Fractal#Natural phenomena with fractal features

The term "fractal" was coined by the mathematician Benoît Mandelbrot in 1975.Benoît Mandelbrot, Objets fractals, 1975, p. 4 Mandelbrot based it on the Latin {{lang|la|frāctus}}, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.{{cite book |title=Mathematical people : profiles and interviews |last1=Albers |first1=Donald J. |last2=Alexanderson |first2=Gerald L. |publisher=AK Peters |year=2008 |isbn=978-1-56881-340-0 |location=Wellesley, MA |page=214 |chapter=Benoît Mandelbrot: In his own words |author2-link=Gerald L. Alexanderson}}{{Cite OED|id=74094|term=fractal}}

File:Simple Fractals.png

File:FractalTree.gif

The word "fractal" often has different connotations for mathematicians and the general public, where the public is more likely to be familiar with fractal art 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 infinite regress in parallel mirrors or the homunculus, 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.{{rp|166; 18}}

This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is rep-tiled 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 Koch curve. 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 3^D = 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).

File:3D Computer Generated Fractal.png

This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable". In a concrete sense, this means fractals cannot be measured in traditional ways. 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 measuring 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.

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File:Von Koch curve.gif is a fractal that begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral bump|alt=|208x208px]]

File:Cantor set in seven iterations.svg

The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, with several notable people contributing canonical fractal forms along the way.{{cite book | last=Edgar | first=Gerald | title=Classics on Fractals | publisher=Westview Press | location=Boulder, CO | year=2004 | isbn=978-0-8133-4153-8 }}{{cite web

|title=A History of Fractal Geometry

|work=MacTutor History of Mathematics

|last=Trochet

|first=Holly

|archive-url=https://web.archive.org/web/20120312153006/http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/fractals.html

|url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/fractals.html

|archive-date=March 12, 2012

|year=2009

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A common theme in traditional African architecture 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.{{cite book| last = Eglash| first = Ron| title = African Fractals Modern Computing and Indigenous Design| year = 1999| publisher = Rutgers University Press| isbn = 978-0-8135-2613-3 }}

According to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).{{cite book |page=310 |url=https://books.google.com/books?id=JrslMKTgSZwC&q=fractal+koch+curve+book&pg=PA310 |first=Clifford A. |last=Pickover

|title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics |year=2009 |publisher=Sterling |isbn=978-1-4027-5796-9 }}

In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them.{{rp|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". Thus, it was not until two centuries had passed that on July 18, 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered a fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable at the Royal Prussian Academy of Sciences.{{rp|7}}

In addition, the quotient difference becomes arbitrarily large as the summation index increases.{{Cite web|url=http://www-history.mcs.st-and.ac.uk/HistTopics/fractals.html|title=Fractal Geometry|website=www-history.mcs.st-and.ac.uk|access-date=April 11, 2017}} Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass, published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals.{{rp|11–24}} Also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called "self-inverse" fractals.{{rp|166}}

{{anchor|julia}}

File:Julia set (indigo).png, a fractal related to the Mandelbrot set|alt=|200x200px]]

File:Fractal tree.gif can be generated by a fractal tree.|200x200px]]

One of the next milestones came in 1904, when Helge von Koch, 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 Koch snowflake.{{rp|25}} Another milestone came a decade later in 1915, when Wacław Sierpiński constructed his famous triangle then, one year later, his carpet. By 1918, two French mathematicians, Pierre Fatou and Gaston Julia, though working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors (i.e., points that attract or repel other points), which have become very important in the study of fractals.

Very shortly after that work was submitted, by March 1918, Felix Hausdorff expanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have non-integer dimensions. The idea of self-similar curves was taken further by Paul Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, the Lévy C curve.

{{anchor|multifractal|Strange attractor}}

File:Karperien Strange Attractor 200.gif that exhibits multifractal scaling|200x200px]]

File:Uniform Triangle Mass Center grade 5 fractal.gif

File:60 degrees 2x recursive IFS.jpg|200x200px]]

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).{{rp|179}}{{cite book | last=Gordon | first=Nigel | title=Introducing fractal geometry | publisher=Icon | location=Duxford | year=2000 | isbn=978-1-84046-123-7 | page=[https://archive.org/details/introducingfract0000lesm/page/71 71] | url=https://archive.org/details/introducingfract0000lesm/page/71 }} That changed, however, in the 1960s, when Benoit Mandelbrot started writing about self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,{{cite journal|author=Mandelbrot, B.|title=How Long Is the Coast of Britain?|journal=Science|date=1967|volume=156|issue=3775|pages=636–638|doi=10.1126/science.156.3775.636|pmid=17837158|bibcode=1967Sci...156..636M|s2cid=15662830|url=http://ena.lp.edu.ua:8080/handle/ntb/52473|access-date=October 31, 2020|archive-date=October 19, 2021|archive-url=https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473|url-status=dead}}{{cite journal |journal=New Scientist |date=April 4, 1985 |title=Fractals – Geometry Between Dimensions |first=Michael |last=Batty |page=31 |volume=105 |issue=1450 }} which built on earlier work by Lewis Fry Richardson.

In 1975, 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 Mandelbrot set, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".{{cite book |url=https://books.google.com/books?id=qDQjyuuDRxUC&pg=PA1 |page=1 |title=Fractal surfaces |volume= 1 |first=John C. |last=Russ |access-date=February 5, 2011 |year=1994 |publisher=Springer |isbn=978-0-306-44702-0 }}

In 1980, Loren Carpenter gave a presentation at the SIGGRAPH where he introduced his software for generating and rendering fractally generated landscapes.{{Cite web|title=Vol Libre, an amazing CG film from 1980|url=https://kottke.org/09/07/vol-libre-an-amazing-cg-film-from-1980|access-date=2023-02-12|website=kottke.org|date=July 29, 2009 |language=en}}

{{anchor|characteristics}}

One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"; 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.{{cite book | last=Gouyet | first=Jean-François | title=Physics and fractal structures | publisher=Masson Springer | location=Paris/New York | year=1996 | isbn=978-0-387-94153-0 }}{{Cite book | last=Falconer | first=Kenneth | title=Fractal Geometry: Mathematical Foundations and Applications | publisher=John Wiley & Sons | year=2003 | pages=xxv | isbn= 978-0-470-84862-3 | no-pp=true }}{{cite book | last=Edgar | first=Gerald | title=Measure, topology, and fractal geometry | publisher=Springer-Verlag | location=New York | year=2008 | isbn=978-0-387-74748-4 |page=1 }}

One point agreed on is that fractal patterns are characterized by fractal dimensions, but whereas these numbers quantify complexity (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns.{{cite book |last=Karperien |first=Audrey |title=Defining microglial morphology: Form, Function, and Fractal Dimension |publisher=Charles Sturt University |year= 2004 |doi=10.13140/2.1.2815.9048 }} In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension. However, this requirement is not met by space-filling curves such as the Hilbert curve.

Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to Falconer, fractals should be only generally characterized by a gestalt of the following features;

• Self-similarity, which may include:

:* Exact self-similarity: identical at all scales, such as the Koch snowflake

:* 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 Mandelbrot set's satellites are approximations of the entire set, but not exact copies.

:* Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals like the well-known example of the coastline of Britain 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.

:* Qualitative self-similarity: as in a time series{{cite book | last=Peters | first=Edgar | title=Chaos and order in the capital markets : a new view of cycles, prices, and market volatility | publisher=Wiley | location=New York | year=1996 | isbn=978-0-471-13938-6 }}

:* Multifractal scaling: characterized by more than one fractal dimension or scaling rule

• Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have emergent properties{{cite book | first1=John |last1=Spencer |first2=Michael S. C. |last2=Thomas |first3=James L. |last3=McClelland |title=Toward a unified theory of development : connectionism and dynamic systems theory re-considered |publisher=Oxford University Press |location=Oxford/New York |year=2009 |isbn=978-0-19-530059-8 }} (related to the next criterion in this list).
• Irregularity locally and globally that cannot easily be described in the language of traditional Euclidean geometry other than as the limit of a recursively 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";see Common techniques for generating fractals.

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.

{{See also|Fractal-generating software}}

{{anchor|L-system}}

File:KarperienFractalBranch.jpg using L-systems principles{{Cite book |editor=Sarker, Ruhul |title=Workshop proceedings: the Sixth Australia-Japan Joint Workshop on Intelligent and Evolutionary Systems, University House, ANU |oclc=224846454|chapter=MicroMod-an L-systems approach to neural modelling |last1=Jelinek |first1=Herbert F. |last2=Karperien |first2=Audrey |last3=Cornforth |first3=David |last4=Cesar |first4=Roberto |last5=Leandro |first5=Jorge de Jesus Gomes |url=https://books.google.com/books?id=FFSUGQAACAAJ |access-date=February 3, 2012 |year=2002 |publisher=University of New South Wales |isbn=978-0-7317-0505-4 |quote=Event location: Canberra, Australia}}|alt=|201x201px]]

{{anchor|algorithms}}

Images of fractals can be created by fractal generating programs. Because of the butterfly effect, a small change in a single variable can have an unpredictable outcome.

• Iterated function systems (IFS) – use fixed geometric replacement rules; may be stochastic or deterministic;{{cite book |editor=Pickover, Clifford A. |title=Chaos and fractals: a computer graphical journey : ten year compilation of advanced research |url=https://books.google.com/books?id=A51ARsapVuUC |access-date=February 4, 2012 |date=August 3, 1998 |publisher=Elsevier |isbn=978-0-444-50002-1 |last=Frame |first=Angus |chapter=Iterated Function Systems |pages=349–351 }} e.g., Koch snowflake, Cantor set, Haferman carpet,{{cite web |title=Haferman Carpet |url=http://www.wolframalpha.com/input/?i=Haferman+carpet |access-date=October 18, 2012 |publisher=WolframAlpha }} Sierpinski carpet, Sierpinski gasket, Peano curve, Harter-Heighway dragon curve, T-square, Menger sponge
• Strange attractors – 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 logistic map)
• L-systems – use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells), blood vessels, pulmonary structure, etc. or turtle graphics patterns such as space-filling curves and tilings
• Escape-time fractals – use a formula or recurrence relation at each point in a space (such as the complex plane); usually quasi-self-similar; also known as "orbit" fractals; e.g., the Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal. 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.
• {{anchor|random}}Random fractals – use stochastic rules; e.g., Lévy flight, percolation clusters, self avoiding walks, fractal landscapes, trajectories of Brownian motion and the Brownian tree (i.e., dendritic fractals generated by modeling diffusion-limited aggregation or reaction-limited aggregation clusters).{{cite book |last=Vicsek |first=Tamás | title=Fractal growth phenomena | publisher=World Scientific | location=Singapore/New Jersey | year=1992 | isbn=978-981-02-0668-0|pages=31; 139–146 }}

File:Finite subdivision of a radial link.png for an alternating link|202x202px]]

• Finite subdivision rules – use a recursive topological algorithm for refining tilingsJ. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196. and they are similar to the process of cell division.{{Cite book|last1=Carbone|first1=Alessandra|url=https://books.google.com/books?id=qZHyqUli9y8C&dq=%2522james+w.+cannon%2522+maths&pg=PA65|title=Pattern Formation in Biology, Vision and Dynamics|last2=Gromov|first2=Mikhael|last3=Prusinkiewicz|first3=Przemyslaw|date=2000|publisher=World Scientific|isbn=978-981-02-3792-9|language=en}} The iterative processes used in creating the Cantor set and the Sierpinski carpet are examples of finite subdivision rules, as is barycentric subdivision.

File:Julia-Set z2+c ani.gif made from a Julia-Set]]

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 fractal analysis. 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 artifacts which are not characteristics of true fractals.

Modeled fractals may be sounds, digital images, electrochemical patterns, circadian rhythms,{{Cite journal | last1=Fathallah-Shaykh | first1=Hassan M. | title=Fractal Dimension of the Drosophila Circadian Clock | doi=10.1142/S0218348X11005476 | journal=Fractals | volume=19 | issue=4 | pages=423–430 | year=2011 }} etc.

