Comment by jlokier

Pi falls out of infinite sums, exponentials and differential equations all over the place. It's a universal constant, first discovered via geometry, but it's independent of geometry.

For example,

pi = the square root of the sum of 6/x^2 over x = (1, 2, 3...).

pi = the smallest positive value of ln(-1)/i.

pi = half the period of non-zero solutions to the differential equation f = -f''.