The first in a series of six new essays commissioned by The Future Tense as part of Mark Andrew Webber’s FORM series, Steven Poole searches for order out of chaos and finds that, with a little creative thinking, even the strictest and most staid of rules can produce wonders.
In his essay “On Prudence” for The Idler, the great Augustan lexicographer, critic, and wit Samuel Johnson explained why slavishly following rules could never result in something admirable: “Rules may obviate faults, but can never confer beauties,” he wrote. “The world is not amazed with prodigies of excellence, but when wit tramples upon rules, and magnanimity breaks the chains of prudence.” Sage advice. Yet two centuries later, a combination of computer and organic genius found what seems a startling exception.
In 1980, in a basement at Harvard, the French mathematician Benoît Mandelbrot was using a Vax supermini computer to investigate the geometry implied by following the rule of repeating a very simple formula. To his amazement, the screen gradually filled with a tremendously complicated vista of psychedelic curlicues, like seashells or spiral galaxies. Initially he thought that there was something wrong with the computer. But when he zoomed in on a part of the image, it looked the same as the full-sized scene, with ever-tinier curls and tendrils as more and more detail was revealed. It seemed to be an infinitely complex shape. In this way, the celebrated “Mandelbrot set” was discovered. Samuel Johnson could not have anticipated it, but in this case rules did confer beauties.
I remember my own thrill at discovering this story while reading James Gleick’s wonderful book Chaos as a teenager. In a footnote, Gleick gives the formula for the Mandelbrot set. I dug out my old Sinclair ZX Spectrum computer from its cardboard box, and wrote a simple program in BASIC that would iterate the formula, plotting a single pixel on screen each time. I ran the program and left it crunching numbers. Many hours later (for the Spectrum, bless it, was about a million times less powerful than your smartphone), I was amazed to see a real Mandelbrot set — monochrome and low-resolution, but recognisable — on the TV screen. The fact that I had recreated the thing myself at home encouraged the Platonic belief that Mandelbrot did not create the Mandelbrot set but rather it was discovered by him. As the mathematician Roger Penrose wrote: “The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there!”
A rule, then, can imply a whole universe. A rule can be fecund in a seemingly magical way, generating results that are impossible to predict merely by inspecting the rule. Before the Mandelbrot set, mathematicians had also seen such emergent phenomena in action with their investigations of “cellular automata”, such as John Conway’s Game of Life, which can be played out (albeit slowly) on graph paper, with eerily biological-seeming results. More generally, as we know, games of (nearly) all types are founded on rules. But the beauty or excitement of any particular game that is played can never be forecast from a consideration of the rules themselves. The world’s greatest chess players have a Platonic sense, too, of their own rule-bound universe. A top-level chess game might be spoken of as an “exploration” of certain ideas; the players, though adversaries, are also cooperating in an attempt to discover the “truth” of a position.
Vastly more powerful computing systems than that on which Benoît Mandelbrot first explored his fractal cosmos now run rule-based simulations for the entertainment of hundreds of millions of people around the world. I am speaking, of course, of videogames. The rule-set of a modern videogame such as Grand Theft Auto V or Far Cry 2 is enormously more complicated than that of chess. But game designers still seek a set of rules that will allow for emergent behaviours they had not thought of. Thoughtful videogame creators such as Clint Hocking or Warren Spector have written fondly of watching players using tactics they had never expected, prompting behaviour by the game’s artificial characters that they had not explicitly programmed in. Such examples show us that the person who designs a set of rules is thereby designing a set of possible worlds. Like a deity designing the laws of nature that will reign in his cosmos.
So it takes creativity to design fruitful rules in the first place. But even once they are in place, they do not nullify the role of human inspiration. Creative artists throughout history have found that rule-based restrictions have liberated their creativity, whether in the strict rules of harmony and counterpoint in Baroque music, or the poetic forms of sonnet and villanelle, or the exotic rules dreamed up by the French literary group Oulipo in the 20th century, which resulted in, for instance, a great novel — La Disparition, by Georges Perec — written without once using the letter “e”.
Even the Mandelbrot set needed a creative mind first to suspect that an apparently simple formula might conceal hidden depths, and then to understand its significance and draw from such discoveries more general lessons about fractal shapes in disciplines as far afield as physics, linguistics, and finance. So it seems Samuel Johnson needs only a mild updating. Rules by themselves cannot confer beauties; but the right rules, allied with the right mind, can produce wonders.
© Steven Poole, 2014
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