Contents 1 Introduction 2 History 3 Characteristics 4 Brownian motion 5 Common techniques for generating fractals 6 Simulated fractals 7 Natural phenomena with fractal features 8 In creative works 9 Physiological responses 10 Ion production capabilities 11 Applications in technology 12 See also 13 Notes 14 References 15 Further reading 16 External links


Introduction[edit] The word "fractal" often has different connotations for laypeople than for mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception. The mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with 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.[1] 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.[3]:166; 18[4][7] This idea of being detailed relates to another feature that can be understood without mathematical background: Having a fractional or fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived. A regular line, for instance, is conventionally understood to be 1-dimensional; if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the Koch snowflake. It is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. The fractal curve divided into parts 1/3 the length of the original line becomes 4 pieces rearranged to repeat the original detail, and this unusual relationship is the basis of its fractal dimension. 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.[3][6][9] 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 a wavy fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the wavy pattern would always re-appear, albeit at a smaller size, 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.[3]


History[edit] 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 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.[11][12] 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).[36] In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them.[3]: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".[9][11][12] 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.[11]:7[12] In addition, the quotient difference becomes arbitrarily large as the summation index increases.[37] Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass,[12] published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals.[11]: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.[3]:166 A Julia set, a fractal related to the Mandelbrot set A Sierpinski triangle 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.[11]:25[12] 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 are 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.[6][11][12] 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 noninteger dimensions.[12] 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).[3]:179[9][12] 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,[38][39] which built on earlier work by Lewis Fry Richardson. In 1975[7] 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".[40] Currently, fractal studies are essentially exclusively computer-based.[9][11][36] In 1980, Loren Carpenter gave a presentation at the SIGGRAPH where he introduced his software for generating and rendering fractally generated landscapes.[41]


Characteristics[edit] 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";[3] 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 an unusual relationship with the space a fractal is embedded in.[2][3][4][6][42] 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.[43] 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.[7] It has been noted that this dimensional requirement is not met by fractal space-filling curves such as the Hilbert curve.[notes 2] According to Falconer, rather than being strictly defined, fractals should, in addition to being nowhere differentiable and able to have a fractal dimension, be generally characterized by a gestalt of the following features;[4] Self-similarity, which may be manifested as: Exact self-similarity: identical at all scales; e.g. 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; 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, for example, the Koch snowflake[6] Qualitative self-similarity: as in a time series[17] 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[44] (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".[8] 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.[3][6]


Brownian motion[edit] A path generated by a one dimensional Wiener process is a fractal curve of dimension 1.5, and Brownian motion is a finite version of this.[45]


Common techniques for generating fractals[edit] Self-similar branching pattern modeled in silico using L-systems principles[26] Images of fractals can be created by fractal generating programs. Because of the butterfly effect a small change in a single variable can have a unpredictable outcome. Iterated function systems – use fixed geometric replacement rules; may be stochastic or deterministic;[46] e.g., Koch snowflake, Cantor set, Haferman carpet,[47] 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[26]), blood vessels, pulmonary structure,[48] 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. 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).[6] A fractal generated by a finite subdivision rule for an alternating link Finite subdivision rules use a recursive topological algorithm for refining tilings[49] and they are similar to the process of cell division.[50] The iterative processes used in creating the Cantor set and the Sierpinski carpet are examples of finite subdivision rules, as is barycentric subdivision.