Fractal patterns have been reconstructed in physical 3-dimensional space{{rp|10}} and virtually, often called "in silico" modeling. Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above. As one illustration, trees, ferns, cells of the nervous system, blood and lung vasculature,{{cite book |chapter=Fractal aspects of three-dimensional vascular constructive optimization

| first1=Horst K. |last1=Hahn |first2=Manfred |last2=Georg |first3=Heinz-Otto |last3=Peitgen| editor1-last=Losa |editor1-first=Gabriele A. |editor2-last=Nonnenmacher |editor2-first=Theo F. | title=Fractals in biology and medicine | url=https://books.google.com/books?id=t9l9GdAt95gC | year=2005 | publisher=Springer | isbn=978-3-7643-7172-2 | pages=55–66 }} and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques.

The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a frond from a fern 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.

{{anchor|fractals in nature}}

{{Further|Patterns in nature}}

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."Hunting the Hidden Dimensional". Nova. PBS. WPMB-Maryland. October 28, 2008. Phenomena known to have fractal features include:

{{div col|colwidth=20em}}

• Actin cytoskeleton{{Cite journal |last=Sadegh |first=Sanaz |date=2017 |title=Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork |journal=Physical Review X |volume=7 |issue=1 |pages=011031 |doi=10.1103/PhysRevX.7.011031 |pmc=5500227 |pmid=28690919|arxiv=1702.03997 |bibcode=2017PhRvX...7a1031S }}
• Algae
• Animal coloration patterns
• Blood vessels and pulmonary vessels
• Brownian motion (generated by a one-dimensional Wiener process).{{Cite book |first1=Kenneth |last1=Falconer |title=Fractals, A Very Short Introduction |publisher=Oxford University Press |year=2013}}
• Clouds and rainfall areas{{Cite journal |last=Lovejoy |first=Shaun |date=1982 |title=Area-perimeter relation for rain and cloud areas|journal=Science |volume=216 |issue=4542 |pages=185–187|doi=10.1126/science.216.4542.185|pmid=17736252 |bibcode=1982Sci...216..185L |s2cid=32255821 }}
• Coastlines {{cite journal | last1=Boffetta | first1=G. | last2=Celani | first2=A. | last3=Dezzani | first3=D. | last4=Seminara | first4=A. | title=How winding is the coast of Britain? Conformal invariance of rocky shorelines | journal=Geophysical Research Letters | volume=35 | issue=3 | date=2008 | issn=0094-8276 | doi=10.1029/2007GL033093 | doi-access=free | arxiv=0712.3076 | bibcode=2008GeoRL..35.3615B }}
• Craters
• Crystals{{cite book |chapter-url=https://books.google.com/books?id=qZHyqUli9y8C&q=crystal+fractals+book&pg=PA78 |first1=James W. |last1=Cannon |first2=William J. |last2=Floyd |first3=Walter R. |last3=Perry |chapter=Crystal growth, biological cell growth and geometry |pages=65–82 |title=Pattern formation in biology, vision and dynamics |year=2000 |publisher=World Scientific |isbn=978-981-02-3792-9 |editor-last1=Carbone|editor-first1=Alessandra|editor-last2=Gromov|editor-first2=Mikhael|editor-last3=Prusinkiewicz|editor-first3=Przemyslaw}}
• DNA
• Dust grains{{citation|title=Electrification in granular gases leads to constrained fractal growth|author=Singh, Chamkor|author2=Mazza, Marco|journal=Scientific Reports|volume=9|year=2019|issue=1|page=9049|doi=10.1038/s41598-019-45447-x|publisher=Nature Publishing Group|pmid=31227758|pmc=6588598|arxiv=1812.06073|bibcode=2019NatSR...9.9049S|doi-access=free}}
• Earthquakes{{Cite journal | last1=Vannucchi | first1=Paola | last2=Leoni | first2=Lorenzo | doi=10.1016/j.epsl.2007.07.056 | title=Structural characterization of the Costa Rica décollement: Evidence for seismically-induced fluid pulsing | journal=Earth and Planetary Science Letters | volume=262 | issue=3–4 | pages=413 | year=2007 |bibcode=2007E&PSL.262..413V | hdl=2158/257208 | s2cid=128467785 | hdl-access=free }}{{cite book |pages=128–140 |title=Critical phenomena in natural sciences: chaos, fractals, selforganization, and disorder: concepts and tools |first=Didier |last=Sornette |year=2004 |publisher=Springer |isbn=978-3-540-40754-6 }}
• Fault lines
• Geometrical optics{{citation |first1=D. |last1=Sweet |first2=E. |last2=Ott |first3=J. A. |last3=Yorke |title=Complex topology in Chaotic scattering: A Laboratory Observation |year=1999 |journal=Nature |volume=399 |pages=315 |doi=10.1038/20573 |issue=6734 |bibcode = 1999Natur.399..315S |s2cid=4361904 }}
• Heart rates{{Cite journal | last1=Tan | first1=Can Ozan | last2=Cohen | first2=Michael A. | last3=Eckberg | first3=Dwain L. | last4=Taylor | first4=J. Andrew | title=Fractal properties of human heart period variability: Physiological and methodological implications | doi=10.1113/jphysiol.2009.169219 | journal=The Journal of Physiology | volume=587 | issue=15 | pages=3929–41 | year=2009 | pmid=19528254| pmc=2746620}}
• Heart sounds
• Lake shorelines and areas{{cite journal |author1=D. Seekell |author2=B. Cael |author3=E. Lindmark |author4=P. Byström |title=The fractal scaling relationship for river inlets to lakes |journal=Geophysical Research Letters |date=2021 |volume=48 |issue=9 |pages=e2021GL093366 |doi=10.1029/2021GL093366| issn=0094-8276 |bibcode=2021GeoRL..4893366S |s2cid=235508504 |url=http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-183511 }}{{cite journal |author1=D. Seekell |author2=M. L. Pace |author3=L. J. Tranvik |author4=C. Verpoorter |title=A fractal-based approach to lake size-distributions |journal=Geophysical Research Letters |date=2013 |volume=40 |issue=3 |pages=517–521 |doi=10.1002/grl.50139 |bibcode=2013GeoRL..40..517S |s2cid=14482711 |url=https://hal.archives-ouvertes.fr/hal-00932495/file/grl.50139.pdf }}{{cite journal |author1=B. B. Cael |author2=D. A. Seekell |title=The size-distribution of Earth's lakes |journal=Scientific Reports |date=2016 |volume=6 |pages=29633 |doi=10.1038/srep29633 |pmid=27388607 |pmc=4937396 |bibcode=2016NatSR...629633C }}
• Lightning bolts
• Mountain-goat horns
• Neurons
• Polymers
• Percolation
• Mountain ranges
• Ocean waves{{cite book |url=https://books.google.com/books?id=l2E4ciBQ9qEC&q=lightning+fractals+book&pg=PA45 |pages=44–46 |title=Fractals and chaos: an illustrated course |first=Paul S. |last=Addison |year=1997 |publisher=CRC Press |access-date=February 5, 2011 |isbn=978-0-7503-0400-9 }}
• Pineapple
• Proteins{{cite journal|last1=Enright|first1=Matthew B.|last2=Leitner|first2=David M.|title=Mass fractal dimension and the compactness of proteins|journal=Physical Review E|date=January 27, 2005|volume=71|issue=1|pages=011912|doi=10.1103/PhysRevE.71.011912|pmid=15697635|bibcode = 2005PhRvE..71a1912E |url=https://zenodo.org/record/895378}}
• Psychedelic Experience
• Purkinje cells{{cite journal|last1=Takeda|first1=T|last2=Ishikawa|first2=A|last3=Ohtomo|first3=K|last4=Kobayashi|first4=Y|last5=Matsuoka|first5=T|title=Fractal dimension of dendritic tree of cerebellar Purkinje cell during onto- and phylogenetic development|journal=Neurosci Research|date=February 1992|volume=13|issue=1|pages=19–31|doi=10.1016/0168-0102(92)90031-7|pmid=1314350|s2cid=4158401}}
• Rings of Saturn{{cite book|last1=Takayasu|first1=H.|title=Fractals in the physical sciences|date=1990|publisher=Manchester University Press|location=Manchester|isbn=978-0-7190-3434-3|page=[https://archive.org/details/fractalsinphysic0000taka_s1f9/page/36 36]|url=https://archive.org/details/fractalsinphysic0000taka_s1f9/page/36}}{{Cite journal|last1=Jun|first1=Li|author-link2=Martin Ostoja-Starzewski|last2=Ostoja-Starzewski|first2=Martin|title=Edges of Saturn's Rings are Fractal|journal=SpringerPlus|date=April 1, 2015|volume=4,158|pages=158|doi=10.1186/s40064-015-0926-6|pmid=25883885|pmc=4392038 |doi-access=free }}
• River networks{{cite journal |last1=Tarboton |first1=David G |last2=Bras |first2=Rafael L |last3=Rodriguez-Iturbe |first3=Ignacio |title=The fractal nature of river networks |journal=Water Resources Research |date=1988 |volume=24 |issue=8 |page=1317|doi=10.1029/WR024i008p01317 |bibcode=1988WRR....24.1317T }}
• Romanesco broccoli
• Snowflakes{{cite book

