Dropping Needles on the Floor Computes Pi

[00:00:00]Take a sheet of paper ruled with evenly spaced parallel lines and drop a needle onto it from a small height. The needle lands somewhere tilted at some angle and one of two things happens. [music] Either it comes to rest lying across one of the lines touching it or it settles in the blank space between two lines [music] touching nothing.

[00:00:20]Drop it once and you learn almost nothing. So drop [music] it again. This one lands flat between the lines, no contact. The next lands nearly upright and clips a line. Another falls at a slant straddling a line cleanly. [music] A fourth lands in a gap and misses.

[00:00:38]Nothing about a single drop is special.

[00:00:40]Whether any one needle crosses a line depends on exactly [music] where its center happens to fall and exactly how it happens to be turned and both of those are pure chance. [music] You cannot predict a single throw. But watch what the whole handful is doing.

[00:00:55]Out of the ones dropped so far, roughly two in every three are lying across a line >> [music] >> and the rest sit in the gaps. That ratio is not an accident and it does not depend on how carefully you throw. It is baked into the geometry of a needle falling on a set of lines. The plan for the rest of this is simple. Keep dropping needles, count the fraction that cross and see what number that fraction is quietly aiming for. Let me fix the setup so the counting means something. The lines are spaced a fixed distance apart. Call that spacing D. The needle has a length equal to that same spacing. So its [music] length also equals D.

[00:01:34]That is the clean case to start with. A needle exactly as long as the gap between lines. Now throw in bulk. After 10 drops the fraction that cross might be six out of [music] 10 or seven or five.

[00:01:47]10 throws is far too few to trust. After a hundred drops the fraction steadies a little. After a thousand it steadies more. Keep going into the tens of thousands, [music] and the fraction stops wandering and parks itself very close to one specific value.

[00:02:03]That value is about 0.637.

[00:02:06]Just under 2/3 of the needles cross a line, >> [music] >> and just over a third land clean. Run the experiment again with a fresh set of throws, and you get the same number back.

[00:02:16]Run it a third [music] time, same again.

[00:02:19]The individual drops are random, but the fraction they pile up to is not.

[00:02:23]>> [music] >> It is a fixed constant of the arrangement, and every batch of throws is just a noisy measurement of it. So, the question sharpens. Where does 0.637 [music] come from?

[00:02:34]It is not a round number. It is not a half, not 2/3 exactly. [music] It is some particular number sitting between them, and it is stable enough that you could hand it to someone as a physical constant. Something in the geometry is choosing it. Here is where it turns strange.

[00:02:51]Work out [music] the crossing probability exactly from the geometry of the line, and the answer is 2 divided by the quantity pi times the spacing, all times the needle length.

[00:03:01]>> [music] >> In our clean case, where the length equals the spacing, the length and spacing cancel, and the probability is simply 2 over pi, 2 over pi.

[00:03:10]The circle constant, the ratio of a circle's circumference to its diameter, sitting inside a problem about straight needles and straight [music] lines.

[00:03:18]There is no circle anywhere in the setup. There is a floor of parallel lines and a thin straight rod, and yet the fraction that cross is governed by pi. Punch 2 over pi into a calculator, and you get 0.6366, which is exactly the number the drops were settling on. And once pi is sitting in the formula, you can turn the whole thing around. Solve for pi instead of for the probability.

[00:03:43]The probability is just the fraction that cross, [music] which is the number of crossings divided by the number of throws. Rearrange, and pi equals 2 times the number of throws divided by the number of crossings. Two times throws over crossings. [music] That is a recipe. Drop a pile of needles, count how many touch a line, plug the two counts into that fraction, >> [music] >> and out comes an estimate of pi.

[00:04:07]Read that again slowly. You can measure pi by dropping needles on the floor and counting.

[00:04:13]No circle drawn, no diameter measured, nothing curved [music] touched at any point.

[00:04:18]Pure randomness, filtered through a floor of straight lines, hands you the circle constant. The rest of this is about why that possibly works. [music] To see where pi sneaks in, follow one needle carefully and describe its landing with just two numbers. The first is how far the needle's center lands from the nearest [music] line.

[00:04:38]Call that distance X. Since the lines are a distance D apart, the nearest line is never more than half of D away. So, X runs from zero, right on a line, up to D over two, dead center in a gap. The second number is the tilt. Call the angle the needle makes with the lines theta. By symmetry, we only need it to run from flat, zero, up to a quarter turn, straight across the lines.

[00:05:02]Now, the crossing condition. A needle tilted at angle theta reaches out from its center toward the nearest line by an amount equal to half its length times the sine of that angle. If that reach is at least as big as X, the distance to the line, the needle touches. So, it crosses exactly when X is less than or equal to half the length times the sine of theta. Lying flat, sine is zero. The reach is nothing, and it only crosses if it was already sitting on a line.

[00:05:30]Standing straight across, sine is one.

