Physicist: There are a couple of different contexts in which the word “dimension” comes up. In the case of fractals the number of dimensions has to do (this is a little handwavy) with the way points in the fractal are distributed. For example, if you have points distributed at random in space you’d say you have a threedimensional set of points, and if they’re all arranged on a flat sheet you’d say you have twodimensional set of points. Way back in the day mathematicians figured that a good way to determine the “dimensionality” of a set is to pick a point in the set, and then put progressively larger and larger spheres of radius R around it. If the number of points contained in the sphere is proportional to R^{d}, then the set is ddimensional.
(Left) as the sphere grows, the number of points from the line that it contains increases like R, so the line is onedimensional. (Right) as the sphere increases in size the number of points from the rectangle that it contains increases like R^{2}, so the square is twodimensional.
However, there’s a problem with this technique. You can have a set that’s really ddimensional, but on a large scale it appears to be a different dimension. For example, a piece of paper is basically 2D, but if you crumple it up into a ball it seems 3D on a large enough scale. A hairball or bundle of cables seems 3D (by the “sphere test”), but they’re really 1D (Ideally at least. Every physical object is always 3D).
A “crumpled up” set seems like it has a higher dimension than it really does. You can get around this by using smaller and smaller spheres. Eventually you’ll get the correct dimension.
This whole “look at the number of points inside of tiny spheres and see how that number scales with size” thing works great for every halfway reasonable sets. However, fractal sets can be “infinitely crumpled”, so no matter how small a sphere you use, you still get a dimension larger than you might expect.
The edge of the Mandelbrot set “should” be onedimensional since it’s just a line. However, it’s infinitely twisty, and no matter how much you zoom in it stays just as messed up.
When the “sphere trick” is applied to tangled messes it doesn’t necessarily have to give you integer numbers until the spheres are small enough. With fractals there is no “small enough” (that should totally be a terrible movie tag line), and you find that they have a dimension that’s often a fraction. The dimension of the Mandelbrot’s boundary (picture above) is 2, which is the highest it can be, but there are more interesting (but less pretty) fractals out there with genuinely fractional dimensions, like the “Koch snowflake” which has a dimension of approximately 1.262.
The Koch snowflake, which is so messed up that it has a dimension greater than 1, but less than 2.
That all said, when somebody (looking at you, all mathematicians) talks about fractional dimensions, they’re really talking about a weird, abstract, and not at all physical notion of dimension. There’s no such thing as “2.5 dimensional universe”. When we talk about the “dimension of space” we’re talking about the number of completely different directions that are available, not the whole “sphere thing”. The dimensions of space are either there or not, so while you could have 4 dimensions, you couldn’t have 3.5.
 Benoit Mandelbrot
 Benoit Mandelbrot
 Chaos theory
 cryptography
 Dimension
 Eugène Delacroix
 finance
 Fractal
 Fractal art
 Fractals
 Gaston Julia
 Gaston Julia
 How Long Is the Coast of Britain? Statistical SelfSimilarity and Fractional Dimension
 IBM
 Institute for Advanced Study
 Isaac Newton
 Leonardo Da Vinci
 machine translation
 Mandelbrot set
 Mathematical analysis
 Mathematics
 Paris
 Science and Technology Foundation of Japan
 Szolem
 telephone lines
 The Fractal Geometry of Nature
 United Kingdom
 Warsaw
 written graphics software
In a world made small and accessible by technology, it is easy to forget the magnitude of nature’s infinite complexity. But sometimes technology reminds us, such as when trawling planet Earth on Google’s Satellite View, zooming across landscapes partitioned by natural and unnatural boundaries.
While searching Google Earth, Paul Bourke, a research associate professor at the University of Western Australia, discovered an amazing sight—the patterns of the Earth seemed to form a delicate geometric pattern when viewed from the sky. Not only delicate, but almost perfect. Bourke was captivated by the geography—lacy tracks of rivers and mountain ranges stretching across the Earth in unison as if digitally cloned.
Fractals are recognized as patterns of selfsimilarity over varying degrees of scale. There are both mathematical fractals as well as natural fractals—the former are idealized and found across a range of scales, while the latter generally only exist across a smaller scale range.
