Fractal Geometry
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Fractal Geometry

Zoom in of the Mandelbrot set

In mathematics, a fractal is a subset of Euclidean space with a fractal dimension that strictly exceeds its topological dimension. Fractals appear the same at different scales, as illustrated in successive magnifications of the Mandelbrot set.[1][2][3][4] Fractals often exhibit similar patterns at increasingly smaller scales, a property called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge,[5] it is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.

One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer.[1] This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.[6]

Analytically, most fractals are nowhere differentiable.[1][4][7] An infinite fractal curve can be conceived of as winding through space differently from an ordinary line - although it is still topologically 1-dimensional, its fractal dimension indicates that it also resembles a surface.[1][6]

Sierpinski carpet (to level 6), a fractal with a topological dimension of 1 and a Hausdorff dimension of 1.893

Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass,[8] and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century.[9][10] The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975. Mandelbrot based it on the Latin fr?ctus, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.[1][11]

There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals."[12] More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."[13] Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a shape made of parts similar to the whole in some way."[14] Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants".[15]

The consensus among mathematicians is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied.[1][2][3] Fractals are not limited to geometric patterns, but can also describe processes in time.[5][4][16][17][18][19] Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media[20] and found in nature,[21][22][23][24][25] technology,[26][27][28][29] art,[30][31] architecture[32] and law.[33] Fractals are of particular relevance in the field of chaos theory because the graphs of most chaotic processes are fractals.[34] Many real and model networks have been found to have fractal features such as self similarity.[35][36][37]

## Introduction

A simple fractal tree

The word "fractal" often has different connotations for the lay public as opposed to mathematicians, 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.[1]:166; 18[2][11]

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 3D = 4. This number is what mathematicians call the fractal dimension of the Koch curve; it is certainly not what is conventionally perceived as the dimension of a curve (this number is not even an integer!). The fact that the Koch curve has a fractal dimension differing from its conventionally understood dimension (that is, its topological dimension) is what makes it a fractal.

3D computer generated fractal

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.[1][4][7] 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.[1]

## History

A Koch snowflake 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
Cantor (ternary) set.

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.[9][10] A common theme in ancient 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.[38] 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).[39]

In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them.[1]: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".[7][9][10] 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.[9]:7[10]

In addition, the quotient difference becomes arbitrarily large as the summation index increases.[40] Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass,[10] published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals.[9]: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.[1]:166

A Julia set, a fractal related to the Mandelbrot set
A Sierpinski gasket can be generated by a fractal tree.

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.[9]:25[10] Another milestone came a decade later in 1915, when Wac?aw Sierpi?ski 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.[4][9][10]

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.[10] 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.[notes 1]

A strange attractor that exhibits multifractal scaling
Uniform mass center triangle fractal
2x 120 degrees recursive IFS

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).[1]:179[7][10] 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,[41][42] which built on earlier work by Lewis Fry Richardson.

In 1975[11] 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".[43][7][9][39]

In 1980, Loren Carpenter gave a presentation at the SIGGRAPH where he introduced his software for generating and rendering fractally generated landscapes.[44]

## Definition and 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";[1] 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.[1][5][2][4][45]

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.[46] 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.[11] However, this requirement is not met by space-filling curves such as the Hilbert curve.[notes 2]

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, in addition to being nowhere differentiable and able to have a fractal dimension, be only generally characterized by a gestalt of the following features;[2]

• 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.[4]
• Qualitative self-similarity: as in a time series[16]
• 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[47] (related to the next criterion in this list).
• Irregularity locally and globally that is not easily described in traditional Euclidean geometric language. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls".[6]
• Simple and "perhaps recursive" definitions; 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, is easily described in Euclidean language, has the same Hausdorff dimension as topological dimension, and is fully defined without a need for recursion.[1][4]. It has been shown that both sets of fractals and non-fractals have the cardinality of aleph-two [48].

## Common techniques for generating fractals

Self-similar branching pattern modeled in silico using L-systems principles[25]

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.

