Opinion 142: Why Is π So Ubiquitous?

By Doron Zeilberger

Written: π day, 2015 (aka 3/14/15)

Eugene Wigner starts out his classic essay, about the "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" by narrating a conversation between a statistician and his friend, showing him a recent paper that he wrote, where the Gaussian distribution,

1/(sqrt(2*Pi))*exp(-x^2/2)

showed up, and he asked what is the meaning of the symbol "Pi", and was amazed that it is the same Pi that he learned in school was the ratio of the circumference of a circle to its diameter.

Wigner used it as a starting point to waxing eloquent about the unreasonable effectiveness of math in science.

A closer look shows that it is not as amazing as it seems at first sight. For the sake of simplicity, let's take the geometrical definition of Pi to be the ratio of the area of a circle to its radius-squared, and the statistical definition of Pi to be the "limit", as n goes to "infinity", of the sequence given by the reciprocal of the square of the probability of getting exactly half of the times Heads if you toss a fair coin 2n times, divided by n. In symbols:

Pi:=lim(1/(binomial(2*n,n)/2^(2*n))^2/n,n=infinity) .

But this is just a "limit" of a (very simple!) counting problem, that of counting sequences of Heads and Tails with certain properties.

As for the Geometrical Pi, it is the limit, of an even simpler counting sequence, namely the number of pairs of integers (x,y) between -n and n, such that x^2+y^2 ≤ n^2, divided by n^2.

Counting problems are done by solving (often obvious) recurrence equations, alias finite difference equations, that come from the combinatorial structure, that in the "limit" turn out to be (often very simple) differential equations. The simplest differential equation is

y'(x)=y(x) , y(0)=1 ,

whose solution, y(x)=exp(x), naturally leads to e:=exp(1), explaining why e is even more ubiquitous than π . The second-simplest differential equation is

y''(x)= -y(x) , y(0)=0, y'(0)=1 ,

whose solution is y(x)=sin(x), and Pi is the smallest positive root of the equation sin(x)=0.

So the reason that both e and Pi are all over the place is that they both arise from finite counting problems, and for the sake of convenience, one replaces Avogardo's number by "infinity" and get "limits", that are really not true "real" numbers (since "real" numbers do not exist), but some equivalence class of sequence-generating algorithms where the equivalence relation is

"converge to the same limit", (or equivalently the sequence of their differences converges to 0)

in analogy with Frege's famous definition of cardinal numbers. [So the notion of "converge to the same limit" is the fundamental one, and it is not a circular definition, since the notion of limit depends on it.]

So Happy π day, but remember, that π is not a number, but an equivalence class of many, often trivial, but nevertheless interesting and important, algorithms.