Pi Is Hiding Everywhere
When someone wishes you a "Happy Pi Day," you probably immediately think of circles—and not only pies. (Pi Day is March 14, or 3.14 if you’re using US date formatting.) That's because if you measure the distance around a circle’s outside (the circumference) and then the distance across it (the diameter), pi is the circumference divided by the diameter.
So anytime you’re dealing with circles, it seems quite logical that the number pi could show up. But many situations where pi appears at first seem to have nothing to do with circles at all. In quantum mechanics, it's in the solution to the Schrödinger equation, the way we model electrons and protons in an atom. It's in the magnetic permeability constant, which is used for calculating magnetic fields. It appears in the motion of a mass swinging on a string, otherwise known as a pendulum. It's in the electric constant, which is used for calculating the electric field due to charges. And it's even in the uncertainty principle, which says you can’t precisely know both the momentum and position of a particle.
Why does it keep showing up? Really, there are two primary reasons: symmetry and oscillations.
Let's talk about symmetry with an example—sunlight. Specifically, let’s consider the sun’s intensity. The easiest way to think about the sun’s power is to think about its rate of energy production, or how much it produces over a certain amount of time. It’s huge. The sun outputs almost 4 x 1026 watts (that’s 4 x 1026 joules) of energy every second.
Since it radiates this power in all directions, we can describe the power per unit area as the solar intensity. As light travels away from the sun, it covers an expanding sphere. As the radius of this sphere increases, the surface area over which the power must be distributed also increases. This means that the solar intensity decreases with distance from the sun. By the time that light has finally reached Earth, its intensity is only around 1,000 watts per square meter. Maybe this 2D diagram will help illustrate the concept:
