Sometimes math and physics conspire in ways that feel too good to be true.
Let’s play a strange sort of mathematical Croquet.
We’ll have two sliding blocks and a wall.
The first block starts by coming in at some velocity from the right, while the second
starts out stationary.
Being overly-idealistic physicists, let’s assume that there is no friction and that
all collisions are perfectly elastic, which means no energy is lost.
The astute among you might complain that such collisions would make now sound, but your
goal will be to count how many collisions take place, so in slight conflict to the assumptions,
I want to leave in a little clack sound to better draw your attention to that count.
The simplest case is when both blocks of the same mass.
The first block hits the second, transferring all of its momentum.
Then the second one bounces off the wall, then it transfers all of its momentum back
to the first, which then sails off towards infinity.
Three total clacks.
What about if that first block has 100 times the mass of the second one?
I promise I’ll explain all the relevant physics in due course, it’s not entirely
obvious how to predict the dynamics here, but in the spirit of getting to the punchline
let’s just watch what happens.
That second one will keep bouncing back and forth between the wall and the first block
with 100 times its mass, like a satisfying game of breakout, slowly and discretely redirecting
the first blocks momentum to point in the opposite direction.
In total, there will be 31 collisions before each block is sliding off to infinity, never
to touch again.
What if the first block 10,000 times the mass of the second one?
In that case, there would be quite a few more clacks, all happening very rapidly at one
point, adding up in all to 313 collisions.
Hang on...wait for it...wait for it...okay 314 clacks.
If it was 1,000,000 times the mass of the second, then again, with all our idealistic
conditions, almost all clacks happen in one big burst, this time resulting in 3,141 total
collisions.
Perhaps you see the pattern here, though it’s forgivable if you don’t since it defies
all expectation.
When the mass of that first block is some power of 100 times the mass of the second,
the number of collisions will have the same digits as the beginning of pi.
This absolutely blew my mind when it was first shared with me.
Credit to the viewer Henry Kavle for introducing me to this fact, which was originally discovered
by the mathematician Gregory Galperin in 1995, and published in 2003.
Part of what I love about this is that if ever there Olympic games for algorithms computing
pi, this one would have to win medals both for being the most elegant, and for being
the most comically inefficient.
I mean, think about the algorithm:
Step 1: Implement a physics engine.
Step 2: Choose the number of digits, d, of pi that you’d like to compute.
Step 3: Set the mass one block to be 100^{d - 1}, and send it traveling on a frictionless
surface towards a block of mass 1.
Step 4: Count the number of collisions.
So for example, to calculate only 20 digits of pi, which fits so nicely on this screen,
one block would have to have 100 billion, billion, billion, billion times the mass of
the other, which if the small block was 1 kilogram means the big one has a mass 10 times
that of the supermassive black hole at the center of the milky way.
That means you’d need to count about 31 billion, billion, clacks, and at one point
in the virtual process, the frequency of clacks would be around 100 billion, billion, billion,
billion clacks per second.
So let’s just say that you’d need very good numerical precision to get this working
accurately, and it would take a very long time to run!
I’ll emphasize again that this process is way over-idealized, quickly departing from
anything that could possibly happen in real physics.
But of course, this is interesting not because of its potential as a pi-computing algorithm,
or as a pragmatic physics demonstration.
It’s mind-boggling because why on earth do the digits of pi show up here!
And it’s such a weird way for pi to show up, too: Its decimal digits are counting something,
whereas usually, its precise value describes something continuous.
I will show you why this is true.
Where there is pi, there is a hidden circle, and in this case, that hidden circle comes
from the conservation of energy.
In fact, you’ll see two separate methods which are each as stunning as the surprising
fact itself.
Delaying gratification, though, I will make you wait until the next video to see what’s
going on.
In the meantime, I highly encourage you to take a stab at it yourself.
And be social about it!
It’s a hard puzzle, so it never hurts to recruit other smart minds to the task.