What happened to fractals?
I still have a copy of Benoit B. Mandelbrot’s The Fractal Geometry of Nature sitting on my shelf. That 1982 hardcover edition is $31.90 on Amazon now. Surprisingly, there is a 2010 Kindle edition, priced at $45.06, technically an eTextbook. I say surprisingly because the original had many graphics pages. I guess you’d better have a Kindle Fire or some other color tablet; you won’t see much on the Kindle paper white. The coffee-table size of the original must have made the Kindle edition difficult too.
That said, I wonder how many millennials know what a fractal is. Computer science types of all ages might, because displaying fractals is often a programming exercise (best seen on the high-res monitors found with graphics workstations). However, even for them, fractals might seem akin to the much simpler Lissajous figures—very intriguing graphics, but so what? Graphics artists might be familiar with fractals as an option when portraying landscapes like mountains and so forth. The origins of these computer applications can be found as wow-content in Mandelbrot’s book.
I once attended a Mandelbrot lecture back in the days when fractals were a fad among theoretical physicists specializing in disordered media (unlike ordinary solid state physics, which takes advantage of the regularities in most crystalline solids, disordered media is a difficult area of research, so scientists look for any tools they can grab). Professor Mandelbrot seemed like a nice enough guy, but I’ve never heard the pronoun “I” used so much in a scientific lecture. His contributions were more applied though, and based on obscure 19th century pure mathematics. Like much of math, that pure study of non-integer-dimensional objects was just hanging around until someone applied it to the physical world. Mandelbrot deserves the credit at least for championing that application.
This often happens at the boundaries of math and science called mathematical physics. Remember the Eightfold Way, that cute scheme for organizing the elementary particle zoo? Solitons and instantons? Probably not, because “the discovery” of the Higgs particle overshadowed much of what came before. But many math and physics students were probably wondering what esoteric topics in algebraic topology and Lie groups and algebras had to do with Nature. Again, the mathematical tools were just hanging around to be used. Today, even EE majors use unitary matrices in signal processing algorithms that might be employed in your cell phone (Gell’man’s Eightfold Way is based on SU(3), the special unitary group with a three-dimensional Lie algebra, its generators corresponding to the three types of quarks).
If your eyes are glazed over (mine aren’t, because I’ve already had my third mug of java), maybe I should talk about a more popular scientific figure, Einstein. Besides being a good role model for my hair disarray when I awake, old Einstein used some pure mathematics that was just hanging around too. While his 1905 papers represent a fantastic tour de force in quintessential modern science, including the special theory of relativity, it’s his general theory that’s most obscure and probably more important for modern astrophysics and cosmology. You might say, “OK, the old fellow conjectured that space-time is curved—so what?” The physics resides in his realization that it’s gravity that causes the curvature. It isn’t really a question of cause and effect, of course. Mass and energy are equivalent; they produce a classical (as opposed to quantum) action-at-a-distance we generally call gravity. But gravity doesn’t really exist independently from mass and energy. The planets move in their respective orbits because of the curvature of space.
It’s hard to get our heads around that concept. Einstein knew he needed a powerful new tool to flesh out his ideas. New to him—he had to learn it. He had the help of a pure mathematician who knew the 19th century theory of differential geometry described by Gauss and Riemann. Today cosmologists learn this language that generalizes vector calculus to tensor calculus. Any physicist who learns the basics of Einstein’s general theory must master these obscure mathematics. And the theory has numerous applications, from understanding why clocks in orbit around the Earth behave differently (GPS makes the corresponding corrections, for example), to understanding the origin of black holes. What’s amazing is that the math was just hanging around waiting to be applied.
Perhaps more interesting is that it can go the other way! Mathematicians can learn mathematics from theoretical physicists. As the only physicist to receive a Fields Medal (1990), mathematics’ Nobel prize, Edward Witten, Princeton professor, used quantum field theory (QFT) techniques to study lower dimensional topologies. He realized that a specific QFT, Chern-Simons theory, said something about knots and 3-manifolds. More glazed eyes? Me too! (In this case, I know another mug of Colombian coffee won’t help.) Witten originally used Feynman integrals (invented by another “intuitive mathematician”), but mathematicians soon caught up with him. Witten went on to prove the Morse inequalities, a conjecture in Morse theory (the study of the topology of manifolds via differentiable functions), using supersymmetric quantum mechanics. He’s also worked in string theory. Both string theory and supersymmetric models of elementary particles are hoped-for solutions to the problem of quantizing general relativity. Whether Witten is the smartest mathematician and/or physicist alive today can be debated, but the tools he’s created will be hanging around for a while, to be sure.
So, what happened to fractals? The fad passed in applying them to disordered media, to my way of thinking. But their importance has gone beyond what anyone could have imagined back when Mandelbrot’s book came out. I’ve seen many applications in computer graphics, of course, but I’ve also seen them applied to chaos theory problems (in fact, some of the examples in the book are simple cases of chaotic systems). I’ve seen them applied in data compression and signal processing too. Sometimes the applier doesn’t even realize he’s applying them. I’d expect them to appear in discussions of weird space-times where integer (1, 2, 3, 4, …) dimensions are insufficient to describe the physical situation. In other words, they’re still hanging around, useful tools from 19th and 20th century mathematics that might have enormous applications in this and following centuries.
So, if your son or daughter is having trouble with Algebra II, for example, blow his or her mind with some fractal applications. Or, he or she can try to blow your mind, and you can say, “Oh, those are those old Mandelbrot things—he called them fractals!”
And so it goes….
