Once in while, a new idea clobbers the brains of physicists and mathematicians. Suddenly, they are forced to look at the universe in new and sometimes counter-intuitive ways. Relativity. Quantum physics. Game theory. Chaos theory.
Abstruse stuff like that isn’t just parlor entertainment for members of Mensa. It has practical applications. Nuclear energy. Space travel. Computers. Even if we don’t get binomial distribution and math equations look to us like the number and letter stickers that my son has randomly plastered on his bedroom door, we should pay attention. Physics explains strange phenomena. Math orders what seems chaotic.
In the 1960’s, a French mathematician named René Thom came up with a way of understanding and describing sudden change. It was a breakthrough because ordinary math presumes that things are fixed and solid like number lines, triangles and cubes. Even the math of motion, calculus, conceives of moving bodies following smooth curves. Throw a stone or fire a rocket into the air and it follows a predictable and elegant path up and back down.
But not everything that happens is so smooth and elegant. The car ahead of you brakes sharply. A corn kernel in a pan of hot oil suddenly pops open. An old wall cracks. Such events may seem unpredictable, measurable only as probabilities. But every sudden change occurs in a particular moment. Think of a dome with a marble perfectly balanced on its apex. That marble isn’t going to stay there for long. At a certain moment, the marble will suddenly start to roll in one direction or another. Wouldn’t it be nifty to know when that moment will occur?
Okay, maybe knowing the precise second when a marble will begin to roll off a dome isn’t all that exciting. But what if math could tell us ahead of time when an airplane engine will fail, the day on which a depressed individual will commit suicide, or when prisoners will riot, or when a government will go to war?
René Thom came up with math which might predict such sudden changes, which he called catastrophes. His insights stirred hope that math could predict the previously unpredictable even in soft-science realms such as biology, psychology and social science. Catastrophe theory, it was thought, might illuminate things as diverse as animals’ territoriality, corporate decisions to raise prices, and humans’ behavior in paying taxes.
The key was Thom’s application of topology which, in plain English, considers what happens to invariant points when the set of which they are a part bends or stretches around that point. According to Thom, there are seven types of unstable curves, simple or three-dimensional, that show how a point will move “suddenly”—but predictably—according to how various forces are warping around it.
The seven types of unstable curves Thom named fold, cusp, swallowtail, butterfly, hyperbolic umbilic, elliptic umbilic, and parabolic umbilic. Don’t stress. There won’t be a quiz. Just have a look at this cusp catastrophe diagram that predicts when a tax scofflaw will suddenly decide to pay his or her taxes:
What you see above is that the space around a point (“A”) can bend and fold as different pressures effect it. When the fold becomes extreme enough, point “A” suddenly leaps from one spot to another. Point “A”, the tax scofflaw, in a startling instant becomes Point “B”, a dutiful taxpayer.
What Thom showed was that such behavior is not random, meaning that it will either happen or not. It will happen, math proves it, but to you and me the change that happens will look sudden and without immediate cause.
So, what has all of that to do with characters in fiction? Well, consider this: characters change. [Read more…]
