Fractal

Inmathematics, afractalis ageometric shapecontaining detailed structure at arbitrarily small scales, usually having afractal dimensionstrictly exceeding thetopological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of theMandelbrot set.This exhibition of similar patterns at increasingly smaller scales is calledself-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in theMenger sponge, the shape is calledaffineself-similar.Fractal geometry lies within the mathematical branch ofmeasure theory.
One way that fractals are different from finitegeometric figuresis how theyscale. Doubling the edge lengths of a filledpolygonmultiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if theradiusof a filled sphere is doubled, itsvolumescales by eight, which is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). 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 anintegerand is in general greater than its conventional dimension.This power is called thefractal dimensionof the geometric object, to distinguish it from the conventional dimension (which is formally called thetopological dimension).
Analytically, many fractals are nowheredifferentiable.An infinitefractal curvecan be conceived of as winding through space differently from an ordinary line – although it is stilltopologically 1-dimensional, its fractal dimension indicates that it locally fills space more efficiently than an ordinary line.
Starting in the 17th century with notions ofrecursion, fractals have moved through increasingly rigorous mathematical treatment to the study ofcontinuousbut notdifferentiablefunctions in the 19th century by the seminal work ofBernard Bolzano,Bernhard Riemann, andKarl Weierstrass,and on to the coining of the wordfractalin the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century.
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."More formally, in 1982 Mandelbrot definedfractalas follows: "A fractal is by definition a set for which theHausdorff–Besicovitch dimensionstrictly exceeds thetopological dimension."Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or fragmentedgeometric shapethat can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole."Still later, Mandelbrot proposed "to usefractalwithout a pedantic definition, to usefractal dimensionas a generic term applicable toallthe variants".
The consensus among mathematicians is that theoretical fractals are infinitely self-similariteratedand detailed mathematical constructs, of which manyexampleshave been formulated and studied.Fractals are not limited to geometric patterns, but can also describe processes in time.Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural mediaand found innature,technology,art,andarchitecture.Fractals are of particular relevance in the field ofchaos theorybecause they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction).
The term "fractal" was coined by the mathematicianBenoît Mandelbrotin 1975.Mandelbrot based it on the Latinfrāctus, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractionaldimensionsto geometricpatterns in nature.
The word "fractal" often has different connotations for mathematicians and the general public, where the public is more likely to be familiar withfractal artthan 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 theinfinite regressin parallel mirrors or thehomunculus, 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.
This idea of being detailed relates to another feature that can be understood without much mathematical background: Having afractal dimensiongreater than its topological dimension, for instance, refers to how a fractal scales compared to how geometricshapesare usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure isrep-tiledinto 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 3= 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 ofrpieces. Now, consider theKoch 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 numberDthat satisfies 3= 4. This number is called thefractal dimensionof 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 theconventionally understooddimension (formally called the topological dimension).
This also leads to understanding a third feature, that fractals as mathematical equations are "nowheredifferentiable". 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 ofmeasuringwith 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.
The history of fractals traces a path from chiefly theoretical studies to modern applications incomputer graphics, with several notable people contributing canonical fractal forms along the way.A common theme in traditionalAfrican architectureis 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.According toPickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopherGottfried Leibnizponderedrecursiveself-similarity(although he made the mistake of thinking that only thestraight linewas self-similar in this sense).
In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them.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, 1872Karl Weierstrasspresented the first definition of afunctionwith agraphthat would today be considered a fractal, having the non-intuitiveproperty of being everywherecontinuousbutnowhere differentiableat the Royal Prussian Academy of Sciences.
In addition, the quotient difference becomes arbitrarily large as the summation index increases.Not long after that, in 1883,Georg Cantor, who attended lectures by Weierstrass,published examples ofsubsetsof the real line known asCantor sets, which had unusual properties and are now recognized as fractals.Also in the last part of that century,Felix KleinandHenri Poincaréintroduced a category of fractal that has come to be called "self-inverse" fractals.