Bourke explains that fractals are found in all parts of life, from the brain sciences and astrophysics to geographic formations and riverbeds. “Fractal and chaotic processes are the norm, not the exception.”
“I always knew these amazing natural patterns would be there,” he said. “They are literally everywhere—it’s just a matter of finding them.”
And find them he did. Bourke, an authority on fractals and visualizations, showcases more than 40 different fractals he’s uncovered while zooming through the satellite views of 25 countries. Through his website, he encourages users to submit examples they’ve found in their own browsing, and provides KMZ coordinate files for each image, allowing users to visit the exact views of the fractal features. Bourke’s collection realizes the power enabled by the openended tools of modern technology and applies them to a practical and popular aesthetic end.
To see more natural fractal patterns, visit Bourke’s website.
 Alaska
 Alec Soth
 Alec Soth
 Australia
 Austria
 China
 Digital art
 Dimension
 earth
 Egypt
 Fractal
 Fractal
 Fractals
 Google Earth
 Greenland
 Landscape
 Lord Soth
 Malaysia
 Man on the Wire
 Mathematical analysis
 Mathematics
 Mexico
 Namibia
 Norway
 pattern
 Paul Bourke
 Russia
 satellite views
 Science
 Selfsimilarity
 space
 Spain
 Topology
 University of Western Australia
 Western Australia
 Western Australia
From this analysis emerged records of 18,520 sub950millisecond crashes and spikes — far more than they, and perhaps almost anyone, expected. Equally as striking as these events’ frequency was their arrangement: While market behavior tends to rise and fall in patterns that repeat themselves, fractalstyle, in periods of days, weeks, months and years, “that only holds down to the time scale at which human stop being able to respond,” said Johnson. “The fractal gets broken.”
http://www.wired.com/wiredscience/2012/02/highspeedtrading/
A piece generated in Apophysis.
Showing breakage in space in non integer Multibrot set
A Fibonacci word fractal by Samuel Monnier
Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still images, animations, and media. Fractal art developed from the mid 1980s onwards.^{[1]} It is a genre of computer art and digital art which are part of new media art. The Julia set and Mandlebrot sets can be considered as icons of fractal art.^{[2]}
Fractal art is not drawn or painted by hand. It is usually created indirectly with the assistance of fractalgenerating software, iterating through three phases: setting parameters of appropriate fractal software; executing the possibly lengthy calculation; and evaluating the product. In some cases, other graphics programs are used to further modify the images produced. This is called postprocessing. Nonfractal imagery may also be integrated into the artwork.^{[3]}
Fractal art could not have developed without computers because of the calculative capabilities they provide.^{[4]} Fractals are generated by applying iterative methods to solving nonlinear equations or polynomial equations. Fractals are any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size.^{[5]}
The Fractal Art Manifesto
As stated by Kerry Mitchell in The Fractal Art Manifesto,^{[6]} "Fractal Art is a subclass of twodimensional visual art, and is in many respects similar to photography—another art form that was greeted by skepticism upon its arrival. Fractal images typically are manifested as prints, bringing fractal artists into the company of painters, photographers, and printmakers. Fractals exist natively as electronic images. This is a format that traditional visual artists are quickly embracing, bringing them into Fractal Art's digital realm. Generating fractals can be an artistic endeavor, a mathematical pursuit, or just a soothing diversion. However, Fractal Art is clearly distinguished from other digital activities by what it is, and by what it is not." According to Mitchell, fractal art is not computerized art, lacking in rules, unpredictable, nor something that any person with access to a computer can do well. Instead, fractal art is expressive, creative, and requires input, effort, and intelligence. Most importantly, "fractal art is simply that which is created by Fractal Artists: ART."
Types
A 3D fractal generated using Visions of Chaos
There are many different kinds of fractal images and can be subdivided into several groups.
 Fractals derived from standard geometry by using iterative transformations on an initial common figure like a straight line (the Cantor dust or the von Koch curve), a triangle (the Sierpinski triangle), or a cube (the Menger sponge). The first fractal figures invented near the end of the 19th and early 20th centuries belong to this group.
 IFS (iterated function systems).
 Strange attractors.
 Fractal flame.
 Lsystem fractals.
 Fractals created by the iteration of complex polynomials: perhaps the most famous fractals.
