I had an incredibly rich mathematical investigation this week, and I’m trying to collect all of the math that I learned from it. I’m also attempting to record it as a sort of memoir of the week, in the hopes that I can work out how to help kids pursue these sorts of investigations. (Since, ultimately, I care more about whether kids can do this sort of noodling around than any of the particular mathematical content that they’ll see in a Geometry course.)
I started by collecting a library of hexagons, and then I discovered something interesting about regular hexagons. After generalizing to regular n-gons, I scribbled down a few questions in my notebook:
- Does the generalization hold for star-polygons?
- Is there a way to use this to find exact values for trig values i.e. cos(72)
- The ratio between a polygon and it’s midpoint-inscribed child approaches 1. What’s a geometric interpretation of this?
- How quickly does it approach 1?
- Does this not just work for regular polygons, but for all convex polygons? Concave polygons?
I tried 2 first, but it didn’t really get me anywhere. Then I thought about 3, and the way that if you had a ton of vertices and edges, it would basically look like a circle, and that it would still look like a circle if you connected its midpoints. I graphed my general formula for n-gons, and I saw visually how quickly it approached 1.
Then I dove into the convex/concave question. I took a concave polygon from my library (an “hourglass”) and connected its midpoints:
This was enough to yield a conjecture: that every sequence of inscribed hexagons constructed in this way ends up being entirely convex. As I struggled with my incredibly shoddy drawings, I realized that this would all be better done on a computer.
So I built a thing in Processing:
[Which is not showing up for some reason. Working on the embedding. Until then, just go here.]
I felt increasingly confident that the conjecture was true, but I still didn’t have a proof. I was able to come up with an argument that every convex polygon would yield a convex polygon, so what I needed to show was that every polygon has a convex polygon somewhere in its sequence.
In the meantime, while playing with my program I landed on a star/pentagram, and I saw that it had a poorly shaped star in the center. I figured that the poorly-shaped star was just because I wasn’t able to get precise with my vertices in my program, so I turned to Geogebra for help:
So this is pretty clearly a counter-example to my conjecture. Once again, we have options. We could try to bar this polygon somehow. Restrict the conjecture to polygons whose edges don’t touch (except at vertices). That would be a shame, though, since the conjecture applies to many non-regular stars. We could allow everything but non-regular stars, but that seems like a downer.
As the week went on, I continued thinking about how to prove this conjecture, in any form. What I wanted to figure out was a way to measure the “concavity” of a polygon, so that I could argue that each step in this sequence reduces the concavity in some way, eventually resulting in a convex polygon. That wasn’t working particularly well, so I turned to twitter for help:
In response, Dave Radcliffe helped out:
The Jason Davies visualization is fantastic:
And there’s even a paper that inspired the visualization, and it claims to prove something akin to my conjecture:
“If this process is repeated…then in the limit the vertices of the polygon iterates converge to an ellipse E that is centered at the origin.”
And that’s great. I haven’t read their paper carefully yet, but I’m looking forward to it. There’s one problem: we already found a counter-example to their theorem in the pentagram!
And that’s the story so far.
Some reflections on this process:
- All of the ideas came in a big spurt on Monday. Then I spent Monday building the programs, and Tuesday and Wednesday trying to organize and clean everything up. Monday was exhilarating; Tuesday and Wednesday were tougher.
- By Thursday I felt like I needed a break from polygons and these iterated sequences, and I couldn’t look at polygons from any other perspective. After a frustrating morning without any real progress, I decided that it was time to reflect and write everything down, and move on.
- This is real math. Even solving posed problems isn’t real like this stuff is. I make no claims that doing real math increases your capacity to solve problems or anything, but I do think that it’s the most fun I’ve personally had with math in a long time. I’d love to share this feeling with students.
- At the same time that I was working on all of this, I was under the sway of Proof and Refutations by Imre Lakatos. (Thanks again, Justin.) That’s some great stuff.
The next step is to figure out what all of this math means for kids. I have two competing impulses:
- Divide up this investigation into a series of good problems that will lead them to some of the cool things.
- Figure out a way to help kids have the sort of open-ended, personal experience that I had.
I’ll sort that out in writing tomorrow.