As a mathematician, I am frequently frustrated with the world’s stubborn refusal to mirror mathematical perfection. No “circle” made of atoms actually has a circumference-to-diameter ratio of π; no population’s growth is exactly an exponential function. The overwhelming approximate-ness of the world generally distresses me, but a recent post on Craig Kaplan’s blog has me looking for creative possibilities in the messiness of the real world.
A solid that doesn’t quite exist. Image: Craig Kaplan.
I met Kaplan, a computer scientist at the University of Waterloo, last year at the Bridges math+art conference, but I didn’t know he had a blog until a friend shared his delightful post about a solid he built. It appears to be 4 dodecagons, 10 decagons, and 28 equilateral triangles, but as he writes, “Unfortunately, there’s a small problem with this polyhedron: it doesn’t exist. Mathematically, you can prove that if you want all the faces to be regular polygons, there’s no way that these shapes will close up into a perfect solid.”
Instead, the solid only appears to exist because of the messy real world: “the real, mathematical error inherent in the solid is comparable to the practical error that comes from working with real-world materials and your imperfect hands.” Kaplan writes beautifully about the serendipity of finding near misses in geometry and closes by asking, “Where else in mathematics or beyond it might we find near misses, once we adopt this mindset?”
I’ve been reading a lot about temperament and tuning recently, so my mind turned to tuning systems. As I have written in the past, no piano can be perfectly in tune because (3/2)12 (twelve perfect fifths) is close, but not quite equal, to 27 (seven octaves). All tuning systems tweak various near misses, especially that one, to create as many intervals as possible that are as close to perfect as possible. Our ears are approximate enough that we can tolerate the little deviations from perfection that make pianos possible.
Recently, I exploited the near miss idea to make bias tape for a sewing project in a new toroidal way. True bias tape has a slope of 1, but I figured 9/8 was close enough and made my bias tape using a closed geodesic on a flat torus. You can read more about my method on Roots of Unity.
I’ll echo Kaplan now and ask you: Where have you found near misses in mathematics? Have you ever used near misses to unlock a new creative possibility in your art?
Ah, excellent! It happens that I met John Baez for the first time a couple of weeks ago, and we had a conversation about near misses. One of the examples he turned to as a demonstration of a near miss was precisely what you suggested: musical temperaments. Every tuning system is really a pattern of compromises: we must decide how to distribute error to make music sound better (or at least less bad?). Nice to see that example come up again here.