-0</a> <a href="#fnref:76" class="footnote-back-ref">↩</a></li>
<li id="fn:77">Gordon, Nigel (2000). Introducing fractal geometry. Duxford: Icon. p. 71. ISBN 978-1-84046-123-7. <a href="978-1-84046-123-7" target="_blank">978-1-84046-123-7</a> <a href="#fnref:77" class="footnote-back-ref">↩</a></li>
<li id="fn:78">Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8. <a href="978-0-8133-4153-8" target="_blank">978-0-8133-4153-8</a> <a href="#fnref:78" class="footnote-back-ref">↩</a></li>
<li id="fn:79">Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling. p. 310. ISBN 978-1-4027-5796-9. <a href="978-1-4027-5796-9" target="_blank">978-1-4027-5796-9</a> <a href="#fnref:79" class="footnote-back-ref">↩</a></li>
<li id="fn:80">"Vol Libre, an amazing CG film from 1980". kottke.org. July 29, 2009. Retrieved February 12, 2023. <a href="https://kottke.org/09/07/vol-libre-an-amazing-cg-film-from-1980" target="_blank">https://kottke.org/09/07/vol-libre-an-amazing-cg-film-from-1980</a> <a href="#fnref:80" class="footnote-back-ref">↩</a></li>
<li id="fn:81">Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:81" class="footnote-back-ref">↩</a></li>
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<li id="fn:84">Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons. xxv. ISBN 978-0-470-84862-3. <a href="978-0-470-84862-3" target="_blank">978-0-470-84862-3</a> <a href="#fnref:84" class="footnote-back-ref">↩</a></li>
<li id="fn:85">Vicsek, Tamás (1992). Fractal growth phenomena. Singapore/New Jersey: World Scientific. pp. 31, 139–146. ISBN 978-981-02-0668-0. <a href="978-981-02-0668-0" target="_blank">978-981-02-0668-0</a> <a href="#fnref:85" class="footnote-back-ref">↩</a></li>
<li id="fn:86">Edgar, Gerald (2008). Measure, topology, and fractal geometry. New York: Springer-Verlag. p. 1. ISBN 978-0-387-74748-4. <a href="978-0-387-74748-4" target="_blank">978-0-387-74748-4</a> <a href="#fnref:86" class="footnote-back-ref">↩</a></li>
<li id="fn:87">Karperien, Audrey (2004). Defining microglial morphology: Form, Function, and Fractal Dimension. Charles Sturt University. doi:10.13140/2.1.2815.9048. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:87" class="footnote-back-ref">↩</a></li>
<li id="fn:88">Albers, Donald J.; Alexanderson, Gerald L. (2008). "Benoît Mandelbrot: In his own words". Mathematical people : profiles and interviews. Wellesley, MA: AK Peters. p. 214. ISBN 978-1-56881-340-0. <a href="978-1-56881-340-0" target="_blank">978-1-56881-340-0</a> <a href="#fnref:88" class="footnote-back-ref">↩</a></li>
<li id="fn:89">The Hilbert curve map is not a homeomorphism, so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert map in R2 are both 2. Note, however, that the topological dimension of the graph of the Hilbert map (a set in R3) is 1. <a href="/wiki/Homeomorphism" target="_blank">/wiki/Homeomorphism</a> <a href="#fnref:89" class="footnote-back-ref">↩</a></li>
<li id="fn:90">Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons. xxv. ISBN 978-0-470-84862-3. <a href="978-0-470-84862-3" target="_blank">978-0-470-84862-3</a> <a href="#fnref:90" class="footnote-back-ref">↩</a></li>
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<li id="fn:92">Peters, Edgar (1996). Chaos and order in the capital markets : a new view of cycles, prices, and market volatility. New York: Wiley. ISBN 978-0-471-13938-6. <a href="978-0-471-13938-6" target="_blank">978-0-471-13938-6</a> <a href="#fnref:92" class="footnote-back-ref">↩</a></li>
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<li id="fn:94">Mandelbrot, Benoît B. (2004). Fractals and Chaos. Berlin: Springer. p. 38. ISBN 978-0-387-20158-0. A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension <a href="978-0-387-20158-0" target="_blank">978-0-387-20158-0</a> <a href="#fnref:94" class="footnote-back-ref">↩</a></li>
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</ol>

Fractal open-in-new

Fractal