A fractal generated by a finite subdivision rule for an alternating link

## Simulated fractals

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,[20] digital images, electrochemical patterns, circadian rhythms,[54] etc. Fractal patterns have been reconstructed in physical 3-dimensional space[28]:10 and virtually, often called "in silico" modeling.[51] Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above.[4][16][28] As one illustration, trees, ferns, cells of the nervous system,[25] blood and lung vasculature,[51] and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques.[25]

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. 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.

## Natural phenomena with fractal features

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.[55] Phenomena known to have fractal features include:

### Fractals in cell biology

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. [75] 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.[76] These processes are crucial in cell physiology and different pathologies.[77]

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 fractals patterns.[61] Similarly Matthias Weiss showed that the endoplasmic reticulum displays fractal features.[78] The current understanding is that fractals are ubiquitous in cell biology, from proteins, to organelles, to whole cells.

## In creative works

Since 1999, more than 10 scientific groups have performed fractal analysis on over 50 of Jackson Pollock's (1912-1956) paintings which were created by pouring paint directly onto his horizontal canvasses[79][80][81][82][83][84][85][86][87][88][89][90][91][excessive citations] Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks.[92] Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals.[93]

Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns.[94] 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.[31][95] Hokky Situngkir also suggested the similar properties in Indonesian traditional art, batik, and ornaments found in traditional houses.[96][97]

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."[98]

In a 1996 interview with Michael Silverblatt, David Foster Wallace admitted 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".[30]

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.

## Physiological responses

Humans appear to be especially well-adapted to processing fractal patterns with D values between 1.3 and 1.5.[99] When humans view fractal patterns with D values between 1.3 and 1.5, this tends to reduce physiological stress.[100][101]

## Applications in technology

### Ion propulsion

When two-dimensional fractals are iterated many times, the perimeter of the fractal increases up to infinity, but the area may never exceed a certain value. A fractal in three-dimensional space is similar; such a fractal may have an infinite surface area, but never exceed a certain volume.[123] This can be utilized to maximize the efficiency of ion propulsion when choosing electron emitter construction and material. If done correctly, the efficiency of the emission process can be maximized.[124]

## Notes

1. ^ The original paper, Lévy, Paul (1938). "Les Courbes planes ou gauches et les surfaces composées de parties semblables au tout". Journal de l'École Polytechnique: 227-247, 249-291., is translated in Edgar, pages 181-239.
2. ^ 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.

