Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
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Okay, other people have already given you nice, simple answers about how circles are fundamental objects, so I hope you’ll permit me to stretch the ELI5 requirement until it writhes in pain.
The true, deep, fundamental reason that pi shows up everywhere is that the *exponential function* shows up everywhere.
The exponential function is a function that is equal to it’s own derivative. If you fix exp(0) = 1, it’s *the only* function equal to it’s own derivative.
This is *huge*. I can’t emphasize enough how fundamental of a fact this is. The exponential function is a *fixed point*, an eigenvector of sorts, of the all-important differentiation operator. That means whenever you’re dealing with *anything* differentiable, any relationship between derivatives of functions, this function exp magically appears. It is often called “the most important function in mathematics”.
The most amazing thing? This exponential function is perfectly well defined on the entire complex plane. Look on the real axis, you get exponential growth/decay. Look on the imaginary axis, you get *cyclic motion*. This function *repeats*, on the imaginary axis, with a period of 2(pi)(i). This period is a fundamental constant of a deeply fundamental function, so therefore pi is a fundamental constant in math.
(The familiar sin and cos functions pop right out from the complex exponential. This is where the famous e^pi ^i +1 = 0 comes from.)
I would say *that’s* the real reason why pi is so universal. That’s why it shows up *everywhere*. It’s because such a vast amount of math boils down to manipulations of the complex exponential function. Sure, pi is also about circles, but that’s because the exponential function is the basic building block of the more fundamental idea of *cyclic motion*, for which circles are a convenient visualization.
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