 Newton fractals, including Nova fractals
 Quaternionic and (recently) hypernionic fractals.
 Fractal terrains generated by random fractal processes.^{[7]}
Fractal expressionism is a term used to differentiate traditional visual art that incorporates fractal elements such as selfsimilarity for example. Perhaps the best example of fractal expressionism is found in Jackson Pollack's dripped patterns. They have been analysed and found to contain a fractal dimension which has been attributed to his technique.^{[8]}
Techniques
Fractal image generated by Electric Sheep
Fractals of all kinds have been used as the basis for digital art and animation. High resolution color graphics became increasingly available at scientific research labs in the mid 1980s. Scientific forms of art, including fractal art, have developed separately from mainstream culture.^{[9]} Starting with 2dimensional details of fractals, such as the Mandelbrot Set, fractals have found artistic application in fields as varied as texture generation, plant growth simulation and landscape generation.
Fractals are sometimes combined with humanassisted evolutionary algorithms, either by iteratively choosing goodlooking specimens in a set of random variations of a fractal artwork and producing new variations, to avoid dealing cumbersome or unpredictable parameters, or collectively, like in the Electric Sheep project, where people use fractal flames rendered with distributed computing as their screensaver and "rate" the flame they are viewing, influencing the server, which reduces the traits of the undesirables, and increases those of the desirables to produce a computergenerated, communitycreated piece of art.
Many fractal images are admired because of their perceived harmony. This is typically achieved by the patterns which emerge from the balance of order and chaos. Similar qualities have been described in Chinese painting and miniature trees and rockeries.^{[10]}
Some of the most popular fractal rendering programs used to make fractal art include Ultra Fractal, Apophysis, Bryce and Sterling. Fractint was the first widely used fractal generating program.
Landscapes
A 3D landscape generated with Terragen
Main article: Fractal landscape
The first fractal image that was intended to be a work of art was probably the famous one on the cover of Scientific American, August 1985. This image showed a landscape formed from the potential function on the domain outside the (usual) Mandelbrot set. However, as the potential function grows fast near the boundary of the Mandelbrot set, it was necessary for the creator to let the landscape grow downwards, so that it looked as if the Mandelbrot set was a plateau atop a mountain with steep sides. The same technique was used a year after in some images in The Beauty of Fractals by HeinzOtto Peitgen and Michael M. Richter.
In this book you can find a formula to estimate the distance from a point outside the Mandelbrot set to the boundary of the Mandelbrot set (and a similar formula for the Julia sets), and one can wonder why the creator did not use this function instead of the potential function, because it grows in a more natural way (see the formula in the articles Mandelbrot set and Julia set).
The three pictures show landscapes formed from the distance function for a family of iterations of the form z^{2} + az^{4} + c. If, in a light from the sun. Then we imagine the rays are parallel (and given by two angles), and we let the colour of a point on the surface be determined by the angle between this direction and the slope of the surface at the point. The intensity (on the earth) is independent of the distance, but the light grows whiter because of the atmosphere, and sometimes the ground looks as if it is enveloped in a veil of mist (second picture). We can also let the light be "artificial", as if it issues from a lantern held by the observer. In this case the colour must grow darker with the distance (third picture).
Artists
The British artist William Latham, has used fractal geometry and other computer graphics techniques in his works.^{[11]} Greg Sams has used fractal designs in postcards, tshirts and textiles. American Vicky BragoMitchell has created fractal art which has appeared in exhibitions and on magazine covers. Scott Draves is credited with inventing flame fractals. Some artists, such as Reginald Atkins, create fractal art for relaxation.^{[3]} Carlos Ginzburg has explored fractal art and developed a concept called "homo fractalus" which is based around the idea that the human is the ultimate fractal.^{[12]} Merrin Parkers from New Zealand specialises in fractal art.^{[13]}
Exhibits
There has been fractal art exhibits at major international art galleries.^{[14]} One of the first exhibitions of fractal art was called Map Art. It was a travelling exhibition of works which originated from researchers at the University of Bremen.^{[15]} Mathematicians HeinzOtto Peitgen and Michael M. Richter discovered the public not only found the images aesthetically pleasing but that they also wanted to understand the scientific background to the images.^{[16]}
In 1989, fractals were part of the subject matter for an art show called Strange Attractors: Signs of Chaos at the New Museum of Contemporary Art.^{[9]} The show consisted of photographs, installations and sculptures designed to provide greater scientific discourse to the field which had already captured the public's attention through colourful and intricate computer imagery.