## References

1. Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5.
2. Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons. xxv. ISBN 978-0-470-84862-3.
3. ^ a b Briggs, John (1992). Fractals:The Patterns of Chaos. London: Thames and Hudson. p. 148. ISBN 978-0-500-27693-8.
4. Vicsek, Tamás (1992). Fractal growth phenomena. Singapore/New Jersey: World Scientific. pp. 31, 139-146. ISBN 978-981-02-0668-0.
5. ^ a b c Gouyet, Jean-François (1996). Physics and fractal structures. Paris/New York: Masson Springer. ISBN 978-0-387-94153-0.
6. ^ a b c Mandelbrot, Benoît B. (2004). Fractals and Chaos. Berlin: Springer. p. 38. ISBN 978-0-387-20158-0. A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension
7. Gordon, Nigel (2000). Introducing fractal geometry. Duxford: Icon. p. 71. ISBN 978-1-84046-123-7.
8. ^ Segal, S. L. (June 1978). "Riemann's example of a continuous 'nondifferentiable' function continued". The Mathematical Intelligencer. 1 (2): 81-82. doi:10.1007/BF03023065. S2CID 120037858.
9. Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8.
10. Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012.
11. ^ a b c d Albers, Donald J.; Alexanderson, Gerald L. (2008). "Benoît Mandelbrot: In his own words". Mathematical people : profiles and interviews. Wellesley, MA: AK Peters. p. 214. ISBN 978-1-56881-340-0.
12. ^ Mandelbrot, Benoit. "24/7 Lecture on Fractals". 2006 Ig Nobel Awards. Improbable Research.
13. ^ Mandelbrot, B. B.: The Fractal Geometry of Nature. W. H. Freeman and Company, New York (1982); p. 15.
14. ^ Feder, Jens (2013). Fractals. Springer Science & Business Media. p. 11. ISBN 978-1-4899-2124-6.
15. ^ Edgar, Gerald (2007). Measure, Topology, and Fractal Geometry. Springer Science & Business Media. p. 7. ISBN 978-0-387-74749-1.
16. ^ a b c 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.
17. ^ Krapivsky, P. L.; Ben-Naim, E. (1994). "Multiscaling in Stochastic Fractals". Physics Letters A. 196 (3-4): 168. Bibcode:1994PhLA..196..168K. doi:10.1016/0375-9601(94)91220-3.
18. ^ Hassan, M. K.; Rodgers, G. J. (1995). "Models of fragmentation and stochastic fractals". Physics Letters A. 208 (1-2): 95. Bibcode:1995PhLA..208...95H. doi:10.1016/0375-9601(95)00727-k.
19. ^ Hassan, M. K.; Pavel, N. I.; Pandit, R. K.; Kurths, J. (2014). "Dyadic Cantor set and its kinetic and stochastic counterpart". Chaos, Solitons & Fractals. 60: 31-39. arXiv:1401.0249. Bibcode:2014CSF....60...31H. doi:10.1016/j.chaos.2013.12.010. S2CID 14494072.
20. ^ a b Brothers, Harlan J. (2007). "Structural Scaling in Bach's Cello Suite No. 3". Fractals. 15 (1): 89-95. doi:10.1142/S0218348X0700337X.
21. ^ a b Tan, Can Ozan; Cohen, Michael A.; Eckberg, Dwain L.; Taylor, J. Andrew (2009). "Fractal properties of human heart period variability: Physiological and methodological implications". The Journal of Physiology. 587 (15): 3929-41. doi:10.1113/jphysiol.2009.169219. PMC 2746620. PMID 19528254.
22. ^ a b c Buldyrev, Sergey V.; Goldberger, Ary L.; Havlin, Shlomo; Peng, Chung-Kang; Stanley, H. Eugene (1995). "Fractals in Biology and Medicine: From DNA to the Heartbeat". In Bunde, Armin; Havlin, Shlomo (eds.). Fractals in Science. Springer.
23. ^ a b 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.
24. ^ a b Karperien, Audrey L.; Jelinek, Herbert F.; Buchan, Alastair M. (2008). "Box-Counting Analysis of Microglia Form in Schizophrenia, Alzheimer's Disease and Affective Disorder". Fractals. 16 (2): 103. doi:10.1142/S0218348X08003880.
