A fractal has been defined as “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,” a property called self-similarity. Roots of the idea of fractals go back to the 17th century, while mathematically rigorous treatment of fractals can be traced back to functions studied by Karl Weierstrass, Georg Cantor and Felix Hausdorff a century later in studying functions that were continuous but not differentiable; however, the term fractal was coined by Benoit Mandelbrot in 1975 and was derived from the latin fractus meaning “broken” or “fractured.” A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. There are several examples of fractals, which are defined as portraying exact self-similarity, quasi self-similarity, or statistical self-similarity. While fractals are a mathematical construct, they are found in nature, which has led to their inclusion in artwork. They are useful in medicine, soil mechanics, seismology and technical analysis.
A fractal often has the following features.
- It has a fine structure at arbitrarily small scales.
- It is too irregular to be easily described in traditional Euclidiean geometric language.
- It is self-similar (at least approximately or stochastically)
- It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling such as the Hilbert curve).
- It has a simple and recursive definition.
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.
Images of fractals can be created using fractal-generating software. Images produced by such software are normally referred to as being fractals even if they do not have the above characteristics, such as when it is possible to zoom into a region of the fractal that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals.
A Koch snowflake, which 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 mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz considered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).
It was not until 1872 that a function appeared whose graph would today be considered fractal, when Karl Weiestrass gave an example of a function with the non-intuitive property of being everywhere continous but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with Weierstrass’s abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch curve. Waclaw Sierpinski constructed his triangle in 1915 and, one year later, his carpet. The idea of self-similar curves was taken further by Paul Pierre Levy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the Levy C curve. Georg Cantor also gave examples of subsets of the real line with unusual properties—these Cantor sets are also now recognized as fractals.
Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henry Poincare, Felix Klein, Pierre Fatau, and Gaston Julia. Without the aid of modern computer graphics, however, they lacked the means to visualize the beauty of many of the objects that they had discovered.
In the 1960s, Benoit Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. which built on earlier work by Lewis Fry Richardson. Finally, in 1975 Mandelbrot coined the word “fractal” to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term “fractal”.
A class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, and Koch curve. Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals can be deterministic (all the above) or stochastic (that is, non-deterministic). For example, the trajectories of the Brownian motion in the plane have a Hausdorff dimension of 2.
Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals (see attractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mendelbrot set. This set contains whole discs, so it has a Hausdorff dimension equal to its topological dimension of 2—but what is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by Mitsuhiro Shishikura in 1991. A closely related fractal is the Julia set.
Five common techniques for generating fractals are:
- Escape-time fractals – (also known as “orbits” fractals) These are defined by a formula or recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrit set, Julia set, the Burning Ship fractal, the Nova fractal and the 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.
- Iterated function systems– These have a fixed geometric replacement rule. Cantor set, Sierpinski set, Sierpinski gasket, peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square sponge, Menger sponge, are some examples of such fractals.
- Random fractals – Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Levy flight, percolation clusters, self avoiding walks, fractal landscape and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggrregation clusters.
- Strange attracors– Generated by iteration of a map or the solution of a system of initial-value differential equations that exhibit chaos.
- L-systems- These are generated by string rewriting and are designed to model the branching patterns of plants.
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 two-dimensional fractals, with a few creating three-dimensional fractal objects, such as a Quaternion. A specific type of three-dimensional fractal, called mandelbulbs, was introduced in 2009.
Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:
- Exact self-similarity – This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity. For example, the Sierpinski triangle and Koch snowflake exhibit exact self-similarity.
- Quasi-self-similarity – This is a looser form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar. The Mandelbrot set is quasi-self-similar, as the satellites are approximations of the entire set, but not exact copies.
- Statistical self-similarity – This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of “fractal” trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar. The coastline of Britain is another example; one cannot expect to find microscopic Britains (even distorted ones) by looking at a small section of the coast with a magnifying glass.
Possessing self-similarity is not the sole criterion for an object to be termed a fractal. Examples of self-similar objects that are not fractals include the logarithmic spiral and straight lines, which do contain copies of themselves at increasingly small scales. These do not qualify, since they have the same Hausdorff dimension as topological dimension.
Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, river networks, fault lines, mountain ranges, craters, snow flakes, crystals, cauliflower, or brocoli, and systems of blood vessels and pulmonary vessels, and ovean waves. DNA and heartbeat can be analyzed as fractals. Even coastlines may be loosely considered fractal in nature.
Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these 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. The connection between fractals and leaves is currently being used to determine how much carbon is contained in trees.
In 1999, certain self similar fractal shapes were shown to have a property of “frequency invariance”—the same electromagnetic properties no matter what the frequency—from Maxwell’s equations (see fractal antenna).
In creative works
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.
Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns. It involves pressing paint between two surfaces and pulling them apart.
Cyberneticist Ron Eglash has suggested that fractal-like 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.
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”.
Main article: Fractal analysis
As described above, random fractals have been used to describe/create many highly irregular real-world objects. Other applications of fractals include:
- Classification of histopathology slides in medicine
- Fractal landscape or Coastline complexity
- Enzyme/enzymology (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, especially computer graphics for organic environments and as part of procedural generation
- Fractography and fracture mechanics
- Fractal antennas– Small size antennas using fractal shapes
- 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 (see Elliot wave principle)
- Fractals in networks