My brother-in-law, a mathematician, told me over a pitcher of beer that the old Spirograph toy we may all have some faint memory of is actually a math problem waiting to be researched. The device creates a combination of circles, each precessing around the other, simply based on a small circle being traced within a larger one. In the Spirograph’s case, they are actually geared, so that the teeth of one hollowed circle are walked by the teeth of another inner circle. There’s a proof to work out for the toothless cases, but the contention and circumstantial evidence suggests that if the ratio of the two radii is rational, then the pattern will repeat; but if the ratio is irrational, then there will never be an overlap of the lines of the pattern, and yet that set of lines will still create “the graph of the corresponding hypocycloid, r, dense in some annulus”. I poached that last part from the slides he presented to students, just to make sure I got it right. And, that’s the part that moved me.
First, he said “annulus,” which is just funny. “Dense in some annulus” is even funnier. Or painful sounding, maybe. But still funny, if you’re an eighth grader, or if you think like one.
Second: holy shit. What this means, I think, is that the Spirograph pattern, composed of circles which have that irrational ratio of radii, will go on forever, never repeating, yet never filling in some of the space. It will become, I think, infinitely dense in some areas, but still empty in others. And that’s just fucking depressing, or amazing, or both — just like a glass is half full and half empty at the same time. I poured from the pitcher into the pint glass to make sure there was no ambiguity.
Thirdly — and this is what continually gets me — I don’t understand irrational numbers. When Uncle M. says “the ratio of these will be irrational,” I just get wigged out because a ratio is what defines a rational number in my mind. But then I think hard about the ratio of the circumference to the diameter of a circle, which gives you 3.1415926535 8979323846 2643383279 5028841971 6939937510 8979323846 2643383279 5028841971 6939937510 … and on and on and on, never ending and irrational, which just messes with my head. The ratio of two numbers is rational unless, as he pointed out to me as I poured the next pint, one of those two numbers in the ratio is itself irrational. “Just take the smaller radius to be 1 in some arbitrary set of units to make it easy,” he suggested to me, the physics educator.
Thanks. Okay, fine: So the other number could be the square root of two. But how the fuck do you measure, with a ruler or a meter stick or a caliper or a compass or anything, a radius that’s the square root of two? Goddam mathematician.
Fortunately there was more beer. I emptied out half the pitcher, and then half again, and then half again, and then . . .