Simulated fractals[edit] A fractal flame 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,[21] digital images, electrochemical patterns, circadian rhythms,[51] etc. Fractal patterns have been reconstructed in physical 3-dimensional space[29]:10 and virtually, often called "in silico" modeling.[48] Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above.[6][17][29] As one illustration, trees, ferns, cells of the nervous system,[26] blood and lung vasculature,[48] and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques.[26] 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[edit] Further information: 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.[52] Phenomena known to have fractal features include: River networks Fault lines Mountain ranges Craters Lightning bolts Coastlines Mountain goat horns Trees Algae Geometrical optics[53] Animal coloration patterns Romanesco broccoli Pineapple Heart rates[22] Heart sounds[23] Earthquakes[30][54] Snowflakes[55] Psychological subjective perception[56] Crystals[57] Blood vessels and pulmonary vessels[48] Ocean waves[58] DNA Soil pores[59] Rings of Saturn[60][61] Proteins[62] Surfaces in turbulent flows[63][64] Frost crystals occurring naturally on cold glass form fractal patterns  Fractal basin boundary in a geometrical optical system[53]  A fractal is formed when pulling apart two glue-covered acrylic sheets  High voltage breakdown within a 4 in (100 mm) block of acrylic creates a fractal Lichtenberg figure  Romanesco broccoli, showing self-similar form approximating a natural fractal  Fractal defrosting patterns, polar Mars. The patterns are formed by sublimation of frozen CO2. Width of image is about a kilometer.  Slime mold Brefeldia maxima growing fractally on wood 


In creative works[edit] Further information: Fractal art and Mathematics and art 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 canvases[65][66][67][68][69][70][71][72][73][74][75][76][77] Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks.[78] Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals.[79] Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns.[80] 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.[32][81] Hokky Situngkir also suggested the similar properties in Indonesian traditional art, batik, and ornaments found in traditional houses.[82][83] 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".[31] A fractal that models the surface of a mountain (animation)  3D recursive image  Recursive fractal butterfly image 


Physiological responses[edit] Humans appear to be especially well-adapted to processing fractal patterns with D values between 1.3–1.5.[84] When humans view fractal patterns with D values between 1.3–1.5, this tends to reduce physiological stress.[85][86]


Ion production capabilities[edit] If a circle boundary is drawn around the two-dimensional view of a fractal, the fractal will never cross the boundary, this is due to the scaling of each successive iteration of the fractal being smaller. When fractals are iterated many times, the perimeter of the fractal increases, while the area will never exceed a certain value. A fractal in three-dimensional space is similar, however, a difference between fractals in two dimensions and three dimensions, is that a three dimensional fractal will increase in surface area, but never exceed a certain volume.[87] 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.[88]


Applications in technology[edit] Main article: Fractal analysis Fractal antennas[89] Fractal transistor[90] Fractal heat exchangers[91] Digital imaging Architecture[33] Urban growth[92][93] Classification of histopathology slides Fractal landscape or Coastline complexity Detecting 'life as we don't know it' by fractal analysis[94] 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[29] Neuroscience[24][25] Diagnostic Imaging[28] Pathology[95][96] Geology[97] Geography[98] Archaeology[99][100] Soil mechanics[27] Seismology[30] Search and rescue[101] Technical analysis[102] Morton order space filling curves for GPU cache coherency in texture mapping,[103][104][105] rasterisation[106][107] and indexing of turbulence data.[108][109]


See also[edit] Mathematics portal Banach fixed point theorem Bifurcation theory Box counting Cymatics Diamond-square algorithm Droste effect Feigenbaum function Form constant Fractal cosmology Fractal derivative Fractalgrid Fractal string Fracton Graftal Greeble Lacunarity List of fractals by Hausdorff dimension Mandelbulb Mandelbox Macrocosm and microcosm Multifractal system Multiplicative calculus Newton fractal Percolation Power law Publications in fractal geometry Random walk Self-reference Strange loop Turbulence Wiener process


Notes[edit] ^ 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. ^ 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.