One of the next milestones came in 1904, whenHelge 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 theKoch snowflake.Another milestone came a decade later in 1915, whenWacław Sierpińskiconstructed his famoustrianglethen, one year later, hiscarpet. By 1918, two French mathematicians,Pierre FatouandGaston Julia, though working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mappingcomplex numbersand iterative functions and leading to further ideas aboutattractors 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 Hausdorffexpanded 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 byPaul Lévy, who, in his 1938 paperPlane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, theLévy C curve.
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).That changed, however, in the 1960s, whenBenoit Mandelbrotstarted writing about self-similarity in papers such asHow Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,which built on earlier work byLewis 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 canonicalMandelbrot set, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".
In 1980,Loren Carpentergave a presentation at theSIGGRAPHwhere he introduced his software for generating and rendering fractally generated landscapes.
One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmentedgeometric shapethat 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 offractal, but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in.
One point agreed on is that fractal patterns are characterized byfractal dimensions, but whereas these numbers quantifycomplexity(i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns.In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whoseHausdorff–Besicovitch dimensionis greater than itstopological dimension.However, this requirement is not met byspace-filling curvessuch as theHilbert curve.
Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According toFalconer, fractals should be only generally characterized by agestaltof the following features;
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.
Images of fractals can be created byfractal generating programs. Because of thebutterfly effect, a small change in a single variable can have anunpredictableoutcome.
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 ornatural phenomena with fractal features. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks forfractal analysis. Some specific applications of fractals to technology are listedelsewhere. 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 displayartifactswhich are not characteristics of true fractals.
Modeled fractals may be sounds,digital images, electrochemical patterns,circadian rhythms,etc.
Fractal patterns have been reconstructed in physical 3-dimensional spaceand virtually, often called "in silico" modeling.Models of fractals are generally created usingfractal-generating softwarethat implements techniques such as those outlined above.As one illustration, trees, ferns, cells of the nervous system,blood and lung vasculature,and other branchingpatterns in naturecan be modeled on a computer by using recursivealgorithmsandL-systemstechniques.
The recursive nature of some patterns is obvious in certain examples—a branch from a tree or afrondfrom afernis 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.
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.Phenomena known to have fractal features include:
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 andbranching.Nerve cellsfunction 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.These processes are crucial in cellphysiologyand differentpathologies.
Multiple subcellular structures also are found to assemble into fractals.Diego Krapfhas shown that through branching processes theactinfilaments in human cells assemble into fractal patterns.Similarly Matthias Weiss showed that theendoplasmic reticulumdisplays fractal features.The current understanding is that fractals are ubiquitous in cell biology, fromproteins, toorganelles, to whole cells.
Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created byJackson Pollockby pouring paint directly onto horizontal canvasses.
Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks.Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals.
Decalcomania, a technique used by artists such asMax Ernst, can produce fractal-like patterns.It involves pressing paint between two surfaces and pulling them apart.
CyberneticistRon Eglashhas suggested that fractal geometry and mathematics are prevalent inAfrican 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.Hokky Situngkiralso suggested the similar properties in Indonesian traditional art,batik, andornamentsfound in traditional houses.
Ethnomathematician Ron Eglash has discussed the planned layout ofBenin cityusing 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."
In a 1996 interview withMichael Silverblatt,David Foster Wallaceexplained that the structure of the first draft ofInfinite Jesthe gave to his editor Michael Pietsch was inspired by fractals, specifically theSierpinski triangle(a.k.a. Sierpinski gasket), but that the edited novel is "more like a lopsided Sierpinsky Gasket".
Some works by the Dutch artistM. C. Escher, such asCircle 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: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.
Humans appear to be especially well-adapted to processing fractal patterns withfractal dimensionbetween 1.3 and 1.5.When humans view fractal patterns with fractal dimension between 1.3 and 1.5, this tends to reduce physiological stress.