Never On Sunday

Man, sometimes math is just plain <i>neat</i>.

As I begin writing this, under the pretense of taking notes as part of my classroom observation, I’m watching a geometry class in progress. The proposition to be demonstrated: that an angle whose apex lies on a circle encloses an arc of that circle measuring twice the angle itself. Thus, a 20° angle encloses 40° of arc.

It’s not the kind of observation that would ever occur to me, but, given the proposition to consider, I quickly run through several test cases in my mind, and yes, it certainly seems to be true. Seeming and being are two different things, however, and test cases do not make a proof. Mr. DeGrassie delivers the key idea before I can work out for myself—something which rarely happens, and when it does, it’s usually a cumbersome, roundabout approach to the conclusion.

But this proof is elegant, depending on the addition of two line segments. Case I: if you’re lucky, one of the sides of angle A intersects the center O of the circle. Draw a radius from O to the point where the other side of angle A meets the circle; call this B. Triangle AOB is an isosceles triangle, with angle OAB equal to angle OBA; together, they add to 180 minus the value of angle AOB. But the external angle formed by lines AO and AB also measures 180 minus the value of angle AOB, so angle A is half angle O, which by definition is also the measure of the arc. Nifty, huh? If you’re less lucky and neither side of angle A intersects O, then angle A can be broken down into the sum of two angles that do (if O lies within angle A) or the difference of two angles that do (if O lies outside angle A), and that sum or difference is half the sum or difference of the two arcs they encompass, which conveniently equal the arc enclosed by A.

It’s a great proof—not so much for the results, but for the demonstration of how you can build up a complete proof from simpler, special cases. Like the pancake lemma, or the Königsberg bridge problem, it’s one of a handful of proofs that show, in an accessible, easily digested fashion, how math is done. Not the mere calculation which math allows, the application of formulas and techniques, but real math: the math that finds patterns and figures out why those patterns exist. Real math isn’t a formulaic business; you can’t simply add this and multiply by the other and prove that a Hamiltonian path exists connecting all ordered n-tuples of a binary variable. Real math isn’t about following the directions, but working out the reasons why, and relies on words as much as numbers.

Real math is also a messy business. False starts and trial-and-error are the norm, until the mathematician starts to get a sense of the problem. Sometimes he has to work upwards from simpler cases, as in the proof above; sometimes he has to settle for the simpler cases alone, because the general case is not, in fact, true. Sometimes a mathematician must build his own tools. All those tidy little proofs that go up on the chalkboard look tidy and brilliant only because the mathematician only reports his final results; as my calc prof liked to say, a mathematician “erases his mistakes.”

I think it’s important to show some of those mistakes from time to time, to reassure students that it isn’t all suposed to be easy or obvious, and that any approach is a good one, as long as it gets you where you’re going without resorting to faulty reasoning. A math teacher, with much to teach and little time to teach it, is under continual pressure to get to the point and move on to the next one. But math is no different from any other creative endeavor in the sense that learning something means operating at or beyond the edges of knowledge, and operating out there invariably means a bit of zig-zagging.