Pi is defined as the ratio of a circle's circumference to its diameter. You would expect to meet it when you measure circles. Instead it turns up almost everywhere, including in places with no circle in sight. It governs the bell curve that describes errors in measurements unrelated to geometry. It appears in the behaviour of springs, the periods of pendulums, the probability that two random numbers share no common factor, the energy levels of the hydrogen atom.

The most famous example is the simplest. Drop a needle on a floor of evenly spaced lines and the probability it crosses a line involves pi. There is no circle anywhere in the setup — just a needle and parallel lines. Pi shows up regardless, which means someone who had never heard of circles could estimate it by dropping needles all afternoon.

Mathematicians have explanations for each appearance. Pi is tied to periodicity, to rotation, to the deep structure of how quantities relate. The explanations are real. They are also, taken together, a little unsettling, because they amount to saying the same constant is threaded through unrelated corners of reality at a level below the surface.

The honest reading is that pi is not really about circles. It is about something more fundamental that circles happen to expose first. A single number wired into the substrate, surfacing wherever you dig deep enough, is the kind of thing you find in a system built on a shared library rather than assembled piece by piece.