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Fractal analysis
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<h2 id="underlying-principles">Underlying principles</h2> <p>Fractals have <a href="/facts/Fractal_dimension/pJ9tVnwn">fractional dimensions</a>, which are a measure of <a href="/facts/Complexity/KjgY905l">complexity</a> that indicates the degree to which the objects fill the available space.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> The fractal dimension measures the change in "size" of a fractal set with the changing observational <a href="/facts/Spatial_scale/Q96HPqcY">scale</a>, and is not limited by <a href="/facts/Integer/PUwwrawx">integer</a> values.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> This is possible given that a smaller section of the fractal resembles the entirety, showing the same <a href="/facts/Statistics/DNPsKGYU">statistical</a> properties at different scales.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> This characteristic is termed <i><a href="/facts/Scale_invariance/S96sxg2R">scale invariance</a></i>, and can be further categorized as <i><a href="/facts/Self-similarity/9Nvo7edY">self-similarity</a></i> or <i><a href="/facts/Self-affinity/9Nvo7edY">self-affinity</a></i>, the latter scaled <a href="/facts/Anisotropy/YMqTQWVB">anisotropically</a> (depending on the direction).<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> Whether the view of the fractal is expanding or contracting, the structure remains the same and appears equivalently complex.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> Fractal analysis uses these underlying properties to help in the understanding and characterization of complex systems. It is also possible to expand the use of fractals to the lack of a single characteristic time scale, or pattern.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p><p><i>Further information on the Origins: <a href="/facts/Fractal/sXj38JeN">Fractal Geometry</a></i> </p> <h2 id="types-of-fractal-analysis">Types of fractal analysis</h2> <p>There are various types of fractal analysis, including <a href="/facts/Box_counting/GtNjNeBe">box counting</a>, <a href="/facts/Lacunarity/2xYyCCM3">lacunarity analysis</a>, mass methods, and <a href="/facts/Multifractal_analysis/Fly3MCTb">multifractal analysis</a>.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a><a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a><a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> A common feature of all types of fractal analysis is the need for <a href="/facts/Pattern/mzD7uLQa">benchmark patterns</a> against which to assess outputs.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> These can be acquired with various types of <a href="/facts/Fractal_generating_software/Qgu3C0eI">fractal generating software</a> capable of generating benchmark patterns suitable for this purpose, which generally differ from software designed to render <a href="/facts/Fractal_art/hjnu1JP2">fractal art</a>. Other types include <a href="/facts/Detrended_fluctuation_analysis/LcLojBnJ">detrended fluctuation analysis</a> and the Hurst absolute value method, which estimate the <a href="/facts/Hurst_exponent/kyxonWEh">hurst exponent</a>.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> </p> <h2 id="applications">Applications</h2> <h3>Ecology and evolution</h3> <p>Unlike theoretical <a href="/facts/Fractal_Curves/jjanzwda">fractal curves</a> which can be easily measured and the underlying <a href="/facts/Property_(mathematics)/CsKlqNSF">mathematical properties</a> calculated; <a href="/facts/Nature/CRC3KQUg">natural</a> systems are sources of heterogeneity and generate complex space-time structures that may only demonstrate partial <a href="/facts/Self-similarity/9Nvo7edY">self-similarity</a>.<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a><a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a> Using fractal analysis, it is possible to analyze and recognize when features of complex <a href="/facts/Ecology/YE8upuJU">ecological</a> systems are altered since fractals are able to characterize the natural complexity in such systems.<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> Thus, fractal analysis can help to quantify patterns in nature and to identify deviations from these natural sequences. It helps to improve our overall understanding of <a href="/facts/Ecosystem/kVE8mSmR">ecosystems</a> and to reveal some of the underlying structural mechanisms of nature.<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a><a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a><a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a> For example, it was found that the structure of an individual tree’s <a href="/facts/Xylem/6lovdKuF">xylem</a> follows the same architecture as the spatial distribution of the trees in the forest, and that the distribution of the trees in the forest shared the same underlying fractal structure as the branches, scaling identically to the point of being able to use the pattern of the trees’ branches mathematically to determine the structure of the forest stand.<a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a><a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> The use of fractal analysis for understanding structures, and spatial and temporal complexity in biological systems has already been well studied and its use continues to increase in ecological research.<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a><a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a><a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a><a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a> Despite its extensive use, it still receives some <a href="/facts/Criticism/AIek6Ke0">criticism</a>.<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a><a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a> </p> <h3>Architecture, urban design and landscape design</h3> <p>In his publication <i>The Fractal Geometry of Nature</i>,<a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a> <a href="/facts/Benoit_Mandelbrot/fa8tKP93">Benoit Mandelbrot</a> suggested fractal theory could be applied to architecture. In this context, Mandelbrot was talking about the self-similar feature of fractal objects, rather than fractal analysis. In 1996, Carl Bovill applied the <a href="/facts/Box_counting/GtNjNeBe">box counting</a> method of fractal analysis to Architecture.