25. Jelinek, Herbert F.; Karperien, Audrey; Cornforth, David; Cesar, Roberto; Leandro, Jorge de Jesus Gomes (2002). "MicroMod-an L-systems approach to neural modelling". In Sarker, Ruhul (ed.). Workshop proceedings: the Sixth Australia-Japan Joint Workshop on Intelligent and Evolutionary Systems, University House, ANU. University of New South Wales. ISBN 978-0-7317-0505-4. OCLC 224846454. Retrieved 2012. Event location: Canberra, Australia
26. ^ a b 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. doi:10.1016/j.apgeog.2011.10.016.
27. ^ a b 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 (Auckland, N.Z.). 2 (1): 109-122. doi:10.2147/OPTH.S1579. PMC 2698675. PMID 19668394.
28. ^ a b c d Losa, Gabriele A.; Nonnenmacher, Theo F. (2005). Fractals in biology and medicine. Springer. ISBN 978-3-7643-7172-2.
29. ^ a b c 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. Bibcode:2007E&PSL.262..413V. doi:10.1016/j.epsl.2007.07.056.
30. ^ a b Wallace, David Foster (August 4, 2006). "Bookworm on KCRW". Kcrw.com. Retrieved 2010.
31. ^ a b Eglash, Ron (1999). "African Fractals: Modern Computing and Indigenous Design". New Brunswick: Rutgers University Press. Archived from the original on January 3, 2018. Retrieved 2010.
32. ^ a b Ostwald, Michael J., and Vaughan, Josephine (2016) The Fractal Dimension of Architecture. Birhauser, Basel. doi:10.1007/978-3-319-32426-5.
33. ^ Baranger, Michael. "Chaos, Complexity, and Entropy: A physics talk for non-physicists" (PDF).
34. ^ a b c C.M. Song, S. Havlin, H.A. Makse (2005). "Self-similarity of complex networks". Nature. 433 (7024): 392-5. arXiv:cond-mat/0503078. Bibcode:2005Natur.433..392S. doi:10.1038/nature03248. PMID 15674285. S2CID 1985935.CS1 maint: multiple names: authors list (link)
35. ^ C.M. Song, S. Havlin, H.A. Makse (2006). "Origins of fractality in the growth of complex networks". Nature Physics. 2 (4): 275. arXiv:cond-mat/0507216. Bibcode:2006NatPh...2..275S. doi:10.1038/nphys266. S2CID 13858090.CS1 maint: multiple names: authors list (link)
36. ^ H.D. Rozenfeld, S. Havlin, D. Ben-Avraham (2007). "Fractal and trans fractal recursive scale-free nets". New J. Phys. 175 (9).CS1 maint: multiple names: authors list (link)
37. ^ Eglash, Ron (1999). African Fractals Modern Computing and Indigenous Design. ISBN 978-0-8135-2613-3.
38. ^ a b Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling. p. 310. ISBN 978-1-4027-5796-9.
39. ^ "Fractal Geometry". www-history.mcs.st-and.ac.uk. Retrieved 2017.
40. ^ Mandelbrot, B. (1967). "How Long Is the Coast of Britain?". Science. 156 (3775): 636-638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158. S2CID 15662830.
41. ^ Batty, Michael (April 4, 1985). "Fractals - Geometry Between Dimensions". New Scientist. 105 (1450): 31.
42. ^ Russ, John C. (1994). Fractal surfaces. 1. Springer. p. 1. ISBN 978-0-306-44702-0. Retrieved 2011.
43. ^ kottke.org. 2009. Vol Libre, an amazing CG film from 1980. [online] Available at: http://kottke.org/09/07/vol-libre-an-amazing-cg-film-from-1980
44. ^ Edgar, Gerald (2008). Measure, topology, and fractal geometry. New York: Springer-Verlag. p. 1. ISBN 978-0-387-74748-4.
45. ^ Karperien, Audrey (2004). Defining microglial morphology: Form, Function, and Fractal Dimension. Charles Sturt University. doi:10.13140/2.1.2815.9048.
46. ^ Spencer, John; Thomas, Michael S. C.; McClelland, James L. (2009). Toward a unified theory of development : connectionism and dynamic systems theory re-considered. Oxford/New York: Oxford University Press. ISBN 978-0-19-530059-8.
