Chapter II · the number that never ends

π

Walk around any circle and you have walked a little more than three times its width. That little more — endlessly precise, forever unfinished — is π.

The "little more" is the same for every circle in the universe. Around 250 BCE, Archimedes trapped it between two nets of straight-sided polygons — a 96-sided shape squeezed outside the circle, another inscribed inside. The circle, he proved, lies between 3 10/71 and 3 1/7.

Later centuries made π an endurance sport — Ludolph van Ceulen spent his life reaching 35 digits and had them carved on his gravestone — and there is a lazier way still: the Monte Carlo method. Throw darts at random at a square, count the fraction inside the inscribed circle, multiply by four. Randomness, given patience, obeys geometry.

Dots fall at random; the solid ones land inside the circle. Watch π̂ wander toward the dashed line, and the Archimedes polygon on the right quietly add sides. Click the board to throw 24 darts · press r to reset.

The digits of π, falling and settling into the sequence — 3.14159… — one new digit every moment, forever.