<a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a> Bovill’s work, using a manual version of box counting, has since been refined by others<a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a> and computational approaches have been developed. <a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a> </p><p>Fractal analysis is one of the few quantitative analysis methods available to architects and designers to understand the visual complexity of buildings, urban areas and landscapes. Typical uses of fractal analysis of the built environment have been to understand the visual complexity of cities and skylines,<a class="footnote-ref" id="fnref:45" href="#fn:45"><sup>45</sup></a> the fractal dimensions of works of different architects <a class="footnote-ref" id="fnref:46" href="#fn:46"><sup>46</sup></a> and the landscape.<a class="footnote-ref" id="fnref:47" href="#fn:47"><sup>47</sup></a> </p><p>Combining the fractal analysis of ecology (see above) with fractal analysis of architecture, fractal dimensions have been used to explore the possible relationship between nature and architecture.<a class="footnote-ref" id="fnref:48" href="#fn:48"><sup>48</sup></a><a class="footnote-ref" id="fnref:49" href="#fn:49"><sup>49</sup></a> Promising results suggest further research is needed in this area. </p> <h3>Animal behaviour</h3> <p>Patterns in animal <a href="/facts/Behavior/0Y1VqmVX">behaviour</a> exhibit fractal properties on spatial and temporal scales.<a class="footnote-ref" id="fnref:50" href="#fn:50"><sup>50</sup></a> Fractal analysis helps in understanding the behaviour of animals and how they interact with their environments on multiple scales in space and time.<a class="footnote-ref" id="fnref:51" href="#fn:51"><sup>51</sup></a> Various animal movement signatures in their respective environments have been found to demonstrate spatially non-linear fractal patterns.<a class="footnote-ref" id="fnref:52" href="#fn:52"><sup>52</sup></a><a class="footnote-ref" id="fnref:53" href="#fn:53"><sup>53</sup></a> This has generated ecological interpretations such as the <a href="/facts/L%C3%A9vy_flight_foraging_hypothesis/yt4uDKSP">Lévy Flight Foraging hypothesis</a>, which has proven to be a more accurate description of animal movement for some species.<a class="footnote-ref" id="fnref:54" href="#fn:54"><sup>54</sup></a><a class="footnote-ref" id="fnref:55" href="#fn:55"><sup>55</sup></a><a class="footnote-ref" id="fnref:56" href="#fn:56"><sup>56</sup></a> </p><p>Spatial patterns and animal behaviour sequences in fractal time have an optimal complexity range, which can be thought of as the homeostatic state on the spectrum where the complexity sequence should regularly fall. An increase or a loss in complexity, either becoming more stereotypical or conversely more random in their behaviour patterns, indicates that there has been an alteration in the functionality of the individual.<a class="footnote-ref" id="fnref:57" href="#fn:57"><sup>57</sup></a><a class="footnote-ref" id="fnref:58" href="#fn:58"><sup>58</sup></a> Using fractal analysis, it is possible to examine the movement sequential complexity of animal behaviour and to determine whether individuals are experiencing deviations from their optimal range, suggesting a change in condition.<a class="footnote-ref" id="fnref:59" href="#fn:59"><sup>59</sup></a><a class="footnote-ref" id="fnref:60" href="#fn:60"><sup>60</sup></a> For example, it has been used to assess welfare of domestic hens,<a class="footnote-ref" id="fnref:61" href="#fn:61"><sup>61</sup></a> stress in bottlenose dolphins in response to human disturbance,<a class="footnote-ref" id="fnref:62" href="#fn:62"><sup>62</sup></a> and parasitic infection in Japanese macaques<a class="footnote-ref" id="fnref:63" href="#fn:63"><sup>63</sup></a> and sheep.<a class="footnote-ref" id="fnref:64" href="#fn:64"><sup>64</sup></a> The research is furthering the field of behavioural ecology by simplifying and quantifying very complex relationships.<a class="footnote-ref" id="fnref:65" href="#fn:65"><sup>65</sup></a> When it comes to animal welfare and <a href="/facts/Conservation_biology/S2693mIF">conservation</a>, fractal analysis makes it possible to identify potential sources of stress on animal behaviour, stressors that may not always be discernible through classical behaviour research.<a class="footnote-ref" id="fnref:66" href="#fn:66"><sup>66</sup></a><a class="footnote-ref" id="fnref:67" href="#fn:67"><sup>67</sup></a><a class="footnote-ref" id="fnref:68" href="#fn:68"><sup>68</sup></a> </p><p>This approach is more objective than classical behaviour measurements, such as <a href="/facts/Frequency/QUSbPaW8">frequency-based</a> observations that are limited by the counts of behaviours, but is able to delve into the underlying reason for the behaviour.<a class="footnote-ref" id="fnref:69" href="#fn:69"><sup>69</sup></a> Another important advantage of fractal analysis is the ability to monitor the health of <a href="/facts/Wildlife/Byh8cAyK">wild</a> and free-ranging animal populations in their natural habitats without invasive measurements. </p> <h2 id="applications-include">Applications include</h2> <p>Applications of fractal analysis include:<a class="footnote-ref" id="fnref:70" href="#fn:70"><sup>70</sup></a> </p> <table><tbody><tr><td><ul><li>Heart rate analysis<a class="footnote-ref" id="fnref:71" href="#fn:71"><sup>71</sup></a></li><li>Human gait, balance, and activity<a class="footnote-ref" id="fnref:72" href="#fn:72"><sup>72</sup></a><a class="footnote-ref" id="fnref:73" href="#fn:73"><sup>73</sup></a></li><li>Human anatomy<a class="footnote-ref" id="fnref:74" href="#fn:74"><sup>74</sup></a></li><li>Diagnostic imaging<a class="footnote-ref" id="fnref:75" href="#fn:75"><sup>75</sup></a></li><li>Cancer research<a class="footnote-ref" id="fnref:76" href="#fn:76"><sup>76</sup></a></li><li>Fractal analysis of complex networks<a class="footnote-ref" id="fnref:77" href="#fn:77"><sup>77</sup></a></li><li><a href="/facts/Categorisation/R6rWYttP">Classification</a> of <a href="/facts/Histopathology/9Eggjrg4">histopathology</a> slides in <a href="/facts/Medicine/2fnp90Je">medicine</a><a