[00:05:33]The reach is at its longest, and it crosses from the farthest away.

[00:05:37]Picture the two numbers as a point in a rectangle. [music] The horizontal axis is the angle theta from zero to a quarter turn. The vertical axis is the distance X from zero to half of D. Every possible landing is one point somewhere in that rectangle spread perfectly evenly.

[00:05:54]The crossings are the landings sitting underneath the curve given by half the length times the sign of the angle. So, the probability of crossing is just the fraction of the rectangle that [music] lies beneath that sine curve.

[00:06:06]A crossing question has become an area question. So, we need the area under a sine curve, and that is exactly where pi walks in.

[00:06:14]The area of the whole rectangle is easy.

[00:06:16]Its width is a quarter turn, and its height is half of D. So, the total area is a quarter turn times half of D. The area under the sine curve takes one small fact. Add up the sine of the angle as it sweeps from flat all the way to a quarter turn, and the total, the area under that arc, comes out to exactly [music] one in the natural units where the reach is scaled by half the length.

[00:06:40]That is a clean exact result. The accumulated sine over a quarter turn is one, and multiplying by the half length [music] gives an area of half the length.

[00:06:50]Now divide. The crossing probability is the area under the sine divided by the area of the rectangle. That is half the length divided by a quarter turn times half the spacing. The quarter turn [music] is pi over two. Push it all through, and the halves cancel. The spacing cancels against the length in the clean case, and what is left is two over pi.

[00:07:12]That is the whole reason pi is here. A quarter turn is a piece of a circle, and a quarter turn measured in the natural way is the number pi over two.

[00:07:20]The moment you average a needle's behavior over every angle it could land at, you are averaging over a slice of a circle, and the size of that slice is measured by pi. Sine is the shadow an angle casts. Averaging that shadow over a quarter turn gives two over pi, and the circle constant drops straight out of the crossings.

[00:07:40]No circle was ever drawn, but a full range of angles is a circle in disguise.

[00:07:45]Now, put the recipe to work honestly with a computer standing in for the hand.

[00:07:49]>> [music] >> Fix a random seed so the run is reproducible. Set the needle length equal to the spacing and simulate drops.

[00:07:56]For each drop, the machine picks a center position and an angle at random, checks whether the distance is under half the sign of the angle, and tallies a crossing or a miss.

[00:08:06]Then it forms two times the throws over the crossings and reports [music] its running guess at pi. After 100 drops, the estimate is 2.82, badly off. The counts are just too small.

[00:08:17]After 1,000 drops, it climbs to [music] 3.09, closer. After 10,000, it reaches 3.15, now agreeing with pi in the first two decimals. After 100,000, it holds near 3.15.

[00:08:31]And after 1 million drops, the estimate is 3.142, sitting a thousandth away from the true value of pi. Plot the running estimate against the number of throws, and you see it home in on the flat line at pi, the swings shrinking as the pile grows.

[00:08:47]But look at how slowly it tightens.

[00:08:49]Going from 10,000 throws to a million, a hundredfold more work, only bought a couple more correct digits. That is the signature of this kind of estimate. The error falls like one over the square root of the number of throws. So, to add one more decimal place of accuracy, you need about a hundred times as many needles. It converges and it is honest, [music] but it is a slow way to compute pi. The point was never speed. The point is that it works at all, that counting crossings pins down the circle constant, which brings us to a famous and slightly embarrassing chapter. In 1901, an Italian named Mario Lazzarini reported doing exactly this experiment by hand.

[00:09:30]He said he threw a needle 3,408 times, counted his crossings, and got an estimate of pi equal to 355 over 113. That fraction, 355 over 113, is a genuinely beautiful approximation of pi. It agrees with the true value to six decimal places, [music] an accuracy people had prized for centuries. And that is precisely the problem. Getting six correct digits of pi from a few thousand random needle drops is wildly better than the one over square root law allows. A few thousand throws should give you a decimal place, maybe two, not six. His answer is far too good to have come from honest random throwing.

[00:10:14]What almost certainly happened is a stopping rule. If you keep throwing needles and only decide to stop at the exact moment your running estimate happens to land on the pretty target you were hoping for, you can hit that target on the nose.

[00:10:27]>> [music] >> You are no longer measuring pi. You are waiting for chance to spell out an answer you already chose, then freezing [music] it.

[00:10:35]The lesson is not that the needle method is broken. The lesson is that when you run a random experiment, you have to fix how much data [music] you will collect before you look at the answer.

[00:10:45]Or the randomness will happily lie to you. There is one more turn, >> [music] >> and it lifts the whole thing to something bigger.

[00:10:52]Go back to what actually determined the crossing rate. It was the needle's length, not its color, [music] not its exact shape, the length.

[00:11:01]And that hints at a deeper rule. Suppose that instead of a straight needle, you drop [music] a bent piece of wire, or a wildly curved one of some total length.

[00:11:10]Break the wire in your mind into many tiny straight segments. