This photo essay is a brief exploration into the mind-boggling and beautiful world of mathematical complexity, by way of the most famous example of fractal geometry, called the Mandelbrot set.
The Mandelbrot set was named after the work of mathematician Benoit Mandelbrot in the 1980's, who was one of the early researchers in the field of dynamic complexity. The Mandelbrot set has a fractal-like geometry, which means that it exhibits self-similarity at multiple scales. However, the small-scale details are not identical to the whole, and in fact, the set is infinitely complex, revealing new geometric surprises at ever increasing magnification. Belying this mind-boggling complexity is the extremely simple mathematic process used to produce it.
The Mandelbrot set is a set of complex numbers (remember in algebra - the points in the 2D complex plane with real and imaginary axes?). Every complex number is either in the set, or outside of it. The set is generated by applying a simple iterative process to each complex number. In mathematical jargon, it is defined by all of the complex numbers C for which the number sequence is bounded, or does not tend towards infinity.
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognizable two-dimensional fractal shape. The set is closely related to the Julia set (which generates similarly complex shapes), and is named after the mathematician Benoît Mandelbrot, who studied and popularized it.
More technically, the Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded.[1] That is, a complex number, c, is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.
For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = i (where i is defined as i2 = −1) gives the sequence 0, i, (−1 + i), −i, (−1 + i), −i, ..., which is bounded and so i belongs to the Mandelbrot set.
Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules, and is one of the best-known examples of mathematical visualization.
In other words, to generate the set, take a complex number, multiply it by itself, and add it to the original number; take that result, multiply it by itself, and add it to the original number; and so on. If the resulting numbers generated during the iteration process grows ever and ever larger, then the original complex number C is not in the Mandelbrot set. If the sequence converges, drifts chaotically, or cycles periodically, then C is in the set. Of course in practice we cannot apply the iteration process an infinite number of times to be sure that C is a member of the set. So instead, the process is applied several hundred or several thousand times, and if the sequence remains relatively small during these iterations, then it is assumed that C is probably a member of the set. Fortunately, there is a very useful criteria for detecting divergence - it can be demonstrated mathematically that if during the iteration process the number sequence exceeds the absolute value of 2, then the sequence will diverge to infinity and the original number C is definitely outside of the set. Many points reach a value that exceeds 2 after only a few iterations.
The typical means of graphically rendering the Mandelbrot fractal is to color points in the complex plane that belong to the set as black, and all other points outside of the set according to how quickly they diverge towards infinity. The speed at which points outside of the set diverge, sometimes referred to as "escape speed", is measured by how many iterations of the number sequence are required until divergence is detected. It turns out that the Mandelbrot set lives very close to the origin of the complex plane, within a radius of 2. The most interesting regions are found near the boundaries of the set, where the escape speeds are relatively higher than those farther away. Zooming down into these regions exposes a remarkable variation of self-similar structures.
In the Mandelbrot set, nature (or is it mathematics) provides us with a powerful visual counterpart of the musical idea of 'theme and variation': the shapes are repeated everywhere, yet each repetition is somewhat different. It would have been impossible to discover this property of iteration if we had been reduced to hand calculation, and I think that no one would have been sufficiently bright or ingenious to 'invent' this rich and complicated theme and variations. It leaves us no way to become bored, because new things appear all the time, and no way to become lost, because familiar things come back time and time again. Because this constant novelty, this set is not truly fractal by most definitions; we may call it a borderline fractal, a limit fractal that contains many fractals. Compared to actual fractals, its structures are more numerous, its harmonies are richer, and its unexpectedness is more unexpected. - Benoit Mandelbrot.
"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." So writes acclaimed mathematician Benoit Mandelbrot in his path-breaking book The Fractal Geometry of Nature. Instead, such natural forms, and many man-made creations as well, are "rough," he says. To study and learn from such roughness, for which he invented the term fractal, Mandelbrot devised a new kind of visual mathematics based on such irregular shapes. Fractal geometry, as he called this new math, is worlds apart from the Euclidean variety we all learn in school, and it has sparked discoveries in myriad fields, from finance to metallurgy, cosmology to medicine.
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 Benoît Mandelbrot in 1975 and was derived from the Latin frāctus 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 Euclidean 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 curves 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.
Frost crystals formed naturally on cold glass illustrate fractal process development in a purely physical system.
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.
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 Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous 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. Wacław Sierpiński constructed his triangle in 1915 and, one year later, his carpet. The idea of self-similar 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. 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 Henri Poincaré, Felix Klein, Pierre Fatou 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, Benoît 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".
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 Mandelbrot 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 carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals.
Random fractals – Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, percolation clusters, self avoiding walks, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.
Strange attractors – 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.
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, lightning, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels, and ocean waves. DNA and heartbeat can be analyzed as fractals. Even coastlines may be loosely considered fractal in nature.
River Fractals
Cactus Fractal
Natural fractal pattern - air displacing a vacuum formed by pulling two glue-covered acrylic sheets apart. The air travels in the direction the horns are pointing, creating dendritic structures into the glue in an effort to equalize the pressures in the room and between the sheets.
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 real-time 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.
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
Seismology
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 Elliott wave principle)
Fractals in networks
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.
Lichtenberg Figure High voltage dielectric breakdown within a block of plexiglas creates a beautiful fractal pattern called a Lichtenberg_figure. The branching discharges ultimately become hairlike, but are thought to extend down to the molecular level.
Is the mathematics intrinsic to nature and nature has arisen because of it, or is the way we see mathematics dependant what exists in the natural world?