|url=https://books.google.com/books?id=aHux78oQbbkC&q=snowflake+fractals+book&pg=PA25 |page=25 |first1=Yves |last1=Meyer |first2=Sylvie |last2=Roques |title=Progress in wavelet analysis and applications: proceedings of the International Conference "Wavelets and Applications", Toulouse, France – June 1992 |year=1993 |publisher=Atlantica Séguier Frontières |access-date=February 5, 2011 |isbn=978-2-86332-130-0 }}

• Soil poresOzhovan M. I., Dmitriev I. E., Batyukhnova O. G. Fractal structure of pores of clay soil. Atomic Energy, 74, 241–243 (1993).
• Surfaces in turbulent flows{{cite journal |last1=Sreenivasan |first1=K. R. |first2=C. |last2=Meneveau |title=The Fractal Facets of Turbulence |journal=Journal of Fluid Mechanics |date=1986 |volume=173 |pages=357–386 |doi=10.1017/S0022112086001209 |bibcode=1986JFM...173..357S|s2cid=55578215 }}{{cite journal |last1=de Silva |first1=C. M. |first2=J. |last2=Philip |first3=K. |last3=Chauhan |first4=C. |last4=Meneveau |first5=I. |last5=Marusic |title=Multiscale Geometry and Scaling of the Turbulent–Nonturbulent Interface in High Reynolds Number Boundary Layers |journal=Phys. Rev. Lett. |date=2013 |volume=111 |issue=6039 |pages=192–196 |doi=10.1126/science.1203223 |pmid=21737736 |bibcode=2011Sci...333..192A|s2cid=22560587 }}
• Trees {{cite journal |last1=Mandelbrot |first1=Benoit B |title=The fractal geometry of trees and other natural phenomena |journal=Geometrical Probability and Biological Structures: Buffon's 200th Anniversary: Proceedings of the Buffon Bicentenary Symposium on Geometrical Probability, Image Analysis, Mathematical Stereology, and Their Relevance to the Determination of Biological Structures |date=1978 |pages=235–249}}

{{div col end}}

File:Frost patterns 2.jpg|Frost crystals occurring naturally on cold glass form fractal patterns

File:Optical Billiard Spheres dsweet.jpeg|Fractal basin boundary in a geometrical optical system

File:Glue1 800x600.jpg|A fractal is formed when pulling apart two glue-covered acrylic sheets

File:Square1.jpg|High-voltage breakdown within a {{convert|4|in|abbr=on}} block of acrylic glass creates a fractal Lichtenberg figure

File:Romanesco broccoli (Brassica oleracea).jpg|Romanesco broccoli, showing self-similar form approximating a natural fractal

File:Fractal defrosting patterns on Mars.jpg|Fractal defrosting patterns, polar Mars. The patterns are formed by sublimation of frozen CO₂. Width of image is about a kilometer.

