Sequences of Inscribed Polygons

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:

  1. Does the generalization hold for star-polygons?
  2. Is there a way to use this to find exact values for trig values i.e. cos(72)
  3. The ratio between a polygon and it’s midpoint-inscribed child approaches 1. What’s a geometric interpretation of this?
  4. How quickly does it approach 1?
  5. 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:notebook1

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:

pentagram

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:

Davies visualization

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:

  1. Divide up this investigation into a series of good problems that will lead them to some of the cool things.
  2. 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.

A Counter-example to the Triangulation Theorem?

triangulation theorem

The triangulation theorem states that every triangulation of an n-gon has exactly n-2 triangles.

But I claim that the 6-gon below can be triangulated into precisely 3 triangles.

triangle 2

So what do you do with this?

1. You can change the theorem, to say that there are at most n-2 triangles in a triangulation.

2. Justin’s approach:

3. Dave’s approach:

I want my students to become genuinely good at proof, and these sort of conversations emerging from counter-examples are one of the main engines of proof. The catch: in order to have practice with these thoughts, students need messy math.

Polygons Make Triangles Fun Again

triangle - regular

This triangle is super typical. There are a couple of interesting things about it, but most Geometry classes would ask kids to prove that the four small triangles are congruent.

But that smaller triangle is special, because it’s formed by connecting the midpoints of the larger triangle. It looks just like the big triangle.

Take a look at a similar situation, but with the 7-gon:

heptagon - regular

You get another 7-gon by connecting the consecutive midpoints, and it’s regular as well. (Why does it have to be regular?)

How much smaller is the inner 7-gon than the larger 7-gon? How does that compare to the difference in size with the 3-gon? What about a 4-gon?

(I got a formula for the n-gon, I think. It involves some square roots, some 1/2’s and cos(360/n).)

I like the math here — I like generalizing. But the larger message for me is that we lose out a lot by neatly dividing up Geometry into the study of triangles, quadrilaterals, circles, etc. The math that I was playing with above is about polygons, and the tired, overstretched triangle gets re-energized by it’s place as just one of many n-gons.

(By the way, in the limit, what happens to the ratio between a polygon and it’s “child”?)

A Gallery of Hexagons

Inspired by one of my favorite posts ever I sat down with an open notebook and started playing around with hexagons, looking for some interesting math.

(Sitting down with a blank page of notebook of paper and just doing some math was an incredibly rich experience for me, and I’m still coasting off of it. It yielded a lot of good ideas.)

One of the things that I quickly realized, as I was looking for interesting things about hexagons, was that in order to do interesting work I needed a nice variety of hexagons to play around with. Creating a gallery of hexagons was a non-trivial exercise for me.

IMG_2924

Creating this gallery and investigating them turned out to be a process with a good deal of positive feedback. Creating shapes yielded interesting ideas, which in turn required the construction of certain types of polygons for investigation. (In particular, as I settled in with inscriptions of polygons I found certain hexagons that persistently showed up. I’ll post this somewhere else soon, but “Lamps” and “Emeralds” seem to switch on and off in the sequence of inscribed polygons.)

Anyway, this is a post to say that I think that I’d like to give kids the chance to poke around with polygons for a while, because I think that it’ll give them a library of examples and counter-examples that will be useful in the sort of investigations that we’ll do throughout the year.

2 Trapezoids make an n-gon. What’s n?

I like this problem. It’s simple. I think that for its full effect some sort of hint should probably be given, something like “Don’t stop until you’ve found at least 5 different values for n.”

I threw this question out on twitter, and got some awesome responses:

The Game of Vish

From Projective Geometry:

J.L Synge has described an amusing and instructive game called Vish (short for “vicious circle”): “The Concise Oxford Dictionary devotes over a column to the word ‘point’…’that which has position but not magnitude.’ This definition passes the buck, as all definitions do. You now have to find out what position and magnitude are. This means further consultation of the Dictionary, and we may as well make the best of it by turning it into a game of Vish. So her goes.

Point = that which has position but not magnitude.

Position = place occupied by a thing.

Place = part of space….

Space = continuous extension …

Extension = extent.

Extent = space over which a thing extends.

Space = continuous extension.

The word “space” is repeated. We have Vish In Seven.”

I like this, but it doesn’t seem like a very fun game to me.  Here are some ideas for ways to spruce this game up:

  • Make it a competition that doesn’t involve a dictionary. Play with a partner. The game starts with “Point,” and the first person gives a definition. Then he asks his partner to define a word in that definition. Then the partner defines the word, and so on. The first person to repeat a defined word loses the game.

I tried playing it with my wife,  but it was sorta boring. We thought it was because defining words is hard/annoying to do, and not in an interesting way. Things get abstract, very quickly, e.g. “Couch” –> “Surface” –> “Layer” –> “Level”.

  • Use a dictionary to find a word with the longest possible “Vish” number. So the competition is to see who can find the longest chain, but then we can do all sorts of interesting math with that. We could find the average vish-length of a word in the dictionary. We could determine the spread, etc. Is it possible to have a vish number of 1? A vish number of 0? What would that mean?

More ideas? Comment it up, people.