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"Fractal Analysis: Revisiting Pollock's Paintings (Reply)". Nature, Brief Communication Arising. 444: E10–11.  ^ Shamar, L. (2015). "What Makes a Pollock Pollock: A Machine Vision Approach" (PDF). International Journal of Arts and Technology. 8: 1–10. doi:10.1504/IJART.2015.067389.  ^ 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.  ^ Frame, Michael; and Mandelbrot, Benoît B.; A Panorama of Fractals and Their Uses ^ Nelson, Bryn; Sophisticated Mathematics Behind African Village Designs Fractal patterns use repetition on large, small scale, San Francisco Chronicle, Wednesday, February 23, 2009 ^ 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 ^ Rulistia, Novia D. (2015-10-06). 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Further reading[edit] 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. Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 978-0-19-850839-7.  Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric; 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." 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."Physics and Fractal Structures" (in French). Jfgouyet.fr. Retrieved 2010-10-17.  Bunde, Armin; Havlin, Shlomo (1996). Fractals and Disordered Systems. Springer.  Bunde, Armin; Havlin, Shlomo (1995). Fractals in Science. Springer.  ben-Avraham, Daniel; Havlin, Shlomo (2000). Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press.  Falconer, Kenneth (2013). Fractals, A Very Short Introduction. Oxford University Press. 


