9x.cl

Plate V of IX

Needle

Finding π by dropping things on the floor

Plate V. Short line segments scattered at random angles across a floor of evenly spaced parallel lines, with crossing needles highlighted. Every control writes itself into the page address, so a figure you like has a link of its own.

What you are looking at

A floor with parallel lines ruled across it, a distance d apart. A needle of length ℓ, no longer than the gap, dropped from a height with no aim at all. The question Buffon asked in 1733 was: what is the chance it lands across a line?

The answer is 2ℓ / πd. There is a π in it, and there is no circle anywhere in the problem — the π arrives because the needle can point in any direction, and integrating over all those directions is where circles get in.

Turn the formula around and you have an instrument. Drop N needles, count the C that cross, and

π ≈ 2ℓN / dC

It works. It is also one of the worst ways to compute π ever devised, and watching it fail is the real lesson of this plate. The error shrinks like 1/√N: to buy one more correct decimal digit you must throw one hundred times as many needles. Four thousand needles gets you to about 3.14. A million might get you 3.1416 if you are lucky with the noise. The running estimate on this page wobbles above and below the true value forever, never converging so much as loitering.

That is not a defect of this particular experiment. It is the central fact about Monte Carlo methods, which are now how a great deal of physics, finance and rendering actually gets done. You accept 1/√N convergence because the alternative — an exact integral in forty dimensions — does not exist.

Try this

  • Let it run to ten thousand needles and watch the lower panel. That red line is not a curve settling down; it is a random walk being slowly squeezed by the shaded band, and it will still be wandering when you get bored.
  • The band is the theoretical 95% interval, computed from the binomial spread of the crossing count — not fitted to the data. The trace should sit inside it about nineteen times out of twenty, and you can watch it fail to.
  • Press New run a few times. Each run is a different wobble. Some are near-perfect at 500 needles by pure luck — which is exactly the trap Lazzarini fell into, or walked into deliberately.
  • Shorten the needle. Short needles cross less often, so you count fewer events and the estimate gets noisier — watch the band widen to match. Long needles, right up against the line spacing, are the most efficient.
  • Push Throws per frame to 800 and leave it. A few hundred thousand throws take under a minute and still will not reliably give you 3.1416.

Notes

Georges-Louis Leclerc, Comte de Buffon, posed the problem in 1733 and published it in 1777, inside a treatise on moral arithmetic. It is generally counted as the first geometric probability problem, and it predates any notion of Monte Carlo integration by nearly two centuries.

The best story attached to it is Mario Lazzarini's. In 1901 he reported dropping a needle 3,408 times and obtaining π = 3.1415929, correct to six decimal places. This is far better than 3,408 throws has any right to produce. The number 3,408 is also suspicious: with his stated needle and line lengths, it is precisely the count that makes the arithmetic collapse to 355/113 — the classic Chinese approximation to π, accurate to six places, known since the fifth century. The consensus is that Lazzarini stopped throwing at the moment the answer looked good. It is the canonical cautionary tale about knowing what you want your experiment to say.

Further reading

Buffon, Essai d'arithmétique morale (1777). For the Lazzarini business, N. T. Gridgeman, "Geometric Probability and the Number π," Scripta Mathematica (1960).