File:Brefeldia maxima plasmodium on wood.jpg|Slime mold Brefeldia maxima growing fractally on wood

File:Dendrit.jpg|Psilomelane dendrites in the Solnhofen Limestone

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 branching.{{Cite journal|last1=Leggett|first1=Susan E.|last2=Neronha|first2=Zachary J.|last3=Bhaskar|first3=Dhananjay|last4=Sim|first4=Jea Yun|last5=Perdikari|first5=Theodora Myrto|last6=Wong|first6=Ian Y.|date=2019-08-27|title=Motility-limited aggregation of mammary epithelial cells into fractal-like clusters|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=35|pages=17298–17306|doi=10.1073/pnas.1905958116|issn=0027-8424|pmc=6717304|pmid=31413194|bibcode=2019PNAS..11617298L|doi-access=free}} Nerve cells 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.{{Cite journal|last1=Jelinek|first1=Herbert F|last2=Fernandez|first2=Eduardo|date=June 1998|title=Neurons and fractals: how reliable and useful are calculations of fractal dimensions?|url=https://linkinghub.elsevier.com/retrieve/pii/S0165027098000211|journal=Journal of Neuroscience Methods|language=en|volume=81|issue=1–2|pages=9–18|doi=10.1016/S0165-0270(98)00021-1|pmid=9696304|s2cid=3811866}} These processes are crucial in cell physiology and different pathologies.{{Cite journal|last=Cross|first=Simon S.|date=1997|title=Fractals in Pathology|journal=The Journal of Pathology|language=en|volume=182|issue=1|pages=1–8|doi=10.1002/(SICI)1096-9896(199705)182:1<1::AID-PATH808>3.0.CO;2-B|pmid=9227334|s2cid=23274235 |issn=1096-9896|doi-access=free}}

Multiple subcellular structures also are found to assemble into fractals. Diego Krapf has shown that through branching processes the actin filaments in human cells assemble into fractal patterns.{{Cite journal |last1=Sadegh |first1=Sanaz |last2=Higgins |first2=Jenny L. |last3=Mannion |first3=Patrick C. |last4=Tamkun |first4=Michael M. |last5=Krapf |first5=Diego |date=2017-03-09 |title=Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork |journal=Physical Review X |language=en |volume=7 |issue=1 |page=011031 |doi=10.1103/PhysRevX.7.011031 |issn=2160-3308 |pmc=5500227 |pmid=28690919|arxiv=1702.03997 |bibcode=2017PhRvX...7a1031S }} Similarly Matthias Weiss showed that the endoplasmic reticulum displays fractal features.{{Cite journal|last1=Speckner|first1=Konstantin|last2=Stadler|first2=Lorenz|last3=Weiss|first3=Matthias|date=2018-07-09|title=Anomalous dynamics of the endoplasmic reticulum network|url=https://link.aps.org/doi/10.1103/PhysRevE.98.012406|journal=Physical Review E|language=en|volume=98|issue=1|pages=012406|doi=10.1103/PhysRevE.98.012406|pmid=30110830|bibcode=2018PhRvE..98a2406S|s2cid=52010780|issn=2470-0045}} The current understanding is that fractals are ubiquitous in cell biology, from proteins, to organelles, to whole cells.

{{Further|Fractal art|Mathematics and art}}

Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by Jackson Pollock by pouring paint directly onto horizontal canvasses.{{cite journal |first=R. P. |last=Taylor |display-authors=et al |title=Fractal Analysis of Pollock's Drip Paintings |journal=Nature |volume=399 |issue=6735 |page=422 |year=1999|bibcode=1999Natur.399..422T |doi=10.1038/20833 |s2cid=204993516 |doi-access=free }}{{cite journal |first1=R. P. |last1=Taylor |display-authors=etal |title=Fractal Analysis: Revisiting Pollock's Paintings (Reply)|journal=Nature |volume=444 |issue=7119 |pages=E10–11 |year=2006 |doi=10.1038/nature05399|bibcode=2006Natur.444E..10T |s2cid=31353634 }}{{cite journal |first1=S. |last1=Lee |first2=S. |last2=Olsen |first3=B. |last3=Gooch |title=Simulating and Analyzing Jackson Pollock's Paintings |journal=Journal of Mathematics and the Arts |volume=1 |issue=2 |pages=73–83 |year=2007 |doi=10.1080/17513470701451253|citeseerx=10.1.1.141.7470 |s2cid=8529592 }}

Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks.{{cite journal |first1=L. |last1=Shamar |title=What Makes a Pollock Pollock: A Machine Vision Approach |journal=International Journal of Arts and Technology |volume=8 |pages=1–10 |year=2015 |doi=10.1504/IJART.2015.067389 |url=http://vfacstaff.ltu.edu/lshamir/publications/wm_pollock.pdf |citeseerx=10.1.1.647.365 |access-date=October 24, 2017 |archive-date=October 25, 2017 |archive-url=https://web.archive.org/web/20171025022609/http://vfacstaff.ltu.edu/lshamir/publications/wm_pollock.pdf |url-status=dead }} Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals.{{cite journal |first1=R. P. |last1=Taylor |first2=B. |last2=Spehar |first3=P. |last3=Van Donkelaar |first4=C. M. |last4=Hagerhall |title=Perceptual and Physiological Responses to Jackson Pollock's Fractals |journal=Frontiers in Human Neuroscience |volume=5 |pages=1–13 |year=2011 |doi=10.3389/fnhum.2011.00060|pmid=21734876 |pmc=3124832 |doi-access=free }}

Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns.Frame, Michael; and Mandelbrot, Benoît B.; [http://classes.yale.edu/Fractals/Panorama/ A Panorama of Fractals and Their Uses] {{Webarchive|url=https://web.archive.org/web/20071223090421/http://classes.yale.edu/Fractals/Panorama/ |date=December 23, 2007 }} It involves pressing paint between two surfaces and pulling them apart.

Cyberneticist Ron Eglash has suggested that fractal geometry and mathematics are prevalent in African art, games, divination, 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.{{cite web |url=http://www.rpi.edu/~eglash/eglash.dir/afractal/afractal.htm |first=Ron |last=Eglash |title=African Fractals: Modern Computing and Indigenous Design |location=New Brunswick |publisher=Rutgers University Press |year=1999 |access-date=October 17, 2010 |archive-url=https://web.archive.org/web/20180103005701/http://homepages.rpi.edu/~eglash/eglash.dir/afractal/afbook.htm |archive-date=January 3, 2018 |url-status=dead }}{{Cite web|last=Nelson|first=Bryn|date=2000-02-23|title=Sophisticated Mathematics Behind African Village Designs / Fractal patterns use repetition on large, small scale|url=https://www.sfgate.com/education/article/Sophisticated-Mathematics-Behind-African-Village-2774181.php|access-date=2023-02-12|website=SFGATE|language=en-US}} Hokky Situngkir also suggested the similar properties in Indonesian traditional art, batik, and ornaments found in traditional houses.Situngkir, Hokky; Dahlan, Rolan (2009). Fisika batik: implementasi kreatif melalui sifat fraktal pada batik secara komputasional. Jakarta: Gramedia Pustaka Utama. {{ISBN|978-979-22-4484-7}}{{cite news |last=Rulistia |first=Novia D. |date=October 6, 2015 |title=Application maps out nation's batik story |url=http://www.thejakartapost.com/news/2015/10/06/application-maps-out-nation-s-batik-story.html |newspaper=The Jakarta Post |access-date=September 25, 2016}}

Ethnomathematician Ron Eglash has discussed the planned layout of Benin city 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."Koutonin, Mawuna (March 18, 2016). "Story of cities #5: Benin City, the mighty medieval capital now lost without trace". Retrieved April 2, 2018.