External links[edit] Wikimedia Commons has media related to Fractal. Wikibooks has a book on the topic of: Fractals Fractals Scaling and Fractals presented by Shlomo Havlin, Bar-Ilan University Hunting the Hidden Dimension, PBS NOVA, first aired August 24, 2011 Benoit Mandelbrot: Fractals and the Art of Roughness, TED, February 2010 Technical Library on Fractals for controlling fluid Equations of self-similar fractal measure based on the fractional-order calculus(2007) v t e Fractals Characteristics Fractal dimensions Assouad Box-counting Correlation Hausdorff Packing Topological Recursion Self-similarity Iterated function system Barnsley fern Cantor set Dragon curve Koch snowflake Menger sponge Sierpinski carpet Sierpinski triangle Space-filling curve T-square n-flake Strange attractor Multifractal system L-system Fractal canopy Space-filling curve H tree Escape-time fractals Burning Ship fractal Julia set Filled Lyapunov fractal Mandelbrot set Newton fractal Random fractals Brownian motion Brownian tree Diffusion-limited aggregation Fractal landscape Lévy flight Percolation theory Self-avoiding walk People Georg Cantor Felix Hausdorff Gaston Julia Helge von Koch Paul Lévy Aleksandr Lyapunov Benoit Mandelbrot Lewis Fry Richardson Wacław Sierpiński Other "How Long Is the Coast of Britain?" Coastline paradox List of fractals by Hausdorff dimension The Beauty of Fractals (1986 book) Fractal art Chaos: Making a New Science (1987 book) The Fractal Geometry of Nature (1982 book) v t e Chaos theory Chaos theory Anosov diffeomorphism Bifurcation theory Butterfly effect Chaos theory in organizational development Complexity Control of chaos Dynamical system Edge of chaos Fractal Predictability Quantum chaos Santa Fe Institute Synchronization of chaos Unintended consequences Chaotic maps (list) Arnold tongue Arnold's cat map Baker's map Complex quadratic map Complex squaring map Coupled map lattice Double pendulum Double scroll attractor Duffing equation Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Kaplan–Yorke map Logistic map Lorenz system Multiscroll attractor Rabinovich–Fabrikant equations Rössler attractor Standard map Swinging Atwood's machine Tent map Tinkerbell map Van der Pol oscillator Zaslavskii map Chaos systems Bouncing ball dynamics Chua's circuit Economic bubble FPUT problem Tilt-A-Whirl Chaos theorists Michael Berry Mary Cartwright Leon O. 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Fractal (disambiguation)Mandelbrot SetSelf-similarMathematicsAbstract ObjectSelf-similarMenger SpongeMandelbrot SetGeometric FiguresScaling (geometry)PolygonAreaVolumeIntegerFractal DimensionTopological DimensionDifferentiable FunctionTopological DimensionEnlargeSierpinski CarpetRecursionContinuous FunctionDifferentiable FunctionBernard BolzanoBernhard RiemannKarl WeierstrassBenoit MandelbrotLatinFractal DimensionPatterns In NatureHausdorff DimensionTopological DimensionFractal DimensionIterationList Of Fractals By Hausdorff DimensionArchitectureChaos TheoryFractal ArtInfinite RegressHomunculusFractal DimensionShapesReasonFractal DimensionDifferentiable FunctionRectifiable CurveEnlargeKoch SnowflakeMathematicsGottfried LeibnizRecursionSelf-similarityStraight LineKarl WeierstrassWeierstrass FunctionGraph Of A FunctionIntuition (knowledge)Continuous FunctionNowhere DifferentiableGeorg CantorSubsetCantor SetFelix KleinHenri PoincaréEnlargeJulia SetEnlargeSierpinski TriangleHelge Von KochKoch SnowflakeWacław SierpińskiSierpinski TriangleSierpinski CarpetPierre FatouGaston JuliaComplex NumbersStrange AttractorsFelix HausdorffPaul Lévy (mathematician)Lévy C CurveEnlargeStrange AttractorMultifractalEnlargeEnlargeBenoit MandelbrotHow Long Is The Coast Of Britain? 