See also
 Algorithmic art
 animation
 Apophysis
 appropriate fractal software
 art
 Carlos Ginzburg
 Chaos theory
 Computer graphics
 distributed computing
 Fractal
 fractalgenerating software
 fractals
 Fractals
 Fractint
 Greg Sams
 Heinz
 humanassisted evolutionary algorithms
 humanassisted evolutionary algorithms
 Kerry Mitchell
 Mandelbrot set
 Mathematicians HeinzOtto Peitgen
 Mathematics
 media art
 Michael M. Richter
 New Zealand
 Otto Peitgen
 Physics
 postprocessing
 Reginald Atkins
 Scientific American
 Scott Draves
 simulation
 Sterling
 The Beauty of Fractals
 University of Bremen
 Vicky BragoMitchell
 William Latham
A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension^{[1]} and may fall between the integers.^{[2]}. Fractals are typically selfsimilar patterns, where selfsimilar means they are "the same from near as from far"^{[3]} Fractals may be exactly the same at every scale, or as illustrated in Figure 1, they may be nearly the same at different scales.^{[4]}^{[5]}^{[2]}^{[6]} The definition of fractal goes beyond selfsimilarity per se to exclude trivial selfsimilarity and include the idea of a detailed pattern repeating itself.^{[2]}^{:166; 18}^{[4]}^{[7]}
As mathematical equations, fractals are usually nowhere differentiable, which means that they cannot be measured in traditional ways.^{[6]}^{[2]}^{[8]} An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1dimensional line yet having a fractal dimension indicating it also resembles a surface. ^{[2]}^{:15}^{[1]}^{:48}
The mathematical roots of the idea of fractals have been traced through a formal path of published works, starting in the 17th century with notions of recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computerbased modelling in the 21st century.^{[9]}^{[10]} The term "fractal" was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.^{[2]}^{:405}^{[7]}
There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. The general consensus is that theoretical fractals are infinitely selfsimilar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth^{[2]}^{[4]}^{[5]}. Fractals are not limited to geometric patterns, but can also describe processes in time.^{[6]}^{[11]}^{[3]} Fractal patterns with various degrees of selfsimilarity have been rendered or studied in images, structures and sounds^{[12]} and found in nature^{[13]}^{[14]}^{[15]}^{[16]}^{[17]}, technology^{[18]}^{[19]}^{[20]}^{[21]}, and art^{[22]}^{[23]}^{[24]}.
Introduction
The word "fractal" often has different connotations for laypeople than mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception. The mathematical concept is difficult to formally define even for mathematicians, but key features can be understood with little mathematical background.
The feature of "selfsimilarity", 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. Selfsimilarity itself is not necessarily counterintuitive (e.g., people have pondered selfsimilarity 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.^{[2]}^{: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 1dimensional; 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 curve in Figure 2. It is also 1dimensional 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.^{[6]}^{[2]}^{[8]} To elaborate, in trying to find the length of a wavy nonfractal 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 one in Figure 2, one would never find a small enough straight segment to conform to the curve, because the wavy pattern would always reappear, 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. This is perhaps counterintuitive, but it is how fractals behave.^{[2]}
History
Figure 2. Koch snowflake, 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.^{[9]}^{[10]} According to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive selfsimilarity (although he made the mistake of thinking that only the straight line was selfsimilar in this sense).^{[25]} In his writings, Leibnitz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them^{[2]}^{: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".^{[10]}^{[9]}^{[8]} Thus, it was not until two centuries had passed that in 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal, having the nonintuitive property of being everywhere continuous but nowhere differentiable.^{[9]}^{:7}^{[10]} 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]}^{:1124} Also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called "selfinverse" fractals.^{[2]}^{:166}
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 curve (see Figure 2)^{[9]}^{:25}^{[10]}. 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 (see Figure 3 and Figure 4).^{[10]}^{[9]}^{[6]} Very shortly after that work was submitted, by March of 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.^{[10]} The idea of selfsimilar curves was taken further by Paul Pierre 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]}
