Fractals are mathematical objects with strange properties. They have been known for many years, but had been relegated to an obscure corner of mathematics. In the beginning fractals were curiosities, very few people thought they had any real applications (Ludwig Boltzmann and Jean Perrin were among the exceptions). All that changed when Benoit Mandelbrot began his career. Mandelbrot discovered that complex phenomenon in a variety of sciences, including astronomy, could be understood in terms of fractals.
Fractal geometry along with several other sciences were motivated by examining human senses. For example, the sense of sight led to the study of electromagnetic radiation and the sense of hearing led to the study of acoustics. However until recently, there had never been any science of roughness.
Starting in the late 1800’s and into the early 1900’s, a number of strange mathematical objects were developed by Georg Cantor, Helge von Koch, David Hilbert, Giuseppe Peano, Carl Ludwig Sierpinski and others. They were called “monster curves” as if they were unruly beasts who needed to be locked up before they did some real damage (the word fractal would come later). Unlike other objects like circles and sine curves which are smooth, these objects are rough and this roughness persists even as the object is magnified. As the object is magnified more and more, the same amount of roughness is present. They are created using a simple process known as aggregate replacement. By repeating this process indefinitely images of these objects form, showing that a complex object can result from a simple procedure.
Even though a group of talented mathematicians had spent a great deal of effort developing these curves, they remained confined to an obscure part of mathematics until Benoit Mandelbrot started his career. Mandelbrot obtained a master’s degree in aeronautical engineering and a Ph.D. in mathematics. After obtaining his Ph.D., Mandelbrot conducted a wide variety of investigations that went, as he described it, “in a number of directions.” He began to realize there was an underlying theme to what he had been working on: these “useless” mathematical objects discovered earlier had a use. He coined the word “fractal” and started applying fractal ideas more systematically. Mandelbrot knew a precise definition of roughness was crucial. There were already several ways to measure roughness (they all came under the vague label “fractal dimension”), but they did not always produce the same result. Mandelbrot realized that the most useful of these was the Hausdorff dimension (though it is the hardest to calculate).
The applications discovered by Mandelbrot and others include problems in biology, chemistry, physics, astronomy, geology, psychology, computer science and economics; however I will limit my discussion to astronomy. One of the applications of fractals in astronomy appears when we examine the orbits of planets. Planets, like all objects in the universe, obey Newton’s law of gravity. However even though Newton’s law is relatively simple, it can be difficult to apply in practice. The so-called N-body problem asks the question “if we have three or more objects each of which exerts a gravitational field, how can we predict the motions of these objects?” No precise answer is known except under a few special cases. You may be wondering at this point “can’t we accurately determine the orbits of the planets in our solar system?” Well we can, sort of. Astronomers use approximations to determine orbits, and these approximations work well enough that we can predict where any of the planets will be next week or twenty years from now. However some astronomers believe we cannot accurately predict where they will be a million years from now (a similar result occurs with any group of objects moving in a gravitational field, such as the particles making up the rings of Saturn or the stars within a globular cluster).
I must make this clear, the orbits of the planets are not fractals: they are very close to perfect ellipses. However if we plot the predicted positions of a particular planet under a number of conditions we find they all fit within a boundary curve called the basin of attraction. This basin is often a fractal.
Fractals built with aggregate replacement are not the only types of fractals known. Other methods of generating fractals go by the names L-system and IFS. It is also possible to use equations involving a mathematical creature known as a complex number to generate what are known as Julia Sets. Julia Sets are intimately linked to another object now known as the Mandelbrot Set. The Mandelbrot Set, named after its inventor and sometimes called the “most complicated object in mathematics,” is also generated using complex numbers.
As an aside, many of these fractals have been used to produce computer images, some of which are quite beautiful. These images are typically very colorful and have caught the attention of some artists. Note that the images are usually based on black and white fractals, the colors are simply added by the computer. While these images have played an important role in making non-specialists aware of the subject, computers played very little role in the development of fractal geometry.
To the best of my knowledge L-Systems, IFSes, Julia Sets and the Mandelbrot Set have no application to the study of astronomy. However there are other types of fractals. The fractals I’ve mentioned so far are all deterministic. In other words, each time they are constructed they look exactly the same as the previous time. On the other hand, a stochastic fractal is generated with a random process (such as tossing a coin). Two stochastic fractals generated with the same process will look different. Most deterministic fractals can be modified to produce a stochastic fractal.
To find more applications of fractal geometry, we need a special type of stochastic fractal known as a “multifractal measure.” In order to explain how multifractal measures are used, I will need to discuss distribution functions (and a particular type of distribution function known as “1/f”). Multifractal measures may prove useful in describing the distribution of craters on the moon as well as the distribution of galaxies in the universe. I will discuss these ideas in part two of this article.
Before ending part one, I need to emphasize something that Mandelbrot has emphasized repeatedly. Fractals have applicability in wide range of scientific disciplines, however fractals are not a panacea. They are useful for describing some phenomenon and even accounting for variability. However they cannot describe all phenomenon of nature. Even when they are useful in describing they rarely are of much help in predicting or explaining. (While multifractals have been used to model stock market fluctuations, I have serious doubts that anyone could use fractals to make specific predictions. For example, it might be useful to predict the price per share of Microsoft stock on January 1st, 2000, but I don’t know how to use fractals to make such a prediction. The same problem occurs in other applications of multifractals, including astronomy).
Most of the ideas for this article came from two talks Dr Mandelbrot gave at the University of Michigan this past October, however I also made use of many additional reference materials. The complete bibliography can be found here.