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Statistical Self-Similarity And Fractional DimensionMultifractalEmergent PropertiesEuclidean GeometryRecursionHausdorff DimensionTopological DimensionWiener ProcessBrownian MotionEnlargeIn SilicoL-systemsFractal-generating SoftwareButterfly EffectPredictabilityIterated Function SystemsKoch SnowflakeCantor SetSierpinski CarpetSierpinski GasketPeano CurveDragon CurveT-Square (fractal)Menger SpongeStrange AttractorLogistic MapL-systemTurtle GraphicsSpace-filling CurvesFormulaRecurrence RelationComplex PlaneMandelbrot SetJulia SetBurning Ship FractalNova FractalLyapunov FractalLévy FlightPercolation TheorySelf-avoiding WalkFractal LandscapesBrownian MotionBrownian TreeDiffusion-limited AggregationEnlargeFinite Subdivision RuleAlternating LinkFinite Subdivision RuleTopologicalCell DivisionCantor SetSierpinski CarpetBarycentric SubdivisionEnlargeFractal FlameFractal AnalysisArtifact (error)Circadian RhythmIn SilicoFractal-generating SoftwarePatterns In NatureAlgorithmL-systemsFrondFernPatterns In NatureRiverFault LineMountainCrater (disambiguation)LightningTreeAlgaAnimal ColorationRomanesco BroccoliPineappleHeart SoundsSnowflakePsychological Subjective PerceptionCrystalBlood VesselPulmonary VesselsWind WaveDNARings Of SaturnProteinsTurbulenceFrost Crystals Occurring Naturally On Cold Glass Form Fractal PatternsFractal Basin Boundary In A Geometrical Optical SystemA Fractal Is Formed When Pulling Apart Two Glue-covered Acrylic SheetsAcryloyl GroupHigh Voltage Breakdown Within A 4 in (100 mm) Block Of Acrylic Creates A Fractal Lichtenberg FigureLichtenberg FigureRomanesco Broccoli, Showing Self-similar Form Approximating A Natural FractalRomanesco BroccoliSelf-similarFractal Defrosting Patterns, Polar Mars. The Patterns Are Formed By Sublimation Of Frozen CO2. Width Of Image Is About A Kilometer.Slime Mold Brefeldia Maxima Growing Fractally On WoodSlime MoldBrefeldia MaximaFractal ArtMathematics And ArtJackson PollockDecalcomaniaMax ErnstRon EglashAfrican ArtGameDivinationTradeArchitectureHokky SitungkirIndonesiaBatikOrnament (art)Michael SilverblattDavid Foster WallaceInfinite JestSierpinski TriangleA Fractal That Models The Surface Of A Mountain (animation)3D Recursive ImageRecursive Fractal Butterfly ImageIon PropulsionFractal AnalysisFractal AntennaCategorisationHistopathologyFractal LandscapeCoastComplexityMichaelis-Menten KineticsAlgorithmic CompositionSignal (information Theory)Fractal CompressionFractal In Soil MechanicsGame DesignComputer GraphicsLifeProcedural GenerationFracture MechanicsSAXST-shirtFashionMARPATDigital SundialFractal Dimension On NetworksMedicineNeuroscienceDiagnostic ImagingPathologyGeologyGeographyArchaeologySoil MechanicsSeismologySearch And RescueTechnical AnalysisMorton OrderGPUCache CoherencyTexture MappingRasterisationPortal:MathematicsBanach Fixed Point TheoremBifurcation TheoryBox CountingCymaticsDiamond-square AlgorithmDroste EffectFeigenbaum FunctionForm ConstantFractal CosmologyFractal DerivativeFractalgridFractal StringFractonGraftalGreebleLacunarityList Of Fractals By Hausdorff DimensionMandelbulbMandelboxMacrocosm And MicrocosmMultifractal SystemMultiplicative CalculusNewton FractalPercolationPower LawList Of Important Publications In MathematicsRandom WalkSelf-referenceStrange LoopTurbulenceWiener ProcessHomeomorphismDigital Object IdentifierInternational Standard Book NumberSpecial:BookSources/978-0-387-94153-0International Standard Book NumberSpecial:BookSources/978-0-7167-1186-5International Standard Book NumberSpecial:BookSources/0-470-84862-6International Standard Book NumberSpecial:BookSources/0-500-27693-5International Standard Book NumberSpecial:BookSources/978-981-02-0668-0Gerald L. AlexandersonInternational Standard Book NumberSpecial:BookSources/978-1-56881-340-0International Standard Book NumberSpecial:BookSources/978-0-387-20158-0International Standard Book NumberSpecial:BookSources/978-1-84046-123-7Digital Object IdentifierInternational Standard Book NumberSpecial:BookSources/978-0-8133-4153-8International Standard Book NumberSpecial:BookSources/978-1-4899-2124-6International Standard Book NumberSpecial:BookSources/978-0-387-74749-1International Standard Book NumberSpecial:BookSources/0-471-13938-6BibcodeDigital Object IdentifierBibcodeDigital Object IdentifierBibcodeDigital Object IdentifierCello Suites (Bach)Digital Object IdentifierDigital Object IdentifierPubMed CentralPubMed IdentifierShlomo HavlinBibcodeDigital