Getting to the bottom of things
Pi is the ratio of a circle's circumference to its diameter, commonly approximated as 3.14159.
For me, Pi points to the epistemological topics of individuation, quantifaction, and problems at the core of logic and reason. Thinking about Pi makes me think about the nature of number, quantity, individual things, and how we know about them. What is number, when we cannot use number to contain Pi? If number is the quantifaction of the individuated, and we assume that individuation gives us useful things, what limit of number's usefulness does calculating Pi show us? While logic also assumes the integrity of absolute individuation, physics can be content with individuation at the level of whatever the technology of the day allows it to find. The integrity of number is at the core of the integrity of our systems of logic.
From Wikipedia on Pi:
"Being an irrational number, Pi cannot be expressed exactly as a common fraction, although fractions such as 22/7 and other rational numbers are commonly used to approximate Pi. Consequently its decimal representation never ends and never settles into a permanent repeating pattern. The digits appear to be randomly distributed; however, to date, no proof of this has been discovered. Also, Pi is a transcendental number, a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of Pi implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge."