I hope you’ll pardon me for being busy. I haven’t had the chance to read all the Pi Day comic strips yet today. But I’d be a fool to let the day pass without something around here. I confess I’m still not sure that Pi Day does anything lasting to encourage people to think more warmly of mathematics. But there is probably some benefit if people temporarily think more fondly of the subject. Certainly I’ll do more foolish things than to point at things and say, “pi, cool, huh?” this week alone.
In Calculating Pi Less Terribly I showed a way to calculate π that’s … well, you see where that sentence was going. This is a method that uses an alternating series. To get π exactly correct you have to do an infinite amount of work. But if you just want π to a certain precision, all right. This will even tell you how much work you have to do. There are other formulas that will get you digits of π with less work, though, and maybe I’ll write up one of those sometime.
And the last of the relevant essays I’ve already written is an A To Z essay about normal numbers. I don’t know whether π is a normal number. No human, to the best of my knowledge, does. Well, anyone with an opinion on the matter would likely say, of course it’s normal. There’s fantastic reasons to think it is. But none of those amount to a proof it is.
That’s my three items. After that I’d like to share … I don’t know whether to classify this as one or three pieces. They’re YouTube videos which a couple months ago everybody in the world was asking me if I’d seen. Now it’s your turn. I apologize if you too got this, a couple months ago, but don’t worry. You can tell people you watched and not actually do it. I’ll alibi you.
It’s a string of videos posted on youTube by 3Blue1Brown. The first lays out the matter with a neat physics problem. Imagine you have an impenetrable wall, a frictionless floor, and two blocks. One starts at rest. The other is sliding towards the first block and the wall. How many times will one thing collide with another? That is, will one block collide with another block, or will one block collide with a wall?
The answer seems like it should depend on many things. What it actually depends on is the ratio of the masses of the two blocks. If they’re the same mass, then there are three collisions. You can probably work that sequence out in your head and convince yourself it’s right. If the outer block has ten times the mass of the inner block? There’ll be 31 collisions before all the hits are done. You might work that out by hand. I did not. You will not work out what happens if the outer block has 100 times the mass of the inner block. That’ll be 314 collisions. If the outer block has 1,000 times the mass of the inner block? 3,141 collisions. You see where this is going.
The second video in the sequence explains why the digits of π turn up in this. And shows how to calculate this. You could, in principle, do this all using Newtonian mechanics. You will not live long enough to finish that, though.
The video shows a way that saves an incredible load of work. But you save on that tedious labor by having to think harder. Part of it is making use of conservation laws, that energy and linear momentum are conserved in collisions. But part is by recasting the problem. Recast it into “phase space”. This uses points in an abstract space to represent different configurations of a system. Like, how fast blocks are moving, and in what direction. The recasting of the problem turns something that’s impossibly tedious into something that’s merely … well, it’s still a bit tedious. But it’s much less hard work. And it’s a good chance to show off you remember the Inscribed Angle Theorem. You do remember the Inscribed Angle Theorem, don’t you? The video will catch you up. It’s a good show of how phase spaces can make physics problems so much more manageable.
The third video recasts the problem yet again. In this form, it’s about rays of light reflecting between mirrors. And this is a great recasting. That blocks bouncing off each other and walls should have anything to do with light hitting mirrors seems ridiculous. But set out your phase space, and look hard at what collisions and reflections are like, and you see the resemblance. The sort of trick used to make counting reflections easy turns up often in phase spaces. It also turns up in physics problems on toruses, doughnut shapes. You might ask when do we ever do anything on a doughnut shape. Well, real physical doughnuts, not so much. But problems where there are two independent quantities, and both quantities are periodic? There’s a torus lurking in there. There might be a phase space using that shape, and making your life easier by doing so.
That’s my promised four or maybe six items. Pardon, please, now, as I do need to get back to reading the comics.