Lab / Experiment
Estimating π
Four fundamentally different methods for computing the same transcendental number. Monte Carlo throws random darts. Buffon drops needles. Leibniz sums a series. Archimedes traps a circle between polygons. All converge to 3.14159265…
Current estimate
3.000000
True π
3.141592653589793
Error
0.141593
Samples
0
Method
Speed
What’s happening
Each method exploits a different relationship between π and geometry or algebra:
- Monte Carlo: Throw random points into a unit square. The fraction landing inside the inscribed quarter-circle estimates π/4. Convergence: O(1/√n).
- Buffon’s needle: Drop needles of length L onto parallel lines spaced D apart. The probability of crossing a line is 2L/(πD). Count crossings, solve for π.
- Leibniz series: π/4 = 1 − 1/3 + 1/5 − 1/7 + … Beautifully simple, agonizingly slow. Each partial sum oscillates around the true value.
- Polygon approximation: Archimedes’ method: compute the perimeters of regular n-gons inscribed in and circumscribed about a unit circle. As n doubles, both bounds converge to π. He used 96-gons to get 3.1408 < π < 3.1429.
The error graph on the right shows convergence rate. Monte Carlo and Buffon converge as 1/√n (slow). Leibniz converges as 1/n (not much better). Archimedes’ polygon method converges exponentially — each doubling of sides roughly doubles the number of correct digits.