47. ^ Soltanifar, M. (2021). "A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals". Mathematics. 9 (13): 1-9. doi:10.3390/math9131546.
48. ^ Frame, Angus (August 3, 1998). "Iterated Function Systems". In Pickover, Clifford A. (ed.). Chaos and fractals: a computer graphical journey : ten year compilation of advanced research. Elsevier. pp. 349-351. ISBN 978-0-444-50002-1. Retrieved 2012.
49. ^ "Haferman Carpet". WolframAlpha. Retrieved 2012.
50. ^ a b c d Hahn, Horst K.; Georg, Manfred; Peitgen, Heinz-Otto (2005). "Fractal aspects of three-dimensional vascular constructive optimization". In Losa, Gabriele A.; Nonnenmacher, Theo F. (eds.). Fractals in biology and medicine. Springer. pp. 55-66. ISBN 978-3-7643-7172-2.
51. ^ J. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153-196.
52. ^ J. W. Cannon, W. Floyd and W. Parry. Crystal growth, biological cell growth and geometry. Pattern Formation in Biology, Vision and Dynamics, pp. 65-82. World Scientific, 2000. ISBN 981-02-3792-8, ISBN 978-981-02-3792-9.
53. ^ Fathallah-Shaykh, Hassan M. (2011). "Fractal Dimension of the Drosophila Circadian Clock". Fractals. 19 (4): 423-430. doi:10.1142/S0218348X11005476.
54. ^ "Hunting the Hidden Dimensional". Nova. PBS. WPMB-Maryland. October 28, 2008.
55. ^ Sadegh, Sanaz (2017). "Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork". Physical Review X. 7 (1): 011031. arXiv:1702.03997. Bibcode:2017PhRvX...7a1031S. doi:10.1103/PhysRevX.7.011031. PMC 5500227. PMID 28690919.
56. ^ Lovejoy, Shaun (1982). "Area-perimeter relation for rain and cloud areas". Science. 216 (4542): 185-187. Bibcode:1982Sci...216..185L. doi:10.1126/science.216.4542.185. PMID 17736252. S2CID 32255821.
57. ^ Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000). Pattern formation in biology, vision and dynamics. World Scientific. p. 78. ISBN 978-981-02-3792-9.
58. ^ C.-K. Peng, S.V. Buldyrev, A.L. Goldberger, S. Havlin, F. Sciortino, M. Simons, H.E. Stanley (1992). "Long-range correlations in nucleotide sequences". Nature. 356 (6365): 168-70. Bibcode:1992Natur.356..168P. doi:10.1038/356168a0. PMID 1301010. S2CID 4334674.CS1 maint: multiple names: authors list (link)
59. ^ Sornette, Didier (2004). Critical phenomena in natural sciences: chaos, fractals, selforganization, and disorder: concepts and tools. Springer. pp. 128-140. ISBN 978-3-540-40754-6.
60. ^ a b c Sweet, D.; Ott, E.; Yorke, J. A. (1999), "Complex topology in Chaotic scattering: A Laboratory Observation", Nature, 399 (6734): 315, Bibcode:1999Natur.399..315S, doi:10.1038/20573, S2CID 4361904
61. ^ S. Havlin, D. Ben-Avraham (1982). "Fractal dimensionality of polymer chains". J. Phys. A. 15 (6): L311-L316. Bibcode:1982JPhA...15L.311H. doi:10.1088/0305-4470/15/6/011.
62. ^ Bunde, Armin; Havlin, Shlomo (1996). Bunde, Armin; Havlin, Shlomo (eds.). Fractals and Disordered Systems. doi:10.1007/978-3-642-84868-1. ISBN 978-3-642-84870-4.CS1 maint: multiple names: authors list (link)
63. ^ Addison, Paul S. (1997). Fractals and chaos: an illustrated course. CRC Press. pp. 44-46. ISBN 978-0-7503-0400-9. Retrieved 2011.
64. ^ Pincus, David (September 2009). "The Chaotic Life: Fractal Brains Fractal Thoughts". psychologytoday.com.
65. ^ Enright, Matthew B.; Leitner, David M. (January 27, 2005). "Mass fractal dimension and the compactness of proteins". Physical Review E. 71 (1): 011912. Bibcode:2005PhRvE..71a1912E. doi:10.1103/PhysRevE.71.011912. PMID 15697635.
