Fractal Theory and Benoit Mandelbrot
Fractal Theory and Benoit Mandelbrot
In 1975 Benoit B. Mandelbrot (1924- ) wanted a word to describe the strange group of mathematical sets he was studying, and looked for inspiration in his son's Latin dictionary. The term he created was "fractal" to describe sets that modeled such diverse phenomena as cloud boundaries, stock market prices, plant growth, and even the distribution of matter in the universe. Mandelbrot's attempts to make the mathematical, scientific, and business communities, as well as the general public, aware of fractals have led some critics to see him as obsessed, both with fractals and his own place in history. While practical uses of fractals have been few, these unusual mathematical sets have left their mark on many areas, from financial analysis to Hollywood special effects.
In the late nineteenth and early twentieth centuries a number of mathematicians described strange mathematical sets that seemed to defy logic. One such set, the Koch curve—named after Nils Fabian von Koch (1870-1924)—is constructed by taking a line segment and replacing the middle third of the line with the other two sides of an equilateral triangle. The result has four line segments, each one-third the length of the original line. The next step takes these four line segments and replaces the middle third of each with the other two sides of an equilateral triangle, and so on to infinity. It is a simple construction but produces unusual results.
The length of the created curve is 4/3 to the power of the number of steps made. So for the first step the length is 4/3, for the second it is 16/9, and so on. Therefore as the number of steps tends to infinity, the length of the Koch curve also tends to infinity, yet mathematically the curve has no area (in the same way that the original line segment has no area).
A further, perhaps more simple, example is the Cantor Set, named after Georg Cantor (1845-1918). Starting again with a line segment, the middle third is removed, creating two line segments. The middle third is then removed from these line segments, and so on to infinity. As the number of steps tends to infinity the overall length of the set tends to zero, as there have been an infinite number of subtractions. Yet the set also has an infinite number of points, for at each step more line segments are created.
Such constructions seem to defy common sense, and to a degree they do, for common sense is based on experiences in our 3-dimensional world. Mathematically, objects are generally analyzed in terms of 0, 1, 2, or 3 dimensions. For example, a cube or sphere is 3-dimensional, a circle or square is 2-dimensional, a line or curve is 1-dimensional, and a point is 0-dimensional.
Mandelbrot's key point was that objects such as the Cantor Set and Koch curve have fractional dimensions, which is why they seem so strange. The Koch curve has a fractal dimension of about 1.262, which is greater than the 1-dimensional line we started with, but smaller than a 2-dimensional square. The Cantor Set has a fractal dimension of 1.585. Just to confuse the issue, not all fractals have a fractional dimension; rather a fractal is defined as having a measured dimension greater than its topological (or, if you like, standard, common sense, or integer) dimension.
Fractals generally have fine structure, visible no matter how far you zoom in, and these details seem similar at all scales. Fractals are often too irregular to be described in traditional geometric terms, and can generally be constructed from simple methods. Unfortunately not all fractals have all of these characteristics; indeed they are hard to generalize, as some are curves, some are surfaces, others are disconnected points, and yet others are so oddly shaped there are no good terms for them.
Mandelbrot has often acknowledged those whose pioneering work led to his own theories on fractals, with a number of his books including short biographies of such individuals. He has a particular fondness for theorists on the edge, or even outside, of mathematics, especially those whose work was forgotten, ignored, or misunderstood. However, Mandelbrot admits that it is only in hindsight that such forerunners were found. While Koch and Cantor are obvious examples, and Felix Hausdorff's (1868-1942) work on measuring dimension has become an important foundation of fractal theory, it was only after-the-fact that many historical precedents were uncovered.
The road to fractals was a long one for Mandelbrot, starting with investigations into the noise in electronic transmission beginning in the 1950s. Mandelbrot noticed that the result always looked similar, no matter how long or short the time duration chosen to measure noise was made. He discovered that the construction of the Cantor Set was a good mathematical approximation of measuring noise, with each construction step being equivalent to a shorter and shorter measuring time.
Mandelbrot then turned to economics, with a 1963 article entitled "The Variation of Certain Speculative Prices." He argued that previous mathematical models did not adequately describe the complexity of the rise or fall of stock prices. Mandelbrot's theories attracted attention, and he was listed in an edition of Who's Who of Economics. However, his writing was hard to follow, his ideas complex, and ultimately they offered no power to predict real stock prices, so interest faded and the next edition of Who's Who dropped his name.