Object IdentifierPubMed CentralPubMed IdentifierDigital Object IdentifierInternational Standard Book NumberSpecial:BookSources/9780731705054OCLCDigital Object IdentifierDigital Object IdentifierPubMed CentralPubMed IdentifierInternational Standard Book NumberSpecial:BookSources/978-3-7643-7172-2BibcodeDigital Object IdentifierDigital Object IdentifierSocial Science Research NetworkInternational Standard Book NumberSpecial:BookSources/978-1-4027-5796-9BibcodeDigital Object IdentifierPubMed IdentifierInternational Standard Book NumberSpecial:BookSources/978-0-306-44702-0International Standard Book NumberSpecial:BookSources/978-0-387-74748-4Digital Object IdentifierInternational Standard Book NumberSpecial:BookSources/978-0-19-530059-8International Standard Book NumberSpecial:BookSources/978-0-444-50002-1International Standard Book NumberSpecial:BookSources/978-3-7643-7172-2International Standard Book NumberSpecial:BookSources/981-02-3792-8International Standard Book NumberSpecial:BookSources/978-981-02-3792-9Digital Object IdentifierBibcodeDigital Object IdentifierInternational Standard Book NumberSpecial:BookSources/978-3-540-40754-6International Standard Book NumberSpecial:BookSources/978-2-86332-130-0International Standard Book NumberSpecial:BookSources/978-981-02-3792-9International Standard Book NumberSpecial:BookSources/978-0-7503-0400-9International Standard Book NumberSpecial:BookSources/9780719034343Digital Object IdentifierBibcodeDigital Object IdentifierBibcodeDigital Object IdentifierBibcodeDigital Object IdentifierBibcodeDigital Object IdentifierArXivBibcodeDigital Object IdentifierDigital Object IdentifierDigital Object IdentifierBibcodeDigital Object IdentifierDigital Object IdentifierBibcodeDigital Object IdentifierBibcodeDigital Object IdentifierBibcodeDigital Object IdentifierDigital Object IdentifierDigital Object IdentifierBibcodeDigital Object IdentifierDigital Object IdentifierInternational Standard Book NumberSpecial:BookSources/978-979-22-4484-7International Standard Book NumberSpecial:BookSources/978-1-4939-3995-4Digital Object IdentifierDigital Object IdentifierDigital Object IdentifierDigital Object IdentifierInternational Standard Book NumberSpecial:BookSources/978-1-4577-1596-9ArXivBibcodeDigital Object IdentifierPubMed CentralPubMed IdentifierDigital Object IdentifierPubMed IdentifierDigital Object IdentifierPubMed IdentifierQiuming ChengDigital Object IdentifierArXivBibcodeDigital Object IdentifierPubMed CentralPubMed IdentifierPubMed IdentifierDigital Object IdentifierInternational Standard Book NumberSpecial:BookSources/978-988-17-0125-1Digital Object IdentifierInternational Standard Book NumberSpecial:BookSources/978-0-387-75888-6ArXivBibcodeDigital Object IdentifierInternational Standard Book NumberSpecial:BookSources/0-12-079061-0International Standard Book NumberSpecial:BookSources/978-3-8376-2829-6International Standard Book NumberSpecial:BookSources/0-471-92287-0Heinz-Otto PeitgenInternational Standard Book NumberSpecial:BookSources/0-387-97903-4Benoit MandelbrotThe Fractal Geometry Of NatureInternational Standard Book NumberSpecial:BookSources/0-7167-1186-9International Standard Book NumberSpecial:BookSources/0-387-96608-0Clifford A. PickoverInternational Standard Book NumberSpecial:BookSources/0-444-50002-2International Standard Book NumberSpecial:BookSources/1-878739-46-8International Standard Book NumberSpecial:BookSources/0-691-08551-XInternational Standard Book NumberSpecial:BookSources/0-691-02445-6International Standard Book NumberSpecial:BookSources/978-0-19-850839-7International Standard Book NumberSpecial:BookSources/0-201-62630-6International Standard Book NumberSpecial:BookSources/1-904555-05-5Arthur C. ClarkeMandelbrot SetInternational Standard Book NumberSpecial:BookSources/9787535722348International Standard Book NumberSpecial:BookSources/2-225-85130-1International Standard Book NumberSpecial:BookSources/978-0-387-94153-0Shlomo HavlinBar-Ilan UniversityPBSNova (TV Series)TED (conference)Template:FractalsTemplate Talk:FractalsFractal DimensionAssouad DimensionMinkowski–Bouligand