66. ^ Takayasu, H. (1990). Fractals in the physical sciences. Manchester: Manchester University Press. p. 36. ISBN 978-0-7190-3434-3.
67. ^ Jun, Li; Ostoja-Starzewski, Martin (April 1, 2015). "Edges of Saturn's Rings are Fractal". SpringerPlus. 4, 158: 158. doi:10.1186/s40064-015-0926-6. PMC 4392038. PMID 25883885.
68. ^ Meyer, Yves; Roques, Sylvie (1993). Progress in wavelet analysis and applications: proceedings of the International Conference "Wavelets and Applications", Toulouse, France - June 1992. Atlantica Séguier Frontières. p. 25. ISBN 978-2-86332-130-0. Retrieved 2011.
69. ^ Ozhovan M. I., Dmitriev I. E., Batyukhnova O. G. Fractal structure of pores of clay soil. Atomic Energy, 74, 241-243 (1993).
70. ^ Sreenivasan, K. R.; Meneveau, C. (1986). "The Fractal Facets of Turbulence". Journal of Fluid Mechanics. 173: 357-386. Bibcode:1986JFM...173..357S. doi:10.1017/S0022112086001209.
71. ^ de Silva, C. M.; Philip, J.; Chauhan, K.; Meneveau, C.; Marusic, I. (2013). "Multiscale Geometry and Scaling of the Turbulent-Nonturbulent Interface in High Reynolds Number Boundary Layers". Phys. Rev. Lett. 111 (6039): 192-196. Bibcode:2011Sci...333..192A. doi:10.1126/science.1203223. PMID 21737736. S2CID 22560587.
72. ^ Singh, Chamkor; Mazza, Marco (2019), "Electrification in granular gases leads to constrained fractal growth", Scientific Reports, Nature Publishing Group, 9 (1): 9049, arXiv:1812.06073, Bibcode:2019NatSR...9.9049S, doi:10.1038/s41598-019-45447-x, PMC 6588598, PMID 31227758
73. ^ Falconer, Kenneth (2013). Fractals, A Very Short Introduction. Oxford University Press.
74. ^ Leggett, Susan E.; Neronha, Zachary J.; Bhaskar, Dhananjay; Sim, Jea Yun; Perdikari, Theodora Myrto; Wong, Ian Y. (August 27, 2019). "Motility-limited aggregation of mammary epithelial cells into fractal-like clusters". Proceedings of the National Academy of Sciences. 116 (35): 17298-17306. Bibcode:2019PNAS..11617298L. doi:10.1073/pnas.1905958116. ISSN 0027-8424. PMC 6717304. PMID 31413194.
75. ^ Jelinek, Herbert F; Fernandez, Eduardo (June 1998). "Neurons and fractals: how reliable and useful are calculations of fractal dimensions?". Journal of Neuroscience Methods. 81 (1-2): 9-18. doi:10.1016/S0165-0270(98)00021-1. PMID 9696304. S2CID 3811866.
76. ^ Cross, Simon S. (1997). "Fractals in Pathology". The Journal of Pathology. 182 (1): 1-8. doi:10.1002/(SICI)1096-9896(199705)182:1<1::AID-PATH808>3.0.CO;2-B. ISSN 1096-9896. PMID 9227334.
77. ^ Speckner, Konstantin; Stadler, Lorenz; Weiss, Matthias (July 9, 2018). "Anomalous dynamics of the endoplasmic reticulum network". Physical Review E. 98 (1): 012406. Bibcode:2018PhRvE..98a2406S. doi:10.1103/PhysRevE.98.012406. ISSN 2470-0045. PMID 30110830.
78. ^ Taylor, R. P.; et al. (1999). "Fractal Analysis of Pollock's Drip Paintings". Nature. 399 (6735): 422. Bibcode:1999Natur.399..422T. doi:10.1038/20833. S2CID 204993516.
79. ^ Mureika, J. R.; Dyer, C. C.; Cupchik, G. C. (2005). "Multifractal Structure in Nonrepresentational Art". Physical Review E. 72 (4): 046101-1-15. arXiv:physics/0506063. Bibcode:2005PhRvE..72d6101M. doi:10.1103/PhysRevE.72.046101. PMID 16383462. S2CID 36628207.
80. ^ Redies, C.; Hasenstein, J.; Denzler, J. (2007). "Fractal-Like Image Statistics in Visual Art: Similarity to Natural Scenes". Spatial Vision. 21 (1): 137-148. doi:10.1163/156856807782753921. PMID 18073055.