class="footnote-ref" id="fnref:78" href="#fn:78"><sup>78</sup></a></li><li><a href="/facts/Fractal_landscape/JTkWfeGm">Fractal landscape</a> or <a href="/facts/Coast/RBD0wxn3">Coastline</a> complexity<a class="footnote-ref" id="fnref:79" href="#fn:79"><sup>79</sup></a><a class="footnote-ref" id="fnref:80" href="#fn:80"><sup>80</sup></a></li><li><a href="/facts/Electrical_engineering/YYXW1KLW">Electrical engineering</a><a class="footnote-ref" id="fnref:81" href="#fn:81"><sup>81</sup></a></li><li>Enzyme/enzymology (<a href="/facts/Michaelis-Menten_kinetics/lYFykW3p">Michaelis-Menten kinetics</a>)<a class="footnote-ref" id="fnref:82" href="#fn:82"><sup>82</sup></a></li><li><a href="/facts/Algorithmic_composition/dGeVh6GP">Generation of new music</a></li><li><a href="/facts/Fractal_art/hjnu1JP2">Generation of various art forms</a></li><li><a href="/facts/Search_and_rescue/pdLTUPgl">Search and rescue</a><a class="footnote-ref" id="fnref:83" href="#fn:83"><sup>83</sup></a></li><li><a href="/facts/Signal_(information_theory)/41YqUJ8c">Signal</a> and <a href="/facts/Fractal_compression/d2qUg7sc">image compression</a></li></ul></td><td></td><td><ul><li>Urban growth<a class="footnote-ref" id="fnref:84" href="#fn:84"><sup>84</sup></a></li><li><a href="/facts/Neuroscience/o5RV8Cas">Neuroscience</a><a class="footnote-ref" id="fnref:85" href="#fn:85"><sup>85</sup></a><a class="footnote-ref" id="fnref:86" href="#fn:86"><sup>86</sup></a><a class="footnote-ref" id="fnref:87" href="#fn:87"><sup>87</sup></a><a class="footnote-ref" id="fnref:88" href="#fn:88"><sup>88</sup></a></li><li><a href="/facts/Diagnostic_imaging/pL6JyJEp">Diagnostic imaging</a><a class="footnote-ref" id="fnref:89" href="#fn:89"><sup>89</sup></a></li><li><a href="/facts/Pathology/U9b18LJG">Pathology</a><a class="footnote-ref" id="fnref:90" href="#fn:90"><sup>90</sup></a><a class="footnote-ref" id="fnref:91" href="#fn:91"><sup>91</sup></a><a class="footnote-ref" id="fnref:92" href="#fn:92"><sup>92</sup></a></li><li><a href="/facts/Geology/WGvVRIYF">Geology</a><a class="footnote-ref" id="fnref:93" href="#fn:93"><sup>93</sup></a></li><li><a href="/facts/Geography/TRd1Toyo">Geography</a><a class="footnote-ref" id="fnref:94" href="#fn:94"><sup>94</sup></a></li><li><a href="/facts/Archaeology/K3Q5P38R">Archaeology</a><a class="footnote-ref" id="fnref:95" href="#fn:95"><sup>95</sup></a><a class="footnote-ref" id="fnref:96" href="#fn:96"><sup>96</sup></a></li><li><a href="/facts/Seismology/Lony1LYX">Seismology</a><a class="footnote-ref" id="fnref:97" href="#fn:97"><sup>97</sup></a><a class="footnote-ref" id="fnref:98" href="#fn:98"><sup>98</sup></a></li><li><a href="/facts/Fractal_in_soil_mechanics/hALDG5FV">Soil studies</a><a class="footnote-ref" id="fnref:99" href="#fn:99"><sup>99</sup></a></li></ul></td><td><ul><li><a href="/facts/Game_design/VayxBcbN">Computer and video game design</a>, especially <a href="/facts/Computer_graphics/lapBE8wv">computer graphics</a> for <a href="/facts/Life/DNw7NUId">organic</a> environments and as part of <a href="/facts/Procedural_generation/I94vq0AQ">procedural generation</a></li><li>Fractography and <a href="/facts/Fracture_mechanics/f9QogJzF">fracture mechanics</a></li><li><a href="/facts/Fractal_antenna/7UKkLXnA">Fractal antennas</a> — Small size antennas using fractal shapes</li><li><a href="/facts/SAXS/Q4tvaRn7">Small angle scattering theory of fractally rough systems</a></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><a href="/facts/Technical_analysis/XQ9LlUgk">Technical analysis</a> of price series (see <a href="/facts/Elliott_wave_principle/rolhxFKD">Elliott wave principle</a>)</li></ul></td></tr></tbody></table> <ul><li>Fractal calculus<a class="footnote-ref" id="fnref:100" href="#fn:100"><sup>100</sup></a></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Multifractal/Fly3MCTb">Multifractal</a></li> <li><a href="/facts/Rescaled_range/77PyfPqT">Rescaled range</a></li> <li><a href="/facts/Analysis_on_fractals/4Q20sTjd">Analysis on fractals</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li><a href="http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm">Fractals and Fractal Analysis</a></li> <li><a href="http://www.fch.vutbr.cz/lectures/imagesci/download_ejournal/01_O.Zmeskal.pdf">Fractal analysis</a></li> <li><a href="http://www.trusoft.netmegs.com/">Benoit – Fractal Analysis Software</a> <a href="https://web.archive.org/web/20080517082415/http://www.trusoft.netmegs.com/">Archived</a> 2008-05-17 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li> <li><a href="http://www.physionet.org/tutorials/fmnc/index.shtml">Fractal Analysis Methods for Human Heartbeat and Gait Dynamics</a> <a href="https://web.archive.org/web/20160506052257/http://www.physionet.org/tutorials/fmnc/index.shtml">Archived</a> 2016-05-06 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Gerges, Firas; Geng, Xiaolong; Nassif, Hani; Boufadel, Michel C. (2021). "Anisotropic Multifractal Scaling of Mount Lebanon Topography: Approximate Conditioning". Fractals. 29 (5): 2150112–2153322. Bibcode:2021Fract..2950112G. doi:10.1142/S0218348X21501127. ISSN 0218-348X. S2CID 234272453. <a href="https://www.worldscientific.com/doi/abs/10.1142/S0218348X21501127" target="_blank">https://www.worldscientific.com/doi/abs/10.1142/S0218348X21501127</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Seuront, Laurent (2009-10-12). Fractals and Multifractals in Ecology and Aquatic Science. CRC Press. doi:10.1201/9781420004243. ISBN 9780849327827. <a href="9780849327827" target="_blank">9780849327827</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Mulligan, R. (2004). "Fractal analysis of highly volatile markets: an application to technology equities". The Quarterly Review of Economics and Finance. 44: 155–179. doi:10.1016/S1062-9769(03)00028-0. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Kamenshchikov, S. (2014). "Transport Catastrophe Analysis as an Alternative to a Monofractal Description: Theory and Application to Financial Crisis Time Series". Journal of Chaos. 2014: 1–8. doi:10.1155/2014/346743. <a href="https://doi.org/10.1155%2F2014%2F346743" target="_blank">https://doi.org/10.1155%2F2014%2F346743</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>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): 3929–3941. 