### Bouncing Billiard Balls Produce Pi

There are many ways of evaluating ${\pi}$, the ratio of the circumference of a circle to its diameter. We review several historical methods and describe a recently-discovered and completely original and ingenious method.

Historical Methods

• Archimedes used inscribed and circumscribed polygons to deduce that

$\displaystyle \textstyle{3\frac{10}{71} < \pi < 3\frac{10}{70}}$

giving roughly ${3.141<\pi<3.143}$. He used polygons of 96 sides.

• A continued fraction expansion gives 22/7, and then 355/113. The latter yields ${\pi}$ to an accuracy of better that one part in a million.
• There are many series expansions that can be used to evaluate ${\pi}$. For example, the Basel series

$\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6}$

It can be summed to arbitrary accuracy, giving ${\pi}$ to any required precision.

• Buffon’s Needle gives a probabilistic way of estimating ${\pi}$. A needle is dropped onto a plane surface ruled by lines. The probability that the needle crosses a line is easily computed, and involves ${\pi}$.
• A more brute-force Monte Carlo method involves throwing darts at a square board with an inscribed circle. The probablity that a random point within the square is also within the circle is ${\pi/4}$. Thus, ${\pi}$ is estimated from the relative frequency of darts falling within the circle.

Galperin’s Billiards Method

A completely new way of evaluating ${\pi}$ was devised not long ago by Gregory Galperin of Eastern Illinois University (Galperin, 2003). It can yield a value of arbitrary precision. Although it is utterly impractical, the method is ingenious!

Galperin considers the configuration illustrated below. We have two billiard balls constrained to roll along a line bounded at one end by a wall. To begin, the left hand ball L, of mass ${m}$, is stationary and the right hand ball R, of mass ${M}$, moves rapidly towards it. All impacts — whether ball/ball or ball/wall — are perfectly elastic. In ball/ball collisions the total momentum is conserved while in ball/wall impacts the momentum of L changes sign but keeps its numerical value. The energy is conserved in all collisions.

The configuration considered by Galperin to evaluate ${\pi}$.

Galperin proved the remarkable result that the total number of hits or impacts, counting both ball/ball and ball/wall collisions, gives the most significant digits of ${\pi}$. The ratio of the masses of the two balls determines how many accurate digits are produced.

The simplest case to consider is when the two balls have equal mass (${\mu = M/m = 1}$). When R strikes L it stops while L moves towards the wall. When it hits the wall, L reverses direction. It moves to the right and stops dead when it hits R. Then R moves away towards infinity.

There are 3 collisions in total. The first significant digit of ${\pi}$ is 3. When ${\mu = 100}$, there are  31 impacts. When ${\mu = 10,000}$, there are  314. You can see a pattern!

If we let ${u}$ be the velocity of L and ${v}$ be the velocity of R, the total momentum and energy are given by

$\displaystyle P = mu + Mv \qquad\qquad E = \textstyle{\frac{1}{2}}(mu^2 + Mv^2) \,.$

We can use these expressions to relate the velocities before and after a ball/ball collision. In this way, we could compute the total number of impacts for a given mass ratio ${\mu}$. However, the calculations rapidly become intricate. Galperin found a much better way, using clever geometric arguments about the trajectory of the motion in configuration space.

We will not describe this approach here but will state the key result. For the full account, see Galperin (2003). See also an illuminating YouTube video by Grant Sanderson in the 3Blue1Brown series.

Geometry for mass ratio ${\mu = 100}$, ${N=1}$ (from 3Blue1Brown video).

The Key Result

The amazing result found by Galperin is that, if the mass ratio ${\mu = M / m }$ is a power of 100, say ${100^N}$, the number of collisions corresponds to the initial N+1 digits of ${\pi}$.

We define ${N}$ to be the common logarithm of the root of the mass ratio: ${N = \log_{10}(\sqrt{\mu})}$. For the case of equal masses (${\mu=1}$) we have ${N=0}$. More generally, we have

$\displaystyle \mu = 10^{2N} = 100^N$

Let ${\Pi(N)}$ denote the number of impacts for a given value of ${N}$. Galperin showed that, whatever the value of the mass ratio ${\mu = M/m}$, there are always a finite number of impacts before ball R ball moves away to the right faster than ball L, and then there are no more collisions. The main theorem states that the total number of hits has ${N+1}$ digits, and they correspond to the first ${N+1}$ digits of ${\pi}$. Specifically, the number of impacts is

$\displaystyle \Pi(N) = \left[ \frac{\pi}{\arctan(10^{-N})} \right]$

For small ${x}$ we have ${\arctan x \approx x}$ and we replace ${\Pi}$ by ${[10^N \pi]}$, yielding the desired result.

Galperin gives fascinating arguments using the phase space of the system. Between collisions, the velocities do not change, so the representative point is fixed in the momentum subspace. For each impact, the point jumps to another location. Energy and momentum conservation confine the geometry of the point. More details are in the references below.

Acknowledgement

The work of Galperin and the 3Blue1Brown video were brought to my attention by Neil Dobbs, School of Mathematics and Statistics, UCD.

Sources

${\bullet}$ Galperin, G., 2003: Playing Pool with ${\pi}$ (the Number ${\pi}$ from a Billiard Point of View). Reg. Chaotic Dynam., 8(4), 375–394. PDF.

${\bullet}$ Grant Sanderson 2018: The most unexpected answer to a counting puzzle. YouTube Video in 3Blue1Brown series.