The fascinating, unexpected appearances of π

For reasons I won’t get into here, I’ve attended a few school physics classes in recent months. Not to learn physics, except as a side-effect. Instead, I’ve been sitting at the back with the teachers’ permission, watching how the teaching happens. Which means I’ve had a substantial dose of physics problems I last heard of when I was in school myself, back in the 1920s.

Maybe you remember them too—problems that start with something like this: “A block of mass m is attached to a spring of mass zero and moves along a frictionless surface …”. Don’t worry, I won’t inflict one or more such on you. I only intend to jog some possibly painful memories.

Now, these are purely theoretical problems. Nowhere on this planet are you going to find, for example, a frictionless surface. Nor a spring that weighs nothing. The point of such problems is to explain to students basic ideas in physics: conservation of momentum, Newton’s laws of motion, like that. And, doing so, becomes easier if you can temporarily ignore inconveniences like friction and springs.

At which point I will digress for a bit to remind you of π. You may remember a column I did here about the sinuosity of rivers—their “bendiness” in a sense, the ratio between a river’s actual length and the distance in a straight line between the source and mouth. In 1996, a scientist reported that when averaged over all rivers in the world, this ratio seems to be π. How on earth does π emerge from the way a river flows? To be sure, more recent research suggests that this scientist was wrong. But even so, it’s a reminder of how π tends to pop up where you’d least expect it to.

There are other examples aplenty. In analysing the swinging of a pendulum, you’ll run into π. A certain scientist in Arizona studies recurrent epidemics, and he believes π is part of the reason some kinds of flu come back every 20 years. Or, take the so-called “normal distribution”, that famous bell-shaped curve that plots the heights of a population, or students’ exam scores, or any number of other statistics. The area under this curve? The square-root of π.

Then there are infinite series that involve π. For example:

1 - (1⁄3) + (1⁄5) - (1⁄7) + (1⁄9) + … = π/4

and

1 + (1/4) + (1⁄9) + (1⁄16) + (1/25) + …= π^2/6

Besides, the remarkable Srinivasa Ramanujan churned out formulae for π, simultaneously showing us how ubiquitous π is and giving us a glimpse into his razor-sharp mind.

So what does all this have to do—what does π have to do—with physics problems featuring blocks?

Well, let me start answering that by reminding you of the value of π. School kids know it as 22/7, but as you no doubt are aware, that’s just an approximation. We can never know the exact value of π—as mathematicians say, it is an irrational number. What we do know is that the first several digits of π are 3.1415926…, and I’d like you to keep that in mind.

Back again to blocks. Imagine two identical 1 kg blocks sitting beside each other on a frictionless surface, and to their left is a wall. Blocks and wall alike are made of some ideal material. “Ideal” in this sense: when two objects made from it collide, it is a totally “elastic” collision, meaning there is no energy or momentum lost (both are “conserved”). That is, if these two blocks are moving towards each other and collide, each will bounce back and move in the opposite direction, having taken on the speed of the other block. Similarly, if a moving block hits the wall, it will bounce off at the same speed in the opposite direction.

What’s more, Newton’s First Law of Motion applies here, which says that a moving object will remain in motion in a straight line unless acted upon by an external force. Friction would be such an external force, but since there’s no friction, a moving block will keep right on moving.

Of course none of this has any bearing on real life, in which friction abounds and collisions are never purely elastic. But remember, this is an idealized, theoretical world that these blocks inhabit. And in this world, let’s say you push the block on the right smartly towards the one on the left. What happens?

It hits the one on the left and is stopped in its tracks. The block on the left starts moving left, at the same speed as its counterpart did before the collision (remember the conservation of energy and momentum). Shortly, it collides with the wall and bounces off, returning the way it came until it hits the (stationary) block on the right.

Now the left block stops in its tracks and the one on the right moves off towards the right at the same speed, and nothing that we know of can stop it.

Question: How many collisions?

Simple, right? There’s one between the two blocks at the start; another when the left block hits the wall; a third between the two blocks at the end. Total: 3.

Now let’s replace the block on the right with another, this one weighing 100 kg, and repeat the experiment. When it hits the smaller block, the smaller one starts moving left, but the larger one doesn’t stop. It only slows down. Again, there’s energy and momentum being conserved here. The smaller block bounces off the wall, moves rightwards and bumps into the still advancing larger block. This slows the larger block some more, while the smaller one now heads back towards the wall again. It bounces this way, back and forth between the larger block and the wall, slowing the larger one until it stops and starts moving to the right.

How many collisions? Take my word for it, it’s 31.

Replace the block on the right with one weighing 10,000 kg and repeat. 314 collisions.1,000,000 kg? 3,141. 100,000,000 kg? 31,415.

Maybe you see what’s going on. Each time we make the block on the right 100 times heavier, we get a collision number that gives us one more digit of … yes, π.

This floored me when I first ran into it. How does π turn up here, in this charming yet peculiar one-digit-at-a-time way? What possible connection is there between π and the way these blocks move and collide?

Answering those questions is a little beyond the ambitions of this column. But you should watch the three short videos here. Not least because of this comment: “If ever there were Olympic Games for algorithms that compute π, this one would have to win medals both for being the most elegant and for being the most comically inefficient.” Inefficient, because to find just the first 13 digits of π, say, we’ll need a block that weighs 1,000,000,000,000,000,000,000,000 kg—one trillion trillion kg. Put a half-dozen of those on some cosmic weighing scale and they’ll outweigh our planet Earth. Try putting one of those on a frictionless surface beside a garden-variety 1kg weight.

You might also read Playing Pool with π, the 2003 paper by the mathematician who first described this perplexing result, Gregory Galperin. Galperin lists “billiards” among his research interests, which may be why his paper uses billiard balls instead of blocks. Balls or blocks, though, the reasoning holds.

But for here and now, I’ll leave you with this. You know that πs the ratio of a circle’s circumference to its diameter. That is, since π is 3.1415926…, the circumference is a little more than three times as long as the diameter. So, suppose you take several strings, each as long as the diameter. Start at some point on the circle and lay them, end-to-end, along the circumference.

Question: How many strings will you lay down before you return to your starting point?

Simple, right? 3. (The fourth will take you past that point.)

Now take strings that are a tenth as long as the diameter and repeat. How many will you lay down? A little thought tells you: 31. A hundredth of the diameter? 314. A thousandth? 3,141.

Maybe you see what’s going on. It’s π again. So digest this: there are ways in which we can convert our experiment with blocks, or Galperin’s with billiard balls, into this exercise with strings. That is, we can transform a problem in physics into one in geometry, and doing so can make it easier to understand. To me, what’s truly fascinating here is this transformation, and what it says about the links between the sciences.

I mean, it’s almost more fascinating than the unexpected appearances of π.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His latest book is Jukebox Mathemagic: Always One More Dance. His Twitter handle is @DeathEndsFun