I am a huge fan of Greek geometry. That’s the real stuff. Real geometry involves an unmarked straightedge and a compass. If it has numbers, you are doing it wrong. One of my colleagues is a geometrical genius–he’s actually one of the most famous origamists in the world (that’s right, I name-dropped an origami master!). I like to tease him when they get into “analytic” stuff late in the year after the proofs are all done. “Oh look, Algebra! I thought you were teaching Geometry this year!”
Doubling the Cube? Squaring the Circle? Neusis Constructions? Now we’re talking. The Greeks thought geometrically in a way that borders on the mystical. Come to think of it, they actually did make a religion out of it.
I also use regular polygons in my Form III doctrine class, the human nature lesson, to illustrate the difference between image and concept, sense and intellect. It’s just game-level stuff, but instead of throwing Descartes’ chiliagon at them right off, I have them work up through imagining regular polygons. It’s neat watching the kids experience the difficulties of maintaining a stable image, especially once I jump to nonagons or other weird-gons.
After straining them for a bit, the idea that concepts are clear and definite in a way that images can never be thoroughly captures the students. You don’t need to stress the difference between a 1000-sided figure and a 999-sided figure, although that works great too. I think it’s enough just to see that you can’t generate an image of a 17-sided or 31-sided regular polygon, even though you know exactly what they are.
The fact that I use weird, prime-sided regular polygons as an illustration in class made this video tickle me all the more:
Bonus history of geometry in this video, too. The moment he said “Gauss,” I knew I was in for it. The only guy after 2000 years who could take geometry to the next level? Hold on to your butts, people. Gauss is the real deal.
“Someone proved you can make a 17-sided regular polygon??? Oh, it’s Gauss. Yeah, that makes sense.”
Math is awesome. Wish I still had me old drafting tools.