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 hardly resembling the image in Figure 3).^{[2]}^{:179}^{[8]}^{[10]} That changed, however, in the 1960s, when Benoît Mandelbrot started writing about selfsimilarity in papers such as How Long Is the Coast of Britain? Statistical SelfSimilarity and Fractional Dimension,^{[26]} 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 computerconstructed visualizations. These images, such as of his canonical Mandelbrot set pictured in Figure 1 captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".^{[27]} Currently, fractal studies are essentially exclusively computerbased.^{[25]}^{[9]}^{[8]}
Figure 4. A strange attractor that exhibits multifractal scaling
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 reducedsize copy of the whole"^{[2]}; this is generally helpful but limited. Authorities disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of selfsimilarity and an unusual relationship with the space a fractal is embedded in.^{[2]}^{[6]}^{[28]} ^{[4]}^{[3]} 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.^{[29]} 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 spacefilling curves such as the Hilbert curve.^{[notes 2]}
According to Falconer, rather than being strictly defined, fractals should, in addition to being differentiable and able to have a fractal dimension, be generally characterized by a gestalt of the following features^{[4]}:

 Selfsimilarity, which may be manifested as:

 Exact selfsimilarity: identical at all scales; e.g. Koch snowflake
 Quasi selfsimilarity: 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, as shown in Figure 1
 Statistical selfsimilarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals; the wellknown 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 selfsimilarity: as in a time series^{[11]}
 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^{[30]} (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"^{[1]}.

 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 selfsimilar without having other typically fractal features. A straight line, for instance, is selfsimilar 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.^{[6]}^{[2]}
Common techniques for generating fractals
Figure 5. Selfsimilar branching pattern modeled in silico using Lsystems principles^{[17]}

 Iterated function systems – use fixed geometric replacement rules; may be stochastic or deterministic^{[31]}; e.g., Koch snowflake, Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, HarterHeighway dragon curve, TSquare, Menger sponge

 Strange attractors – use iterations of a map or solutions of a system of initialvalue differential equations that exhibit chaos (e.g., see multifractal image)

 Lsystems  use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells^{[17]}), blood vessels, pulmonary structure^{[32]}, etc. (e.g., see Figure 5) or turtle graphics patterns such as spacefilling curves and tilings

 Escapetime fractals – use a formula or recurrence relation at each point in a space (such as the complex plane); usually quasiselfsimilar; 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 escapetime 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 diffusionlimited aggregation or reactionlimited aggregation clusters).^{[6]}
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^{[12]}, digital images, electrochemical patterns, circadian rhythms^{[33]}, etc. Fractal patterns have been reconstructed in physical 3dimensional space^{[20]}^{:10} and virtually, often called "in silico" modeling^{[32]}. Models of fractals are generally created using fractalgenerating software that implements techniques such as those outlined above.^{[20]}^{[6]}^{[11]} As one illustration, trees, ferns, cells of the nervous system^{[17]}, blood and lung vasculature,^{[32]} and other branching patterns in nature can be modeled on a computer by using recursive algorithms and Lsystems techniques^{[17]}. 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 realworld 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 algorithm.
Natural phenomena with fractal features
Approximate fractals found in nature display selfsimilarity 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.^{[34]}
Examples of phenomena known or anticipated to have fractal features are listed below:
 clouds
 river networks
 fault lines
 mountain ranges
 craters
 lightning bolts
 coastlines
 snow flakes
 various vegetables (cauliflower and broccoli)
 animal coloration patterns.
 heart rates^{[13]}
 heartbeat^{[14]}
 earthquakes^{[21]}^{[35]}
 snow flakes^{[36]}
 crystals^{[37]}
 blood vessels and pulmonary vessels^{[32]},
 ocean waves^{[38]}
 DNA
In creative works
A fractal that models the surface of a mountain (animation)
Further information: Fractal art
Fractal patterns have been found in the paintings of American artist Jackson Pollock. While Pollock's paintings appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in his work.^{[24]}
Decalcomania, a technique used by artists such as Max Ernst, can produce fractallike patterns.^{[39]} It involves pressing paint between two surfaces and pulling them apart.