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:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Zappasodi, Filippo; Olejarczyk, Elzbieta; Marzetti, Laura; Assenza, Giovanni (2014). "Fractal Dimension of EEG Activity Senses Neuronal Impairment in Acute Stroke". PLOS ONE. 9 (6): 3929–3941. Bibcode:2014PLoSO...9j0199Z. doi:10.1371/journal.pone.0100199. PMC 4072666. PMID 24967904. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4072666" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4072666</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Hisonothai, M.; Nakagawa, M. (2008). "EEG signal classification method based on fractal features and neural network". 2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society. Vol. 2008. pp. 3880–3. doi:10.1109/IEMBS.2008.4650057. ISBN 978-1-4244-1814-5. PMID 19163560. S2CID 22136019. <a href="978-1-4244-1814-5" target="_blank">978-1-4244-1814-5</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Fractal Analysis of Digital Images http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm <a href="http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm" target="_blank">http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>"Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society". Fractals: An Interdiscipinary Journal on the Complex Geometry of Nature. ISSN 1793-6543. <a href="http://www.worldscinet.com/fractals/fractals.shtml" target="_blank">http://www.worldscinet.com/fractals/fractals.shtml</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Khalili Golmankhaneh, Alireza (2022). Fractal Calculus and its Applications. Singapore: World Scientific Pub Co Inc. p. 328. doi:10.1142/12988. ISBN 978-981-126-110-7. S2CID 248575991. <a href="978-981-126-110-7" target="_blank">978-981-126-110-7</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Mandelbrot, B. (1967-05-05). "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. ISSN 0036-8075. PMID 17837158. S2CID 15662830. Archived from the original on 2021-10-19. Retrieved 2020-12-21. <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:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Seuront, Laurent (2009-10-12). Fractals and Multifractals in Ecology and Aquatic Science. CRC Press. doi:10.1201/9781420004243. ISBN 9780849327827. <a href="9780849327827" target="_blank">9780849327827</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Seuront, Laurent (2009-10-12). Fractals and Multifractals in Ecology and Aquatic Science. CRC Press. doi:10.1201/9781420004243. ISBN 9780849327827. <a href="9780849327827" target="_blank">9780849327827</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Mandelbrot, B. (1967-05-05). "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. ISSN 0036-8075. PMID 17837158. S2CID 15662830. Archived from the original on 2021-10-19. Retrieved 2020-12-21. <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:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Goldberger, Ary L; Peng, C.-K; Lipsitz, Lewis A (January 2002). "What is physiologic complexity and how does it change with aging and disease?". Neurobiology of Aging. 23 (1): 23–26. doi:10.1016/S0197-4580(01)00266-4. PMID 11755014. S2CID 17022186. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Gerges, Firas; Geng, Xiaolong; Nassif, Hani; Boufadel, Michel C. (2021). "Anisotropic Multifractal Scaling of Mount Lebanon Topography: Approximate Conditioning". Fractals. 29 (5): 2150112–2153322. Bibcode:2021Fract..2950112G. doi:10.1142/S0218348X21501127. ISSN 0218-348X. S2CID 234272453. <a href="https://www.worldscientific.com/doi/abs/10.1142/S0218348X21501127" target="_blank">https://www.worldscientific.com/doi/abs/10.1142/S0218348X21501127</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>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:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>"Digital Images in FracLac". ImageJ. Archived from the original on 2011-10-20. Retrieved 2012-02-08. <a href="http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Images.htm" target="_blank">http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Images.htm</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>MacIntosh, Andrew J. J.; Pelletier, Laure; Chiaradia, Andre; Kato, Akiko; Ropert-Coudert, Yan (December 2013). "Temporal fractals in seabird foraging behaviour: diving through the scales of time". Scientific Reports. 3 (1): 1884. Bibcode:2013NatSR...3.1884M. doi:10.1038/srep01884. ISSN 2045-2322. PMC 3662970. PMID 23703258. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3662970" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3662970</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Frontier, Serge (1987). "Applications of Fractal Theory to Ecology". Developments in Numerical Ecology. Springer Berlin Heidelberg. pp. 335–378. doi:10.1007/978-3-642-70880-0_9. ISBN 9783642708824. <a href="9783642708824" target="_blank">9783642708824</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Scheuring, István; Riedi, Rudolf H. (August 1994). "Application of multifractals to the analysis of vegetation pattern". Journal of Vegetation Science. 5 (4): 489–496. Bibcode:1994JVegS...5..489S. doi:10.2307/3235975. JSTOR 3235975. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>Seuront, Laurent; Lagadeuc, Yvan (1998). "Spatio-temporal structure of tidally mixed coastal waters: variability and heterogeneity". Journal of Plankton Research. 20 (7): 1387–1401. doi:10.1093/plankt/20.7.1387. ISSN 0142-7873. <a href="https://doi.org/10.1093%2Fplankt%2F20.7.1387" target="_blank">https://doi.org/10.1093%2Fplankt%2F20.7.1387</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Rutherford, Kenneth M.D.; Haskell, Marie J.; Glasbey, Chris; Jones, R.Bryan; Lawrence, Alistair B. (September 2003). "Detrended fluctuation analysis of behavioural responses to mild acute stressors in domestic hens". Applied Animal Behaviour Science. 83 (2): 125–139. doi:10.1016/S0168-1591(03)00115-1. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Mandelbrot, B. (1967-05-05). "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. ISSN 0036-8075. PMID 17837158. S2CID 15662830. Archived from the original on 2021-10-19. Retrieved 2020-12-21. <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:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>Bradbury, Rh; Reichelt, Re (1983). "Fractal Dimension of a Coral Reef at Ecological Scales". Marine Ecology Progress Series. 