# Dictionary of Mathematical Eponymy: The Wallis Sieve

### What is the Wallis Sieve?

Well, since $\sin(x) = x - \frac{1}{3}x^3 + \frac{1}{5}x^5- \dots$, it clearly has a factor of $x$ you can divide out:

$S(x) = \frac{\sin(x)}{x} = 1 - \frac{1}{3}x^2 + \frac{1}{5}x^4 - \dots$.

Where does that expression have zeros? Wherever $x$ is a non-zero multiple of $\pi$ – so $S(x) = k\left(\pi^2 - x^2\right)\left((2\pi)^2 - x^2\right)\dots$

We know that $S(0)=1$, so we want to get rid of the $k$ by dividing all of those brackets through by whatever their constant is:

$S(x) = \left(1 - \frac{x^2}{\pi^2}\right)\left(1 -\frac{x^2}{(2\pi)^2}\right)\left(1 - \frac{x^2}{(3\pi)^2}\right)\dots$

Consider $S\left(\frac{\pi}{2}\right)$. We can evaluate it directly as $\frac{2}{\pi}$; the expansion gives $\left(1-\frac{1}{4(1)^2}\right)\left(1 - \frac{1}{4(2)^2}\right)\dots$

Now, $1 - \frac{1}{4n^2}$, by the difference of two squares, is $\left(1-\frac{1}{2n}\right)\left(1 + \frac{1}{2n}\right)$, or $\frac{2n-1}{2n}\cdot \frac{2n+1}{2n}$.

Putting it all together gives $\frac{2}{\pi} = \left(\frac{1}{2}\cdot\frac{3}{2}\right)\left(\frac{3}{4}\cdot\frac{5}{4}\right)\cdots$

So $\pi = 2\cdot \frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\dots$

This is the Wallis *product*. So what is the Wallis sieve?

Start with a unit square. Divide it into a 3-by-3 grid and remove the middle square. You now have eight squares. Divide each into a 5-by-5 grid and remove the middle square. Keep repeating this process.

The resulting shape has area $\frac{8}{9}\cdot \frac{24}{25}\cdot\frac{48}{49}\dots = \frac{\pi}{4}$.

### Why is it interesting?

I like the Wallis Sieve because it applies a 17th-century technique (the Wallis product) to a 20th-century idea (the Sierpinski carpet) and you end up with an unexpected $\pi$. (The Sierpinski carpet has an area of zero).

Incidentally, you can use the expansion of $S(x)$ to solve the Basel problem, but this is not the Dictionary of Mathematical Toponymy.

### Who was John Wallis?

John Wallis was born in Ashford, Kent in late 1616 (December 3rd, if you’re using new-style calendars) and died in Oxford late in 1703 (October 8th). Wallis did a great deal of the work that eventually led to calculus, invented the infinity symbol and introduced the words “momentum” and “continued fraction” to the English language. Oh, and introduced negative and fractional powers. And extended the number line into the negative domain.

Plus he was a mathematical ninja: in one fit of insomnia ((True Mathematical Ninjas never sleep)), he took the square root of a 53-digit number in his head, and recited the result from memory in the morning.