A hundred forty years back, Karl Weierstrass presented a function that was continuous everywhere yet differentiable nowhere. In geometric terms, he described a curve without a defined tangent anywhere. A point on the curve is where it is, surrounded by other points of the curve as close nearby as may be wished, but all the same there is no telling which way in the vicinity the curve is headed. (For more on this try the very readable Master’s Thesis by Johan Thim, “Continuous Nowhere Differentiable Functions.” The graph above is an approximation of the Katsuura function.)
More recently, last week actually, taking in a high school basketball game, old Weierstrass came back to mind at half-time. Mrs. Mansfield had taken the littlest Mansfields out for some snacks and missed the pom squad’s routine, and asked me how it was after she returned. And since I was asked I unloaded why I mostly had to avert my eyes. “It was the usual. Lots of rapid-fire motion this way and that, frenetic jerking to a tune of static electricity discharging. Almost spastic. They executed it very well. They were precise, synchronized. Not a bit of flowing motion. The sort of thing that somehow girls like these have been stuck in a rut with for about twenty-five years now. When their mothers were girls, this style took hold, and it’s still going strong. I heard that Michael Jackson died a couple years ago.” A memory of watching a half-time routine like this around 1990, and being tired of it then, ran through my mind. That evening, Youtube salved my wounded, dance-loving mind with some deliciously differentiable clips of Cyd Charisse in motion. Why do we get stuck with some things and not the things we wish we were stuck with?