In a 1996 interview with Michael Silverblatt, David Foster Wallace explained that the structure of the first draft of Infinite Jest he gave to his editor Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle (a.k.a. Sierpinski gasket), but that the edited novel is "more like a lopsided Sierpinsky Gasket".{{cite web |url=http://www.kcrw.com/etc/programs/bw/bw960411david_foster_wallace |first=David Foster |last=Wallace |title=Bookworm on KCRW |date=August 4, 2006 |publisher=Kcrw.com |access-date=October 17, 2010 |archive-date=November 11, 2010 |archive-url=https://web.archive.org/web/20101111033857/http://www.kcrw.com/etc/programs/bw/bw960411david_foster_wallace |url-status=dead }}

Some works by the Dutch artist M. C. Escher, such as Circle Limit III, 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:{{Cite journal |last1=Robles |first1=Kelly E. |last2=Roberts |first2=Michelle |last3=Viengkham |first3=Catherine |last4=Smith |first4=Julian H. |last5=Rowland |first5=Conor |last6=Moslehi |first6=Saba |last7=Stadlober |first7=Sabrina |last8=Lesjak |first8=Anastasija |last9=Lesjak |first9=Martin |last10=Taylor |first10=Richard P. |last11=Spehar |first11=Branka |last12=Sereno |first12=Margaret E. |date=2021 |title=Aesthetics and Psychological Effects of Fractal Based Design |journal=Frontiers in Psychology |volume=12 |doi=10.3389/fpsyg.2021.699962 |pmid=34484047 |pmc=8416160 |issn=1664-1078 |doi-access=free }} 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.

{{anchor|fractals in technology}}

File:Animated fractal mountain.gif|A fractal that models the surface of a mountain (animation)

File:FRACTAL-3d-FLOWER.jpg|3D recursive image

File:Fractal-BUTTERFLY.jpg|Recursive fractal butterfly image

File:Apophysis-100303-104.jpg|A fractal flame

Humans appear to be especially well-adapted to processing fractal patterns with fractal dimension between 1.3 and 1.5.{{cite book |chapter=Fractal Fluency: An Intimate Relationship Between the Brain and Processing of Fractal Stimuli |last=Taylor |first=Richard P. |pages=485–496 |title=The Fractal Geometry of the Brain |editor-last=Di Ieva |editor-first=Antonio |date=2016 |publisher=Springer |series=Springer Series in Computational Neuroscience |isbn=978-1-4939-3995-4}} When humans view fractal patterns with fractal dimension between 1.3 and 1.5, this tends to reduce physiological stress.{{cite journal | last=Taylor | first=Richard P. | title=Reduction of Physiological Stress Using Fractal Art and Architecture | journal=Leonardo | volume=39 | issue=3 | year=2006 | pages=245–251 | doi=10.1162/leon.2006.39.3.245| s2cid=8495221 | url=https://zenodo.org/record/894740 }}For further discussion of this effect, see {{cite journal | last1=Taylor | first1=Richard P. | last2=Spehar | first2=Branka | last3=Donkelaar | first3=Paul Van | last4=Hagerhall | first4=Caroline M. | title=Perceptual and Physiological Responses to Jackson Pollock's Fractals | journal=Frontiers in Human Neuroscience | volume=5 | pages=60 | year=2011 | doi=10.3389/fnhum.2011.00060| pmid=21734876 | pmc=3124832 | doi-access=free }}

{{Main|Fractal analysis}}

{{div col|colwidth=25em}}

• Fractal antennas{{cite journal |last1=Hohlfeld |first1=Robert G. |last2=Cohen |first2=Nathan |title=Self-similarity and the geometric requirements for frequency independence in Antennae |journal=Fractals |volume=7 |issue=1 |pages=79–84 |year=1999 |doi=10.1142/S0218348X99000098 }}
• Fractal transistor{{cite book| last1=Reiner |first1=Richard |first2=Patrick |last2=Waltereit |first3=Fouad |last3=Benkhelifa |first4=Stefan |last4=Müller |first5=Herbert |last5=Walcher |first6=Sandrine |last6=Wagner |first7=Rüdiger |last7=Quay |first8=Michael |last8=Schlechtweg |first9=Oliver | last10=Ambacher| first10=O.|last9=Ambacher |date=2012 | title=2012 24th International Symposium on Power Semiconductor Devices and ICs | chapter=Fractal structures for low-resistance large area AlGaN/GaN power transistors |isbn=978-1-4577-1596-9 |doi=10.1109/ISPSD.2012.6229091 |pages=341–344 | s2cid=43053855 }}
• Fractal heat exchangers{{cite web|author1=Zhiwei Huang|author2=Yunho Hwang|author3=Vikrant Aute|author4=Reinhard Radermacher|title=Review of Fractal Heat Exchangers|date=2016|url=http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=2724&context=iracc|format=PDF|postscript= International Refrigeration and Air Conditioning Conference. Paper 1725}}
• Digital imaging
• Architecture
• Urban growth{{Cite journal | last1=Chen | first1=Yanguang | title=Modeling Fractal Structure of City-Size Distributions Using Correlation Functions | doi=10.1371/journal.pone.0024791| journal=PLOS ONE | volume=6 | issue=9 | pages=e24791 | year=2011 | pmid=21949753 | pmc=3176775|arxiv = 1104.4682 |bibcode = 2011PLoSO...624791C | doi-access=free }}{{cite web |url=http://library.thinkquest.org/26242/full/ap/ap.html |title=Applications |access-date=October 21, 2007 |url-status=dead |archive-url=https://web.archive.org/web/20071012223212/http://library.thinkquest.org/26242/full/ap/ap.html |archive-date=October 12, 2007 }}
• Classification of histopathology slides
• Fractal landscape or Coastline complexity
• Detecting 'life as we don't know it' by fractal analysis{{cite journal| url = http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9012687&fileId=S1473550413000177| title = "Detecting 'life as we don't know it' by fractal analysis"| journal = International Journal of Astrobiology| date = October 2013| volume = 12| issue = 4| pages = 314–320| doi = 10.1017/S1473550413000177| last1 = Azua-Bustos| first1 = Armando| last2 = Vega-Martínez| first2 = Cristian| hdl = 11336/26238| s2cid = 122793675| hdl-access = free}}
• Enzymes (Michaelis–Menten kinetics)
• Generation of new music
• Signal and image compression
• Creation of digital photographic enlargements
• Fractal in soil mechanics
• Computer and video game design
• Computer Graphics
• Organic environments
• Procedural generation
• Fractography and fracture mechanics
• Small angle scattering theory of fractally rough systems
• T-shirts and other fashion
• Generation of patterns for camouflage, such as MARPAT
• Digital sundial
• Technical analysis of price series
• Fractals in networks
• Medicine
• Neuroscience
• Diagnostic Imaging
• Pathology{{Cite journal | last1=Smith | first1=Robert F. | last2=Mohr | first2=David N. | last3=Torres | first3=Vicente E. | last4=Offord | first4=Kenneth P. | last5=Melton III | first5=L. Joseph