Cyberneticist Ron Eglash has suggested that fractallike structures are prevalent in African art 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.^{[23]}^{[40]}
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 (aka Sierpinski gasket) but that the edited novel is "more like a lopsided Sierpinsky Gasket".^{[22]}
Applications in technology
Main article: Fractal analysis
 fractal antennas^{[41]}
 digital imaging
 urban growth^{[42]}^{[43]}
 Classification of histopathology slides
 Fractal landscape or Coastline complexity
 Enzyme/enzymology (MichaelisMenten kinetics)
 Generation of new music
 Signal and image compression
 Creation of digital photographic enlargements
 Seismology
 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
 Tshirts and other fashion
 Generation of patterns for camouflage, such as MARPAT
 Digital sundial
 Technical analysis of price series
 Fractals in networks
 medicine^{[20]}
 neuroscience^{[16]}^{[15]}
 diagnostic imaging^{[19]}
 pathology^{[44]}^{[45]}
 geology^{[46]}
 geography^{[47]}
 archaeology^{[48]}^{[49]}
 soil mechanics^{[18]}
 seismology^{[21]}
 search and rescue^{[50]}
 technical analysis^{[51]}
See also
 Banach fixed point theorem
 Bifurcation theory
 Box counting
 Butterfly effect
 Complexity
 Constructal theory
 Cymatics
 Diamondsquare algorithm
 Droste effect
 Feigenbaum function
 Fractal compression
 Fractal cosmology
 Fractal networks
 Fractint
 Fracton
 Golden ratio
 Graftal
 Greeble
 Lacunarity
 List of fractals by Hausdorff dimension
 Publications in fractal geometry
 Mandelbulb
 Multifractal system
 Newton fractal
 Percolation
 Power law
 Random walk
 Sacred geometry
 Self avoiding walk
 Selfreference
 Strange loop
 Turbulence
Fractalgenerating programs
There are many fractal generating programs available, both free and commercial. Some of the fractal generating programs include:
 Apophysis  open source software for Microsoft Windows based systems
 Electric Sheep  open source distributed computing software
 Fractint  freeware with available source code
 Sterling  Freeware software for Microsoft Windows based systems
 SpangFract  For Mac OS
 Ultra Fractal  A proprietary fractal generator for Microsoft Windows based systems
 XaoS  A cross platform open source realtime fractal zooming program
Most of the above programs make twodimensional fractals, with a few creating threedimensional fractal objects, such as a Quaternion. A specific type of threedimensional fractal, called mandelbulbs, was introduced in 2009.
 art
 Beno
 Britain
 created using fractalgenerating software
 David Foster Wallace
 Felix Hausdorff
 Felix Klein
 Fractal
 Fractal antenna
 fractals
 Fractals
 Gaston Julia
 Georg Cantor
 Gottfried Leibniz
 Hausdorff dimension
 Henri Poincar
 How Long Is the Coast of Britain? Statistical SelfSimilarity and Fractional Dimension
 Index of fractalrelated articles
 Jackson Pollock
 Koch snowflake
 Lsystem
 Lewis Fry Richardson
 Mandelbrot set
 Mathematics
 Max Ernst
 Michael Pietsch
 Michael Silverblatt
 modeling algorithm
 paint
 Pattern
 Paul Pierre
 Physics
 Pierre Fatou
 pressing paint
 recursive algorithms
 Ron Eglash
 Selfsimilarity
 Weierstrass function
According to Falconer, one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension.^{[1]} Presented here is a list of fractals ordered by increasing Hausdorff dimension, with the purpose of visualizing what it means for a fractal to have a low or a high dimension.
In this tutorial we show how to use two important methods of the BitmapData class,
perlinNoise and ColorMatrixFilter. By combining the two, we obtain a realistic
dynamic cloud effect. The color schemes of the sky and the clouds can easily be changed.
 Alpha
 America
 animation
 Barbara Kaskosz
 barbara@flashandmath.com
 complicated algorithm
 Computer science
 Dan Gries
 dan@flashandmath.com
 Digital photography
 Doug Ensley
 doug@flashandmath.com
 Fractals
 Image noise
 Ken Perlin
 Mathematical Association of America
 Mathematics
 mobile devices
 National Science Foundation
 Noise
 Perlin
 Perlin noise
 Sound