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.
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.
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:
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”?)
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.
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.
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:
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.