81. ^ Lee, S.; Olsen, S.; Gooch, B. (2007). "Simulating and Analyzing Jackson Pollock's Paintings". Journal of Mathematics and the Arts. 1 (2): 73-83. CiteSeerX 10.1.1.141.7470. doi:10.1080/17513470701451253. S2CID 8529592.
82. ^ Alvarez-Ramirez, J.; Ibarra-Valdez, C.; Rodriguez, E.; Dagdug, L. (2008). "1/f-Noise Structure in Pollock's Drip Paintings". Physica A. 387 (1): 281-295. Bibcode:2008PhyA..387..281A. doi:10.1016/j.physa.2007.08.047.
83. ^ Graham, D. J.; Field, D. J. (2008). "Variations in Intensity for Representative and Abstract Art, and for Art from Eastern and Western Hemispheres" (PDF). Perception. 37 (9): 1341-1352. CiteSeerX 10.1.1.193.4596. doi:10.1068/p5971. PMID 18986061. S2CID 2794724.
84. ^ Alvarez-Ramirez, J.; Echeverria, J. C.; Rodriguez, E. (2008). "Performance of a high-dimensional R/S method for Hurst exponent estimation". Physica A. 387 (26): 6452-6462. Bibcode:2008PhyA..387.6452A. doi:10.1016/j.physa.2008.08.014.
85. ^ Coddington, J.; Elton, J.; Rockmore, D.; Wang, Y. (2008). "Multifractal Analysis and Authentication of Jackson Pollock Paintings". Proceedings of SPIE. 6810 (68100F): 1-12. Bibcode:2008SPIE.6810E..0FC. doi:10.1117/12.765015. S2CID 7650553.
86. ^ Al-Ayyoub, M.; Irfan, M. T.; Stork, D. G. (2009). "Boosting Multi-Feature Visual Texture Classifiers for the Authentification of Jackson Pollock's Drip Paintings". SPIE Proceedings on Computer Vision and Image Analysis of Art II. Computer Vision and Image Analysis of Art II. 7869 (78690H): 78690H. Bibcode:2011SPIE.7869E..0HA. doi:10.1117/12.873142. S2CID 15684445.
87. ^ Mureika, J. R.; Taylor, R. P. (2013). "The Abstract Expressionists and Les Automatistes: multi-fractal depth?". Signal Processing. 93 (3): 573. doi:10.1016/j.sigpro.2012.05.002.
88. ^ Taylor, R. P.; et al. (2005). "Authenticating Pollock Paintings Using Fractal Geometry". Pattern Recognition Letters. 28 (6): 695-702. doi:10.1016/j.patrec.2006.08.012.
89. ^ Jones-Smith, K.; et al. (2006). "Fractal Analysis: Revisiting Pollock's Paintings". Nature. 444 (7119): E9-10. Bibcode:2006Natur.444E...9J. doi:10.1038/nature05398. PMID 17136047. S2CID 4413758.
90. ^ Taylor, R. P.; et al. (2006). "Fractal Analysis: Revisiting Pollock's Paintings (Reply)". Nature. 444 (7119): E10-11. Bibcode:2006Natur.444E..10T. doi:10.1038/nature05399. S2CID 31353634.
91. ^ Shamar, L. (2015). "What Makes a Pollock Pollock: A Machine Vision Approach" (PDF). International Journal of Arts and Technology. 8: 1-10. CiteSeerX 10.1.1.647.365. doi:10.1504/IJART.2015.067389.
92. ^ Taylor, R. P.; Spehar, B.; Van Donkelaar, P.; Hagerhall, C. M. (2011). "Perceptual and Physiological Responses to Jackson Pollock's Fractals". Frontiers in Human Neuroscience. 5: 1-13. doi:10.3389/fnhum.2011.00060. PMC 3124832. PMID 21734876.
93. ^ Frame, Michael; and Mandelbrot, Benoît B.; A Panorama of Fractals and Their Uses
94. ^ Nelson, Bryn; Sophisticated Mathematics Behind African Village Designs Fractal patterns use repetition on large, small scale, San Francisco Chronicle, Wednesday, February 23, 2009
95. ^ 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
96. ^ Rulistia, Novia D. (October 6, 2015). "Application maps out nation's batik story". The Jakarta Post. Retrieved 2016.
97. ^ Koutonin, Mawuna (March 18, 2016). "Story of cities #5: Benin City, the mighty medieval capital now lost without trace". Retrieved April 2, 2018.