10: 169–171. Bibcode:1983MEPS...10..169B. doi:10.3354/meps010169. ISSN 0171-8630. <a href="https://doi.org/10.3354%2Fmeps010169" target="_blank">https://doi.org/10.3354%2Fmeps010169</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>Hastings, Harold M.; Pekelney, Richard; Monticciolo, Richard; Vun Kannon, David; Del Monte, Diane (January 1982). "Time scales, persistence and patchiness". Biosystems. 15 (4): 281–289. Bibcode:1982BiSys..15..281H. doi:10.1016/0303-2647(82)90043-0. ISSN 0303-2647. PMID 7165795. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> <li id="fn:33"><p>West, G. B. (1997-04-04). "A General Model for the Origin of Allometric Scaling Laws in Biology". Science. 276 (5309): 122–126. doi:10.1126/science.276.5309.122. PMID 9082983. S2CID 3140271. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li> <li id="fn:34"><p>West, G. B.; Enquist, B. J.; Brown, J. H. (2009-04-28). "A general quantitative theory of forest structure and dynamics". Proceedings of the National Academy of Sciences. 106 (17): 7040–7045. Bibcode:2009PNAS..106.7040W. doi:10.1073/pnas.0812294106. ISSN 0027-8424. PMC 2678466. PMID 19363160. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2678466" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2678466</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li> <li id="fn:35"><p>Rieu, Michel; Sposito, Garrison (1991). "Fractal Fragmentation, Soil Porosity, and Soil Water Properties: II. Applications". Soil Science Society of America Journal. 55 (5): 1239. Bibcode:1991SSASJ..55.1239R. doi:10.2136/sssaj1991.03615995005500050007x. ISSN 0361-5995. <a href="/w/index.php?title=Garrison_Sposito&action=edit&redlink=1" target="_blank">/w/index.php?title=Garrison_Sposito&action=edit&redlink=1</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li> <li id="fn:36"><p>Morse, D. R.; Lawton, J. H.; Dodson, M. M.; Williamson, M. H. (April 1985). "Fractal dimension of vegetation and the distribution of arthropod body lengths". Nature. 314 (6013): 731–733. Bibcode:1985Natur.314..731M. doi:10.1038/314731a0. ISSN 0028-0836. S2CID 4362382. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li> <li id="fn:37"><p>Li, Xiaoyan; Passow, Uta; Logan, Bruce E (January 1998). "Fractal dimensions of small (15–200 μm) particles in Eastern Pacific coastal waters". Deep Sea Research Part I: Oceanographic Research Papers. 45 (1): 115–131. doi:10.1016/s0967-0637(97)00058-7. ISSN 0967-0637. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></p></li> <li id="fn:38"><p>Lovejoy, S.; Schertzer, D. (May 2006). "Multifractals, cloud radiances and rain". Journal of Hydrology. 322 (1–4): 59–88. Bibcode:2006JHyd..322...59L. doi:10.1016/j.jhydrol.2005.02.042. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></p></li> <li id="fn:39"><p>Halley, J. M.; Hartley, S.; Kallimanis, A. S.; Kunin, W. E.; Lennon, J. J.; Sgardelis, S. P. (2004-02-24). "Uses and abuses of fractal methodology in ecology". Ecology Letters. 7 (3): 254–271. Bibcode:2004EcolL...7..254H. doi:10.1111/j.1461-0248.2004.00568.x. ISSN 1461-023X. S2CID 6059069. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></p></li> <li id="fn:40"><p>Bryce, R. M.; Sprague, K. B. (December 2012). "Revisiting detrended fluctuation analysis". Scientific Reports. 2 (1): 315. Bibcode:2012NatSR...2..315B. doi:10.1038/srep00315. ISSN 2045-2322. PMC 3303145. PMID 22419991. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3303145" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3303145</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></p></li> <li id="fn:41"><p>Mandelbrot, Benoit (1982). The Fractal Geometry of Nature. San Francisco: Freeman. <a href="#fnref:41" class="footnote-back-ref">↩</a></p></li> <li id="fn:42"><p>Bovill, Carl (1996). Fractal Geometry in Architecture and Design. Boston: Birkhauser. <a href="#fnref:42" class="footnote-back-ref">↩</a></p></li> <li id="fn:43"><p>Vaughan, Josephine; Ostwald, Michael J. (2010). "Refining a computational fractal method of analysis: Testing Bovill's architectural data". New Frontiers - Proceedings of the 15th International Conference on Computer-Aided Architectural Design in Asia, CAADRIA 2010. doi:10.52842/conf.caadria.2010.029. hdl:1959.13/927195. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></p></li> <li id="fn:44"><p>Lorenz, Wolfgang. "Fractal Geometry of Architecture Implementation of the Box-Counting Method in a CAD-Software". 27th Conference on Education and Research in Computer Aided Architectural Design in Europe, eCAADe 2009. doi:10.52842/conf.ecaade.2009.697. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></p></li> <li id="fn:45"><p>Rodin, Vladimir; Rodin, Elena (2000). "The fractal dimension of Tokyo's streets". Fractals. 8 (4). doi:10.1142/S0218348X00000457. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></p></li> <li id="fn:46"><p>Ostwald, Michael J.; Vaughan, Josephine; Tucker, Chris (2015). Characteristic visual complexity: Fractal dimensions in the architecture of frank lloyd wright and le corbusier. In:Architecture and Mathematics from Antiquity to the Future: Volume II: The 1500s to the Future. Springer International Publishing. pp. 339–354. ISBN 978-331900143-2. <a href="978-331900143-2" target="_blank">978-331900143-2</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></p></li> <li id="fn:47"><p>Patuano, A; Tara, A (2020). "Fractal geometry for landscape architecture: Review of methodologies and interpretations". Journal of Digital Landscape Architecture. 5: 72–80. doi:10.14627/537690008. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></p></li> <li id="fn:48"><p>Vaughan, Josephine; Ostwald, Michael J. (2022). "Measuring the geometry of nature and architecture: comparing the visual properties of Frank Lloyd Wright's Fallingwater and its natural setting". Open House International. 47 (1): 51–67. doi:10.1108/OHI-01-2021-0011. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></p></li> <li id="fn:49"><p>Bourchtein, A; Bourchtein, L; Naoumova, N (2014). "On fractal complexity of built and natural landscapes". ) Lecture Notes in Computer Science. 2 (8580). doi:10.1007/978-3-319-09129-7_33. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></p></li> <li id="fn:50"><p>MacIntosh, Andrew J. J.; Pelletier, Laure; Chiaradia, Andre; Kato, Akiko; Ropert-Coudert, Yan (December 2013). "Temporal fractals in seabird foraging behaviour: diving through the scales of time". Scientific Reports. 3 (1): 1884. Bibcode:2013NatSR...3.1884M. doi:10.1038/srep01884. ISSN 2045-2322. PMC 3662970. PMID 23703258. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3662970" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3662970</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></p></li> <li id="fn:51"><p>Seuront, Laurent (2009-10-12). Fractals and Multifractals in Ecology and Aquatic Science. CRC Press. doi:10.1201/9781420004243. ISBN 9780849327827. <a href="9780849327827" target="_blank">9780849327827</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></p></li> <li id="fn:52"><p>Catalan, Jordi; Marrasé, Cèlia; Pueyo, Salvador; Peters, Francesc; Bartumeus, Frederic (2003-10-28). "Helical Lévy walks: Adjusting searching statistics to resource availability in microzooplankton". Proceedings of the National Academy of Sciences. 100 (22): 12771–12775. Bibcode:2003PNAS..10012771B. doi:10.1073/pnas.2137243100. ISSN 0027-8424. PMC 240693. PMID 14566048. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC240693" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC240693</a> <a href="#fnref:52" class="footnote-back-ref">↩</a></p></li> <li id="fn:53"><p>Garcia, F.; Carrère, P.; Soussana, J.F.; Baumont, R. (September 2005). "Characterisation by fractal analysis of foraging paths of ewes grazing heterogeneous swards". Applied Animal Behaviour Science. 93 (1–2): 19–37. doi:10.1016/j.applanim.2005.01.001. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:53" class="footnote-back-ref">↩</a></p></li> <li id="fn:54"><p>Humphries, N. E.; Weimerskirch, H.; Queiroz, N.; Southall, E. J.; Sims, D. W. (2012-05-08). "Foraging success of biological Levy flights recorded in situ". Proceedings of the National Academy of Sciences. 109 (19): 7169–7174. Bibcode:2012PNAS..109.7169H. doi:10.1073/pnas.1121201109. ISSN 0027-8424. PMC 3358854. PMID 22529349. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3358854" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3358854</a> <a href="#fnref:54" class="footnote-back-ref">↩</a></p></li> <li id="fn:55"><p>Raposo, E P; Buldyrev, S V; da Luz, M G E; Viswanathan, G M; Stanley, H E (2009-10-30). "Lévy flights and random searches". Journal of Physics A: Mathematical and Theoretical. 42 (43): 434003. Bibcode:2009JPhA...42Q4003R. doi:10.1088/1751-8113/42/43/434003. ISSN 1751-8113. S2CID 13887492. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:55" class="footnote-back-ref">↩</a></p></li> <li id="fn:56"><p>Viswanathan, G.M; Afanasyev, V; Buldyrev, Sergey V; Havlin, Shlomo; da Luz, M.G.E; Raposo, E.P; Stanley, H.Eugene (June 2001). "Lévy flights search patterns of biological organisms". Physica A: Statistical Mechanics and Its Applications. 295 (1–2): 85–88. Bibcode:2001PhyA..295...85V. doi:10.1016/S0378-4371(01)00057-7. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:56" class="footnote-back-ref">↩</a></p></li> <li id="fn:57"><p>Goldberger, Ary L; Peng, C.-K; Lipsitz, Lewis A (January 2002). "What is physiologic complexity and how does it change with aging and disease?". Neurobiology of Aging. 23 (1): 23–26. doi:10.1016/S0197-4580(01)00266-4. PMID 11755014. S2CID 17022186. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:57" class="footnote-back-ref">↩</a></p></li> <li id="fn:58"><p>MacIntosh, Andrew James Jonathan (2014). "The Fractal Primate". Primate Research. 30 (1): 95–119. doi:10.2354/psj.30.011. ISSN 1880-2117. <a href="https://doi.org/10.2354%2Fpsj.30.011" target="_blank">https://doi.org/10.2354%2Fpsj.30.011</a> <a href="#fnref:58" class="footnote-back-ref">↩</a></p></li> <li id="fn:59"><p>Burgunder, Jade; Petrželková, Klára J.; Modrý, David; Kato, Akiko; MacIntosh, Andrew J.J. (August 2018). "Fractal measures in activity patterns: Do gastrointestinal parasites affect the complexity of sheep behaviour?". Applied Animal Behaviour Science. 205: 44–53. doi:10.1016/j.applanim.2018.05.014. S2CID 53475196. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:59" class="footnote-back-ref">↩</a></p></li> <li id="fn:60"><p>MacIntosh, A. J. J.; Alados, C. L.; Huffman, M. A. (2011-10-07). "Fractal analysis of behaviour in a wild primate: behavioural complexity in health and disease". Journal of the Royal Society Interface. 8 (63): 1497–1509. doi:10.1098/rsif.2011.0049. ISSN 1742-5689. PMC 3163426. PMID 21429908. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3163426" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3163426</a> <a href="#fnref:60" class="footnote-back-ref">↩</a></p></li> <li id="fn:61"><p>Rutherford, Kenneth M.D.; Haskell, Marie J.; Glasbey, Chris; Jones, R.Bryan; Lawrence, Alistair B. (September 2003). "Detrended fluctuation analysis of behavioural responses to mild acute stressors in domestic hens". Applied Animal Behaviour Science. 83 (2): 125–139. doi:10.1016/S0168-1591(03)00115-1. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:61" class="footnote-back-ref">↩</a></p></li> <li id="fn:62"><p>Cribb, Nardi; Seuront, Laurent (September 2016). "Changes in the behavioural complexity of bottlenose dolphins along a gradient of anthropogenically-impacted environments in South Australian coastal waters: Implications for conservation and management strategies". Journal of Experimental Marine Biology and Ecology. 482: 118–127. Bibcode:2016JEMBE.482..118C. doi:10.1016/j.jembe.2016.03.020. ISSN 0022-0981. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:62" class="footnote-back-ref">↩</a></p></li> <li id="fn:63"><p>MacIntosh, A. J. J.; Alados, C. L.; Huffman, M. A. (2011-10-07). "Fractal analysis of behaviour in a wild primate: behavioural complexity in health and disease". Journal of the Royal Society Interface. 8 (63): 1497–1509. doi:10.1098/rsif.2011.0049. ISSN 1742-5689. PMC 3163426. PMID 21429908. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3163426" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3163426</a> <a href="#fnref:63" class="footnote-back-ref">↩</a></p></li> <li id="fn:64"><p>Burgunder, Jade; Petrželková, Klára J.; Modrý, David; Kato, Akiko; MacIntosh, Andrew J.J. (August 2018). "Fractal measures in activity patterns: Do gastrointestinal parasites affect the complexity of sheep behaviour?". Applied Animal Behaviour Science. 