| title=Renal insufficiency in community patients with mild asymptomatic microhematuria | journal=Mayo Clinic Proceedings | volume=64 | issue=4 | pages=409–414 | year=1989 | pmid=2716356 |doi=10.1016/s0025-6196(12)65730-9}}{{Cite journal | last1=Landini | first1=Gabriel | title=Fractals in microscopy | doi=10.1111/j.1365-2818.2010.03454.x | journal=Journal of Microscopy | volume=241 | issue=1 | pages=1–8 | year=2011 | pmid= 21118245| s2cid=40311727 }}

• Geology{{Cite journal | last1=Cheng | first1=Qiuming | author-link=Qiuming Cheng| title=Multifractal Modeling and Lacunarity Analysis | journal=Mathematical Geology | volume=29 | issue=7 | pages=919–932 | doi=10.1023/A:1022355723781 | year=1997 | bibcode=1997MatG...29..919C | s2cid=118918429 }}
• Geography{{Cite journal | last1=Chen | first1=Yanguang | title=Modeling Fractal Structure of City-Size Distributions Using Correlation Functions | doi=10.1371/journal.pone.0024791 | journal=PLOS ONE | volume=6 | issue=9 | pages=e24791 | year=2011 | pmid=21949753 | pmc=3176775|arxiv = 1104.4682 |bibcode = 2011PLoSO...624791C | doi-access=free}}
• Archaeology{{cite journal | last1=Burkle-Elizondo | first1=Gerardo

| last2=Valdéz-Cepeda | first2=Ricardo David | title=Fractal analysis of Mesoamerican pyramids | journal=Nonlinear Dynamics, Psychology, and Life Sciences | volume=10 | issue=1 | pages=105–122 | year=2006

| pmid=16393505}}{{Cite journal | last1=Brown | first1=Clifford T. | last2=Witschey | first2=Walter R. T. | last3=Liebovitch | first3=Larry S. | title=The Broken Past: Fractals in Archaeology | doi=10.1007/s10816-005-2396-6 | journal=Journal of Archaeological Method and Theory | volume=12 | pages=37–78 | year=2005 | s2cid=7481018 }}

• Soil mechanics
• Seismology
• Search and rescue{{cite journal| title=An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents| first1=Panteha |last1=Saeedi |first2=Soren A. |last2=Sorensen | journal=Proceedings of the World Congress on Engineering 2009 | year=2009 |pages=93–98 | isbn=978-988-17-0125-1|url=http://www.iaeng.org/publication/WCE2009/WCE2009_pp93-98.pdf}}
• Morton order space filling curves for GPU cache coherency in texture mapping,{{cite web|title=GPU internals|url=http://fileadmin.cs.lth.se/cs/Personal/Michael_Doggett/pubs/doggett12-tc.pdf}}{{cite web|title=sony patents|url=https://www.google.ch/patents/US20150287166?dq=morton+order+texture+swizzling}}{{cite web|title = description of swizzled and hybrid tiled swizzled textures|url=https://news.ycombinator.com/item?id=2239173}} rasterisation{{cite web | title=US8773422B1 - System, method, and computer program product for grouping linearly ordered primitives | website=Google Patents | date=December 4, 2007 | url=https://patents.google.com/patent/US8773422 | access-date=December 28, 2019}}{{cite web | title=US20110227921A1 - Processing of 3D computer graphics data on multiple shading engines | website=Google Patents | date=December 15, 2010 | url=http://www.google.ch/patents/US20110227921 | access-date=December 27, 2019}} and indexing of turbulence data.{{cite web |url=http://turbulence.pha.jhu.edu |title=Johns Hopkins Turbulence Databases}}{{cite journal|last1=Li|first1=Y.|first2=E.|last2=Perlman|first3=M.|last3=Wang|first4=y.|last4=Yang|first5=C.|last5=Meneveau|first6=R.|last6=Burns|first7=S.|last7=Chen|first8=A.|last8=Szalay|first9=G.|last9=Eyink|title=A Public Turbulence Database Cluster and Applications to Study Lagrangian Evolution of Velocity Increments in Turbulence|journal=Journal of Turbulence|date=2008|volume=9|pages=N31|doi=10.1080/14685240802376389|arxiv=0804.1703|bibcode=2008JTurb...9...31L|s2cid=15768582}}