98. ^ Taylor, Richard P. (2016). "Fractal Fluency: An Intimate Relationship Between the Brain and Processing of Fractal Stimuli". In Di Ieva, Antonio (ed.). The Fractal Geometry of the Brain. Springer Series in Computational Neuroscience. Springer. pp. 485-496. ISBN 978-1-4939-3995-4.
99. ^ Taylor, Richard P. (2006). "Reduction of Physiological Stress Using Fractal Art and Architecture". Leonardo. 39 (3): 245-251. doi:10.1162/leon.2006.39.3.245. S2CID 8495221.
100. ^ For further discussion of this effect, see Taylor, Richard P.; Spehar, Branka; Donkelaar, Paul Van; Hagerhall, Caroline M. (2011). "Perceptual and Physiological Responses to Jackson Pollock's Fractals". Frontiers in Human Neuroscience. 5: 60. doi:10.3389/fnhum.2011.00060. PMC 3124832. PMID 21734876.
101. ^ Hohlfeld, Robert G.; Cohen, Nathan (1999). "Self-similarity and the geometric requirements for frequency independence in Antennae". Fractals. 7 (1): 79-84. doi:10.1142/S0218348X99000098.
102. ^ Reiner, Richard; Waltereit, Patrick; Benkhelifa, Fouad; Müller, Stefan; Walcher, Herbert; Wagner, Sandrine; Quay, Rüdiger; Schlechtweg, Michael; Ambacher, Oliver; Ambacher, O. (2012). "Fractal structures for low-resistance large area AlGaN/GaN power transistors". Proceedings of ISPSD: 341-344. doi:10.1109/ISPSD.2012.6229091. ISBN 978-1-4577-1596-9. S2CID 43053855.
103. ^ Zhiwei Huang; Yunho Hwang; Vikrant Aute; Reinhard Radermacher (2016). "Review of Fractal Heat Exchangers" (PDF) International Refrigeration and Air Conditioning Conference. Paper 1725CS1 maint: postscript (link)
104. ^ 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.
105. ^ "Applications". Archived from the original on October 12, 2007. Retrieved 2007.
106. ^ "Detecting 'life as we don't know it' by fractal analysis"
107. ^ 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.
108. ^ 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.
109. ^ Cheng, Qiuming (1997). "Multifractal Modeling and Lacunarity Analysis". Mathematical Geology. 29 (7): 919-932. doi:10.1023/A:1022355723781. S2CID 118918429.
110. ^ 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.
111. ^ 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.
112. ^ 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.
113. ^ Saeedi, Panteha; Sorensen, Soren A. (2009). "An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents" (PDF). Proceedings of the World Congress on Engineering 2009: 93-98. ISBN 978-988-17-0125-1.
114. ^ Bunde, A.; Havlin, S. (2009). "Fractal Geometry, A Brief Introduction to". Encyclopedia of Complexity and Systems Science. p. 3700. doi:10.1007/978-0-387-30440-3_218. ISBN 978-0-387-75888-6.
115. ^ "GPU internals" (PDF).
116. ^
117. ^
118. ^ "US8773422B1 - System, method, and computer program product for grouping linearly ordered primitives". Google Patents. December 4, 2007. Retrieved 2019.
119. ^ "US20110227921A1 - Processing of 3D computer graphics data on multiple shading engines". Google Patents. December 15, 2010. Retrieved 2019.
120. ^
121. ^ Li, Y.; Perlman, E.; Wang, M.; Yang, y.; Meneveau, C.; Burns, R.; Chen, S.; Szalay, A.; Eyink, G. (2008). "A Public Turbulence Database Cluster and Applications to Study Lagrangian Evolution of Velocity Increments in Turbulence". Journal of Turbulence. 9: N31. arXiv:0804.1703. Bibcode:2008JTurb...9...31L. doi:10.1080/14685240802376389. S2CID 15768582.
122. ^ "Introduction to Fractal Geometry". www.fractal.org. Retrieved 2017.
123. ^ DeFelice, David (August 18, 2015). "NASA - Ion Propulsion". NASA. Retrieved 2017.