I was sorry to see the obituary in the New York Times of Benoit Mandelbrot, the mathematician and one of the founders of the field of fractal geometry.
Dr. Mandelbrot coined the term “fractal” to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature.
When we study mathematics, and many of its applications in the physical sciences, we tend to focus on functions that are “smooth”, those that can be described mathematically as continuous, and perhaps continuously differentiable. These functions certainly have many important uses, but one of the reasons we spend a lot of time studying them is that they tend to be analytically tractable; we find that we can discover their properties (relatively) easily. That should not blind us to the reality that many objects in nature do not exhibit such nice characteristics.
As the Times article mentions, one stimulus that led Dr. Mandelbrot to the concept of fractals was the question, “How long is the coast of Britain?” At first glance, this might seem like a simple problem in geometry, but it gets tricky. If we look at a map of Britain in an atlas, we can in principle trace the coastline, and convert our scaled measurement to miles, kilometers, or whatever. However, if we take a larger scale map — say one that occupies a page ten times the size of the atlas page — we will find that it shows small irregularities in the coastline that were not visible in the first map; and when we trace this coastline and convert the scaled measurement, we will get a larger number (longer coastline) that we did in the first case. As we continue to look at finer and finer details, we will notice that small features tend to look like larger features, and that our measure of the coastline keeps getting longer.
This kind of self-similar pattern is actually very common in nature. It is exhibited by snowflakes, ferns, and geologic features. If you take a “head” of broccoli, and break off one of its constituent florets, you will have something that, in a smaller size, looks very much like the original head. Dr. Mandelbrot’s idea was to assign these objects a fractal dimension, as a way of quantifying their “raggedness”. He found applications of the concept in many fields.
Over nearly seven decades, working with dozens of scientists, Dr. Mandelbrot contributed to the fields of geology, medicine, cosmology and engineering. He used the geometry of fractals to explain how galaxies cluster, how wheat prices change over time and how mammalian brains fold as they grow, among other phenomena.
Many of these complex patterns can be generated from relatively simple non-linear functions. The Mandelbrot Set, which has an extremely complex fractal boundary, is the set of values of C for which iteration of the complex quadratic,
Zn+1 = Zn2 + C
remains bounded. The set, graphed below, has an odd appeal.
Fractals have been used to generate natural looking scenery, for example in the Star Wars films. Dr. Mandelbrot’s wonderful book, The Fractal Geometry of Nature, contains many more examples.
The BBC News site also has an article on Mandelbrot. It also mentions his observations on the applicability of fractals to finance.
Mandelbrot was also highly critical of the world banking system, arguing the economic model it used was unable to cope with its own complexity.
I had the great good fortune to meet Dr. Mandelbrot and hear him speak, during the time he was working at IBM’s Thomas Watson Research Center in Yorktown, NY. He was truly a fascinating person, and thought-provoking in the best possible sense.