205: 44–53. doi:10.1016/j.applanim.2018.05.014. S2CID 53475196. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:64" class="footnote-back-ref">↩</a></p></li> <li id="fn:65"><p>Bradbury, J. W.; Vehrencamp, S. L. (2014-05-01). "Complexity and behavioral ecology". Behavioral Ecology. 25 (3): 435–442. doi:10.1093/beheco/aru014. ISSN 1045-2249. <a href="https://doi.org/10.1093%2Fbeheco%2Faru014" target="_blank">https://doi.org/10.1093%2Fbeheco%2Faru014</a> <a href="#fnref:65" class="footnote-back-ref">↩</a></p></li> <li id="fn:66"><p>Rutherford, Kenneth M.D.; Haskell, Marie J.; Glasbey, Chris; Jones, R.Bryan; Lawrence, Alistair B. (September 2003). "Detrended fluctuation analysis of behavioural responses to mild acute stressors in domestic hens". Applied Animal Behaviour Science. 83 (2): 125–139. doi:10.1016/S0168-1591(03)00115-1. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:66" class="footnote-back-ref">↩</a></p></li> <li id="fn:67"><p>Alados, C.L.; Escos, J.M.; Emlen, J.M. (February 1996). "Fractal structure of sequential behaviour patterns: an indicator of stress". Animal Behaviour. 51 (2): 437–443. Bibcode:1996AnBeh..51..437A. doi:10.1006/anbe.1996.0040. S2CID 53184132. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:67" class="footnote-back-ref">↩</a></p></li> <li id="fn:68"><p>Rutherford, K. M. D.; Haskell, M. J.; Glasbey, C.; Jones, R. B.; Lawrence, A. B. (February 2004). "Fractal analysis of animal behaviour as an indicator of animal welfare". Animal Welfare. 13 (1): 99–103. doi:10.1017/S0962728600014433. S2CID 146350786. Retrieved 2019-03-27. <a href="https://www.ingentaconnect.com/contentone/ufaw/aw/2004/00000013/a00101s1/art00014" target="_blank">https://www.ingentaconnect.com/contentone/ufaw/aw/2004/00000013/a00101s1/art00014</a> <a href="#fnref:68" class="footnote-back-ref">↩</a></p></li> <li id="fn:69"><p>MacIntosh, Andrew James Jonathan (2014). "The Fractal Primate". Primate Research. 30 (1): 95–119. doi:10.2354/psj.30.011. ISSN 1880-2117. <a href="https://doi.org/10.2354%2Fpsj.30.011" target="_blank">https://doi.org/10.2354%2Fpsj.30.011</a> <a href="#fnref:69" class="footnote-back-ref">↩</a></p></li> <li id="fn:70"><p>"Applications". Archived from the original on 2007-10-12. Retrieved 2007-10-21. <a href="https://web.archive.org/web/20071012223212/http://library.thinkquest.org/26242/full/ap/ap.html" target="_blank">https://web.archive.org/web/20071012223212/http://library.thinkquest.org/26242/full/ap/ap.html</a> <a href="#fnref:70" class="footnote-back-ref">↩</a></p></li> <li id="fn:71"><p>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): 3929–3941. 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:71" class="footnote-back-ref">↩</a></p></li> <li id="fn:72"><p>Costa, Isis da Silva; Gamundí, Antoni; Miranda, José G. Vivas; França, Lucas G. Souza; Santana, De; Novaes, Charles; Montoya, Pedro (2017). "Altered Functional Performance in Patients with Fibromyalgia". Frontiers in Human Neuroscience. 11: 14. doi:10.3389/fnhum.2017.00014. ISSN 1662-5161. PMC 5266716. PMID 28184193. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5266716" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5266716</a> <a href="#fnref:72" class="footnote-back-ref">↩</a></p></li> <li id="fn:73"><p>França, L. G. S.; Montoya, Pedro; Miranda, J. G. V. (2017). "On multifractals: a non-linear study of actigraphy data". Physica A: Statistical Mechanics and Its Applications. 514: 612–619. arXiv:1702.03912. doi:10.1016/j.physa.2018.09.122. S2CID 18259316. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:73" class="footnote-back-ref">↩</a></p></li> <li id="fn:74"><p>Kędzia, A.; Derkowski, W. (2024). "Modern Methods of Neuroanatomical and Neurophysiological Research". MethodsX. 13 (December). doi:10.1016/j.mex.2024.102881. PMC 11340600. PMID 39176151. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11340600" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11340600</a> <a href="#fnref:74" class="footnote-back-ref">↩</a></p></li> <li id="fn:75"><p>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:75" class="footnote-back-ref">↩</a></p></li> <li id="fn:76"><p>Kam, Y.; Karperien, A.; Weidow, B.; Estrada, L.; Anderson, A. R.; Quaranta, V. (2009). "Nest expansion assay: A cancer systems biology approach to in vitro invasion measurements". BMC Research Notes. 2: 130. doi:10.1186/1756-0500-2-130. PMC 2716356. PMID 19594934. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2716356" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2716356</a> <a href="#fnref:76" class="footnote-back-ref">↩</a></p></li> <li id="fn:77"><p>Xiao, Xiongye; Chen, Hanlong; Bogdan, Paul (25 November 2021). "Deciphering the generating rules and functionalities of complex networks". Scientific Reports. 11 (1): 22964. Bibcode:2021NatSR..1122964X. doi:10.1038/s41598-021-02203-4. PMC 8616909. PMID 34824290. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8616909" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8616909</a> <a href="#fnref:77" class="footnote-back-ref">↩</a></p></li> <li id="fn:78"><p>Losa, Gabriele A.; Nonnenmacher, Theo, eds. (2005). Fractals in biology and medicine. Springer. ISBN 978-3-7643-7172-2. Retrieved 1 February 2012. <a href="978-3-7643-7172-2" target="_blank">978-3-7643-7172-2</a> <a href="#fnref:78" class="footnote-back-ref">↩</a></p></li> <li id="fn:79"><p>Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012. <a href="978-0-7167-1186-5" target="_blank">978-0-7167-1186-5</a> <a href="#fnref:79" class="footnote-back-ref">↩</a></p></li> <li id="fn:80"><p>Mandelbrot, B. (1967). "How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". 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 2021-10-19. Retrieved 2020-12-21. <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:80" class="footnote-back-ref">↩</a></p></li> <li id="fn:81"><p>Li, H. (2013). "Fractal analysis of side channels for breakdown structures in XLPE cable insulation". J Mater Sci: Mater Electron. 24 (5): 1640–1643. doi:10.1007/s10854-012-0988-y. S2CID 136564926. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:81" class="footnote-back-ref">↩</a></p></li> <li id="fn:82"><p>Reuveni, Shlomi; Granek, Rony; Klafter, Joseph (2008). "Proteins: Coexistence of Stability and Flexibility". Physical Review Letters. 