DimensionCorrelation DimensionHausdorff DimensionPacking DimensionLebesgue Covering DimensionRecursionSelf-similarityBarnsley FernIterated Function SystemBarnsley FernCantor SetDragon CurveKoch SnowflakeMenger SpongeSierpinski CarpetSierpinski TriangleSpace-filling CurveT-square (fractal)N-flakeStrange AttractorMultifractal SystemL-systemFractal CanopySpace-filling CurveH TreeFractalBurning Ship FractalJulia SetFilled Julia SetLyapunov FractalMandelbrot SetNewton FractalBrownian MotionBrownian TreeDiffusion-limited AggregationFractal LandscapeLévy FlightPercolation TheorySelf-avoiding WalkGeorg CantorFelix HausdorffGaston JuliaHelge Von KochPaul Lévy (mathematician)Aleksandr LyapunovBenoit MandelbrotLewis Fry RichardsonWacław SierpińskiHow Long Is The Coast Of Britain? Statistical Self-Similarity And Fractional DimensionCoastline ParadoxList Of Fractals By Hausdorff DimensionThe Beauty Of FractalsFractal ArtChaos: Making A New ScienceThe Fractal Geometry Of NatureTemplate:Chaos TheoryTemplate Talk:Chaos TheoryChaos TheoryAnosov DiffeomorphismBifurcation TheoryButterfly EffectChaos Theory In Organizational DevelopmentComplexityControl Of ChaosDynamical SystemEdge Of ChaosPredictabilityQuantum ChaosSanta Fe InstituteSynchronization Of ChaosUnintended ConsequencesConus Textile ShellCircle Map With Black Arnold TonguesChaos TheoryList Of Chaotic MapsArnold TongueArnold's Cat MapBaker's MapComplex Quadratic PolynomialComplex Squaring MapCoupled Map LatticeDouble PendulumDouble Scroll AttractorDuffing EquationDuffing MapDyadic TransformationDynamical BilliardsOuter BilliardExponential Map (discrete Dynamical Systems)Gauss Iterated MapGingerbreadman MapHénon MapHorseshoe MapIkeda MapInterval Exchange TransformationKaplan–Yorke MapLogistic MapLorenz SystemMultiscroll AttractorRabinovich–Fabrikant EquationsRössler AttractorStandard MapSwinging Atwood's MachineTent MapTinkerbell MapVan Der Pol OscillatorZaslavskii MapBouncing Ball DynamicsChua's CircuitEconomic BubbleFermi–Pasta–Ulam–Tsingou ProblemTilt-A-WhirlMichael Berry (physicist)Mary CartwrightLeon O. 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C. EscherCircle Limit IIIPrint Gallery (M. C. Escher)Relativity (M. C. Escher)Reptiles (M. C. Escher)Waterfall (M. C. Escher)Martin DemaineErik DemaineScott DravesJan DibbetsJohn ErnestHelaman FergusonPeter ForakisBathsheba GrossmanGeorge W. HartDesmond Paul HenryJohn A. HiigliAnthony Hill (artist)Charles JencksGarden Of Cosmic SpeculationRobert LonghurstIstván OroszHinke OsingaHamid Naderi YeganehA Bird In FlightBoat (drawing)Tony RobbinOliver SinHiroshi SugimotoDaina TaiminaRoman VerostkoPolykleitosVitruviusDe ArchitecturaLuca PacioliDe Divina ProportionePiero Della FrancescaDe Prospectiva PingendiLeon Battista AlbertiDe PicturaDe Re AedificatoriaSebastiano SerlioAndrea PalladioI Quattro Libri Dell'architetturaAlbrecht DürerFrederik Macody LundJay HambidgeSamuel ColmanOwen Jones (architect)The Grammar Of OrnamentErnest Hanbury HankinG. H. HardyA Mathematician's ApologyGeorge David BirkhoffDouglas HofstadterGödel, Escher, BachNikos SalingarosJournal Of Mathematics And The ArtsArs Mathematica (organization)The Bridges OrganizationEuropean Society For Mathematics And The ArtsGoudreau Museum Of Mathematics In Art And ScienceInstitute For FiguringMuseum Of MathematicsDroste EffectMathematical BeautyPatterns In NatureSacred GeometryCategory:Mathematics And ArtTemplate:Patterns In NaturePatterns In NatureFractureDuneFoamMeanderPhyllotaxisSoap BubbleSymmetryCrystal SymmetryQuasicrystalsFloral SymmetrySymmetry In BiologyTessellationVortex StreetWaveWidmanstätten PatternPattern FormationBiologyNatural SelectionCamouflageMimicrySexual SelectionMathematicsChaos TheoryLogarithmic SpiralPhysicsCrystalFluid DynamicsPlateau's LawsSelf-organizationPlatoPythagorasEmpedoclesLeonardo FibonacciLiber AbaciAdolf ZeisingErnst HaeckelJoseph PlateauWilson BentleyD'Arcy Wentworth ThompsonOn Growth And FormAlan TuringThe Chemical Basis Of MorphogenesisAristid LindenmayerBenoit MandelbrotHow Long Is The Coast Of Britain? 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