Moving on from finance, Mandelbrot turned to more theoretical and mathematical approaches for his coalescing ideas. In 1967 he published a paper entitled "How Long Is the Coast of Britain?" in which he showed that while "common sense" suggests you could, for instance, take an aerial photograph and measure the outline, in actuality the result obtained depends on the smallest resolution of measurement used. As the scale becomes finer, bays and peninsulas reveal new sub-bays and sub-peninsulas. The Koch curve is a reasonable approximation of a coastline, with each step in the construction being equivalent to using a smaller and smaller resolution of measurement. Mathematically any coast, or any river for that matter, is of infinite length if you use an infinitely small resolution of measurement. This helped explain why some countries give different lengths than others for international borders, as they may have used different resolutions of measurement.
In 1973-74 Mandelbrot gave a series of lectures at the Collège de France in which he refined his earlier works into a more coherent whole. In 1975 he coined the word "fractal," and gathered the lectures together as a book, Les objets fractals: forme, hasard et dimension. In 1977 an English version, Fractals: Form, Chance, and Dimension, appeared with further revisions and expansions. The book was not overly mathematical yet was still difficult to follow and did not generate much interest.
In 1982 Mandelbrot tried again, releasing The Fractal Geometry of Nature, which was a further reworking of his previous material; indeed portions were identical to earlier works. However, the 1982 book included something his previous writings had been lacking: colorful computer-generated pictures. Fractal landscapes, planetscapes, and unusual patterns showed the beauty and potential of fractals.
Quickly fractals entered the "popular" science press with articles often containing bright, colorful illustrations of fractal designs. Computer magazines offered code to display fractals on the home computer screen, and many programs and screen savers used fractal effects. Fractals have appeared on t-shirts, beach towels, calendars, posters, in music videos, and have been displayed as art. The most popular fractal image is named the Mandelbrot Set, once called the most complicated object in mathematics, which consists of "bubbles" with infinitely fine surface detail.
The high profile of fractals in the 1980s led to a rediscovery of Mandelbrot in fields that had overlooked his early work. Reprints of Mandelbrot's economic articles were updated to include new terms and diagrams. The language of many of Mandelbrot's early papers was difficult to read, as English was not his first language, and the re-editing of papers has made them far more accessible and readable.
The popular success of fractals was based on the appeal of colorful images, which has provided fuel for critics. Many pure mathematicians have found Mandelbrot's lack of solid proofs frustrating. Some opponents have seen Mandelbrot as obsessed with a field that is no more than a novelty, and obsessed with his place in history. The way Mandelbrot has popped in and out of many disciplines, leaving behind only partly formed ideas for others to sift through, has been a source of criticism for some, but a source of opportunity for others.
Practical applications for fractals are few, but at the same time surprising in their range. Fractals have contributed to computer image compression techniques. Realistic movie special effects, such as landscapes and planets, have been generated by fractals. However, it is in the mathematical modeling and graphical analysis of irregular natural phenomena that fractals flourish: snowflakes, coastlines, rivers, cloud edges, mountains, galactic dust, turbulent fluids, moon craters, and the list could go on and on.
Yet there are no true fractals in nature, with "natural fractals" like coastlines breaking down somewhere, often at the atomic level (although it should be remembered that there are no true straight lines in nature either). Just as a sphere is only an approximation of the shape of the Earth, and an ellipse is only an approximation for the orbit of planets, so a fractal model is only an approximation within a given range of scale. An elliptical model for an orbit allows for accurate predictions of the location of a planet, and a spherical earth can successfully model phenomena such as its magnetic field. However, a fractal model of stock prices, crater impacts, or broccoli growth cannot predict how prices will change, where the next meteor will strike, or the final shape of a broccoli floret—all they can do is offer an imaginary version, with no predictive power at all.
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Falconer, Kenneth J. Fractal Geometry: MathematicalFoundations and Applications. John Wiley & Sons, 1990.
Mandelbrot, Benoit B. Fractals: Form, Chance, and Dimension. San Francisco, 1977.
Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York, 1982.
Mandelbrot, Benoit B. Fractals and Scaling in Finance. New York, 1996.