100 (20): 208101. Bibcode:2008PhRvL.100t8101R. doi:10.1103/PhysRevLett.100.208101. ISSN 0031-9007. PMID 18518581. S2CID 16203048. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:82" class="footnote-back-ref">↩</a></p></li> <li id="fn:83"><p>Panteha Saeedi, and Soren A. Sorensen (2009). "An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents". Proceedings of the World Congress on Engineering 2009 (PDF). pp. 93–98. ISBN 978-988-17-0125-1. <a href="978-988-17-0125-1" target="_blank">978-988-17-0125-1</a> <a href="#fnref:83" class="footnote-back-ref">↩</a></p></li> <li id="fn:84"><p>Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions". PLOS ONE. 6 (9): e24791. arXiv:1104.4682. Bibcode:2011PLoSO...624791C. doi:10.1371/journal.pone.0024791. PMC 3176775. PMID 21949753. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3176775" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3176775</a> <a href="#fnref:84" class="footnote-back-ref">↩</a></p></li> <li id="fn:85"><p>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): 103–107. doi:10.1142/S0218348X08003880. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:85" class="footnote-back-ref">↩</a></p></li> <li id="fn:86"><p>França, Lucas G. Souza; Miranda, José G. Vivas; Leite, Marco; Sharma, Niraj K.; Walker, Matthew C.; Lemieux, Louis; Wang, Yujiang (2018). "Fractal and Multifractal Properties of Electrographic Recordings of Human Brain Activity: Toward Its Use as a Signal Feature for Machine Learning in Clinical Applications". Frontiers in Physiology. 9: 1767. arXiv:1806.03889. Bibcode:2018arXiv180603889F. doi:10.3389/fphys.2018.01767. ISSN 1664-042X. PMC 6295567. PMID 30618789. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6295567" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6295567</a> <a href="#fnref:86" class="footnote-back-ref">↩</a></p></li> <li id="fn:87"><p>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:87" class="footnote-back-ref">↩</a></p></li> <li id="fn:88"><p>Nikolić, D.; Moca, V.V.; Singer, W.; Mureşan, R.C. (2008). "Properties of multivariate data investigated by fractal dimensionality". Journal of Neuroscience Methods. 172 (1): 27–33. doi:10.1016/j.jneumeth.2008.04.007. PMID 18495248. S2CID 12268410. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:88" class="footnote-back-ref">↩</a></p></li> <li id="fn:89"><p>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:89" class="footnote-back-ref">↩</a></p></li> <li id="fn:90"><p>Smith, Robert F.; Mohr, David N.; Torres, Vicente E.; Offord, Kenneth P.; Melton III, L. Joseph (1989). "Renal insufficiency in community patients with mild asymptomatic microhematuria". Mayo Clinic Proceedings. 64 (4): 409–414. doi:10.1016/s0025-6196(12)65730-9. PMID 2716356. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:90" class="footnote-back-ref">↩</a></p></li> <li id="fn:91"><p>Al-Kadi O.S, Watson D. (2008). "Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images" (PDF). IEEE Transactions on Biomedical Engineering. 55 (7): 1822–1830. doi:10.1109/tbme.2008.919735. PMID 18595800. S2CID 14784161. Archived from the original (PDF) on 2014-04-13. Retrieved 2014-04-10. <a href="https://web.archive.org/web/20140413124458/http://sro.sussex.ac.uk/1919/1/tbme.pdf" target="_blank">https://web.archive.org/web/20140413124458/http://sro.sussex.ac.uk/1919/1/tbme.pdf</a> <a href="#fnref:91" class="footnote-back-ref">↩</a></p></li> <li id="fn:92"><p>Landini, Gabriel (2011). "Fractals in microscopy". Journal of Microscopy. 241 (1): 1–8. doi:10.1111/j.1365-2818.2010.03454.x. PMID 21118245. S2CID 40311727. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:92" class="footnote-back-ref">↩</a></p></li> <li id="fn:93"><p>Cheng, Qiuming (1997). "Multifractal Modeling and Lacunarity Analysis". Mathematical Geology. 29 (7): 919–932. Bibcode:1997MatG...29..919C. doi:10.1023/A:1022355723781. S2CID 118918429. <a href="/wiki/Qiuming_Cheng" target="_blank">/wiki/Qiuming_Cheng</a> <a href="#fnref:93" class="footnote-back-ref">↩</a></p></li> <li id="fn:94"><p>Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions". PLOS ONE. 6 (9): e24791. arXiv:1104.4682. Bibcode:2011PLoSO...624791C. doi:10.1371/journal.pone.0024791. PMC 3176775. PMID 21949753. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3176775" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3176775</a> <a href="#fnref:94" class="footnote-back-ref">↩</a></p></li> <li id="fn:95"><p>Burkle-Elizondo, Gerardo; Valdéz-Cepeda, Ricardo David (2006). "Fractal analysis of Mesoamerican pyramids". Nonlinear Dynamics, Psychology, and Life Sciences. 10 (1): 105–122. PMID 16393505. <a href="/wiki/PMID_(identifier)" target="_blank">/wiki/PMID_(identifier)</a> <a href="#fnref:95" class="footnote-back-ref">↩</a></p></li> <li id="fn:96"><p>Brown, Clifford T.; Witschey, Walter R. T.; Liebovitch, Larry S. (2005). "The Broken Past: Fractals in Archaeology". Journal of Archaeological Method and Theory. 12: 37–78. doi:10.1007/s10816-005-2396-6. S2CID 7481018. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:96" class="footnote-back-ref">↩</a></p></li> <li id="fn:97"><p>Vannucchi, Paola; Leoni, Lorenzo (2007). "Structural characterization of the Costa Rica décollement: Evidence for seismically-induced fluid pulsing". Earth and Planetary Science Letters. 262 (3–4): 413–428. Bibcode:2007E&PSL.262..413V. doi:10.1016/j.epsl.2007.07.056. hdl:2158/257208. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:97" class="footnote-back-ref">↩</a></p></li> <li id="fn:98"><p>Didier Sornette (2004). Critical phenomena in natural sciences: chaos, fractals, self-organization, and disorder: concepts and tools. Springer. pp. 128–140. ISBN 978-3-540-40754-6. <a href="978-3-540-40754-6" target="_blank">978-3-540-40754-6</a> <a href="#fnref:98" class="footnote-back-ref">↩</a></p></li> <li id="fn:99"><p>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:99" class="footnote-back-ref">↩</a></p></li> <li id="fn:100"><p>Khalili Golmankhaneh, Alireza (2022). Fractal Calculus and its Applications. Singapore: World Scientific Pub Co Inc. p. 328. doi:10.1142/12988. ISBN 978-981-126-110-7. S2CID 248575991. <a href="978-981-126-110-7" target="_blank">978-981-126-110-7</a> <a href="#fnref:100" class="footnote-back-ref">↩</a></p></li> </ol>