{{div col end}}

{{Portal|Mathematics|Systems science}}

{{div col|colwidth=20em|small=yes}}

• {{Annotated link|Banach fixed point theorem}}
• {{Annotated link|Bifurcation theory}}
• {{Annotated link|Box counting}}
• {{Annotated link|Cymatics}}
• {{Annotated link|Determinism}}
• {{Annotated link|Diamond-square algorithm}}
• {{Annotated link|Droste effect}}
• {{Annotated link|Feigenbaum function}}
• {{Annotated link|Form constant}}
• {{Annotated link|Fractal cosmology}}
• {{Annotated link|Fractal derivative}}
• {{Annotated link|Fractalgrid}}
• {{Annotated link|Fractal sequence}}
• {{Annotated link|Fractal string}}
• {{Annotated link|Fracton}}
• {{Annotated link|Graftal}}
• {{Annotated link|Greeble}}
• {{Annotated link|H tree}}
• {{Annotated link|Infinite regress}}
• {{Annotated link|Lacunarity}}
• {{Annotated link|List of fractals by Hausdorff dimension}}
• {{Annotated link|Mandelbulb}}
• {{Annotated link|Mandelbox}}
• {{Annotated link|Macrocosm and microcosm}}
• {{Annotated link|Matryoshka doll}}
• {{Annotated link|Menger Sponge}}
• {{Annotated link|Multifractal system}}
• {{Annotated link|Newton fractal}}
• {{Annotated link|Percolation}}
• {{Annotated link|Power law}}
• {{Annotated link|List of important publications in mathematics#Geometry|Publications in fractal geometry}}
• {{Annotated link|Random walk}}
• {{Annotated link|Self-reference}}
• {{Annotated link|Self-similarity}}
• {{Annotated link|Systems theory}}
• {{Annotated link|Strange loop}}
• {{Annotated link|Turbulence}}
• {{Annotated link|Wiener process}}

{{div col end}}

{{Reflist

|group=notes

|refs=

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.

The original paper, {{cite journal|last=Lévy |first=Paul |title=Les Courbes planes ou gauches et les surfaces composées de parties semblables au tout |journal=Journal de l'École Polytechnique |year=1938 |pages=227–247, 249–291 }}, is translated in Edgar, pages 181–239.

}}

{{Reflist}}

• Stanley, Eugene H, Ostrowsky, N. (editors); On Growth and Fractal Form Fractal and Non-Fractal Patterns in Physics, Martinus Nijhoff Publisher, 1986. {{ISBN|0-89838-850-3}}
• Barnsley, Michael F.; and Rising, Hawley; Fractals Everywhere. Boston: Academic Press Professional, 1993. {{isbn|0-12-079061-0}}
• Duarte, German A.; Fractal Narrative. About the Relationship Between Geometries and Technology and Its Impact on Narrative Spaces. Bielefeld: Transcript, 2014. {{isbn|978-3-8376-2829-6}}
• Falconer, Kenneth; Techniques in Fractal Geometry. John Wiley and Sons, 1997. {{isbn|0-471-92287-0}}
• Jürgens, Hartmut; Peitgen, Heinz-Otto; and Saupe, Dietmar; Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992. {{isbn|0-387-97903-4}}
• Mandelbrot, Benoit B.; The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1982. {{isbn|0-7167-1186-9}}
• Peitgen, Heinz-Otto; and Saupe, Dietmar; eds.; The Science of Fractal Images. New York: Springer-Verlag, 1988. {{isbn|0-387-96608-0}}
• Pickover, Clifford A.; ed.; Chaos and Fractals: A Computer Graphical Journey – A 10 Year Compilation of Advanced Research. Elsevier, 1998. {{isbn|0-444-50002-2}}
• Jones, Jesse; Fractals for the Macintosh, Waite Group Press, Corte Madera, CA, 1993. {{isbn|1-878739-46-8}}.
• Lauwerier, Hans; Fractals: Endlessly Repeated Geometrical Figures, Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991. {{isbn|0-691-08551-X}}, cloth. {{isbn|0-691-02445-6}} paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.
• {{Cite book | last=Sprott | first=Julien Clinton | title=Chaos and Time-Series Analysis | publisher=Oxford University Press | year=2003 | isbn=978-0-19-850839-7 }}
• Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric; [http://www.fractalexplorer.com Exploring Fractals on the Macintosh], Addison Wesley, 1995. {{isbn|0-201-62630-6}}
• Lesmoir-Gordon, Nigel; The Colours of Infinity: The Beauty, The Power and the Sense of Fractals. 2004. {{isbn|1-904555-05-5}} (The book comes with a related DVD of the Arthur C. Clarke documentary introduction to the fractal concept and the Mandelbrot set.)
• Liu, Huajie; Fractal Art, Changsha: Hunan Science and Technology Press, 1997, {{isbn|9787535722348}}.
• Gouyet, Jean-François; Physics and Fractal Structures (Foreword by B. Mandelbrot); Masson, 1996. {{isbn|2-225-85130-1}}, and New York: Springer-Verlag, 1996. {{isbn|978-0-387-94153-0}}. Out-of-print. Available in PDF version at.{{cite web |url=http://www.jfgouyet.fr/fractal/fractauk.html |title=Physics and Fractal Structures |language=fr |publisher=Jfgouyet.fr |access-date=October 17, 2010 }}
• {{Cite book |first1=Kenneth |last1=Falconer |title=Fractals, A Very Short Introduction |publisher=Oxford University Press |year=2013 }}

{{Commons}}

{{Wikibooks|Fractals}}

• {{webarchive |url=http://webarchive.loc.gov/all/20011116035117/http://dmoz.org/science/math/chaos_and_fractals/ |title=Fractals |date=November 16, 2001}}
• "[https://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html Hunting the Hidden Dimension]", PBS NOVA, first aired August 24, 2011
• [https://www.ted.com/talks/benoit_mandelbrot_fractals_and_the_art_of_roughness Benoit Mandelbrot: Fractals and the Art of Roughness] ({{Webarchive|url=https://web.archive.org/web/20140217132541/http://www.ted.com/talks/benoit_mandelbrot_fractals_the_art_of_roughness.html |date=February 17, 2014 }}), TED, February 2010
• [http://adsabs.harvard.edu/abs/2007PrGeo..22..451Y Equations of self-similar fractal measure based on the fractional-order calculus]（2007）

{{Fractals}}

{{Chaos theory}}

{{Mathematical art}}

{{Patterns in nature}}

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Category:Computational fields of study

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