Here’s a way to compute π with no circles at all — just drop matchsticks on a floor of evenly spaced planks. Count what fraction land crossing a crack between planks. That fraction, flipped, hands you π. It feels like magic: a number about circles falling out of random tossing. It works because a needle’s chance of crossing depends on the sine of its angle, and integrating sine drags π in. Slide to drop more and watch the estimate settle.
Buffon’s needle: drop a needle of length L on a plane ruled with parallel lines spaced t apart (L ≤ t). The probability it crosses a line is P = 2L/(πt). With L = t this is 2/π ≈ 0.6366, so π ≈ 2 / (fraction that cross) — a Monte Carlo estimate of π from pure geometry, one of the earliest (Buffon, 1777). The π enters through integrating the crossing condition over the needle’s uniform angle: the crossing chance is proportional to |sin θ|, and ∫|sin| brings in π. Convergence is slow (~1/√N), the hallmark of Monte Carlo. A fail-loud self-check throws unless 2 / (2/π) recovers π. ◆ real geometric probability, node-verified.
The relation P = 2/π (for L=t) is exact; the on-screen estimate is a deterministic illustration of convergence, not a fresh random trial each frame — a true Monte Carlo run would wobble and tighten like ~1/√N.