Math for Love

3n + 1

This weekend I gave a middle school version of a keynote lecture at the local Mathcounts tournament, as a group of 5th to 8th graders finished slices of pizza and waited for the contest results to be posted. The talk actually went quite well. It was my first ever powerpoint, and I must admit there is some real power to including video in a presentation.

I talked about the 3n+1 problem, a lovely little unsolved problem that is incredibly simple to relate, but virtually impossible to get anywhere with. It goes like this: say you pick a number, and generate a sequence based on the following two rules:

1. If your number is odd, multiply it by 3 and add 1.

2. If your number is even, divide it by 2.

Then you repeat that process with your new number. So for example, if you start with 5, you get the sequence

5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1 …

and the 4, 2, 1 loop repeats.

Now the absolutely astonishing thing is that no matter what number you start with, you always seem to come back to 1 eventually. But nobody knows if it’s true for ALL numbers, and no one can prove it. Try it… even some simple numbers can take a long time, but they all get to one eventually.

The nature of these unsolved problems, especially when their statement is so simple, makes me want to know their answers. Why does this happen? Does it even happen, for sure, always? I need to know why, why multiplying by 3 and adding 1 has this strange behavior.

Incidentally, I tried multiplying by 3 and subtracting 1 and didn’t get anything like this. Only about 34% of the numbers I tried came back to 1 in that case.

If I chose a and n in some sufficiently random way, and built some an+1 sequence in a sensible manner, what is the chance the sequence generated by those choices would return to 1?

Sometime I feel we don’t even understand how multiplication and addition are truly related, at the deepest level.

And no one can satisfy my desire to know. We live with mysteries.

Planning inspiration judiciously

I have begun teaching two, not one, but two sections of differential equations this quarter, and immediately, the classes are different from each other. In one, the students contribute, respond, emote; in the other, I feel like I’m facing a mute wall. This is natural, of course, and not anything to worry about, but what does occur to me is that what works with one won’t work for the other.

For example, I was thinking of what I might say to the first class about the complex numbers. Just a teaser—I won’t teach them in earnest for a few weeks, when we need them. But still, it’s such a natural question to ask: if we invent complex numbers to have a place for the square root of one (which we call the imaginary number i), what is the square root of i? Such a natural question. And then, I think of Slaughterhouse 5, and the aliens description of our 3-dimensional life as comparable to riding a roller coaster with only a tiny pinhole to see out of. That’s exactly what the real numbers feel like once you’ve gone to the complex. They’re just a tiny slice of the whole picture, and once you know what the whole picture is, and how beautiful it is, you can’t imagine living without that knowledge.

This is what I thought of telling my students. But maybe just the one class. There has to be a naturalness to it, improvising off the script, and I have to talk to each of these classes in the best way for them.

Forgetting and learning

One of the remarkable things about learning math is that steps forward in understanding require a kind of forgetting. Everything always looks simple in retrospect; it’s letting go of your biases that prevent you from learning that is difficult. There are great examples of this in the book Ender’s Game: the main character goes into zero gravity for the first time and realizes that he can let go of his conception of “down.” Simple, but difficult.

I’ve often wondered about why this is so often the case. After all, learning seems constructive in nature, but forgetting and letting go of bias is deconstructive, and seems like it’s the opposite process. Today I had another thought on the topic though. I’ve long suspected that much learning involves a refinement of vision and categorization. Toddlers sometimes go through phases of calling all animals “doggies,” for example, because their categories aren’t sufficiently refined. (I heard one story of a child passing a field and point to the “moo doggies” there.) But this means that we learn through differentiating more deeply between things. In other words, we forget that they seemed alike. They’re actually different.

Of course, so much of math is about seeing similarities between things that look totally dissimilar. In effect, both skills are crucial: we need to disconnect ideas from each other (forget our biases) in order to connect them up to each other in new ways. Learning seems to involve the constructive and deconstructive in equal measure.

Meanwhile, I’m getting pretty hopeful about my thesis. I think I may actually be on the verge of calculating the number of Fourier-Mukai partners of certain K3 surfaces. Just have to go over (and over) the details.

Progress and Simplicity

I had a little progress on my thesis work, recently. Essentially, I was able to prove what form a composition of transformations would take in the most general case. I had a hunch (and a hope) that it would be the simplest thing I could think of: given two transformations defined by two numbers, the composition of them should be defined by the product of those numbers. And indeed, that’s the way it is, in the simplest case.

This is one of the reasons I like math: things sometimes work out like they’re supposed to. If you study biology, it seems like everything is a hopeless mess; any process in the body is affected by every other process, so you can barely ever get a clear look. In math, there’s this magical way that things end up being surprisingly simple when you look at them the right way. That’s what seems to be happening with my work: a complicated transformation involving geometric structures connected via an abstract algebraic process and the whole thing boils down to multiplication.

For the moment. I have a feeling the subsequent cases will be a bit rougher.

Turtles all the Way Down

I’ve been terribly delinquent about posting here recently.

I’ve just finished my second week of my summer course, Turtles All the Way Down, which I’ve been teaching through the Robinson Center at the University of Washington. I have to say, it’s been a thrilling and tiring ride so far. We meet from 9-2:20, Monday, Tuesday, and Thurdsay, which adds up to more than fifteen hours a week in class. The students, each of whom just finished seventh, eighth, or ninth grade, are a pretty impressive bunch.

My goal in the class has been to get the kids asking their own questions. I would say they’re on a roll at this point. Many of the issues that get raised in class are deeply mathematically substantial. What’s also fun is that I get to learn from my students and my TA.

Two of the gems from the course:

1. There are “bigger” infinites than the infinity we think of. We’ve discussed the two smallest in class so far. My students are clamoring to know if there are more (there are), and what they look like.

2. There’s a more general way to think about dimension that allows you to measure the dimension of fractals as being nonnatural numbers. We’ve seen fractals with dimension close to 1.77, for example. This was actually new to me, and pretty cool.

I’ll try to update with more highlights soon!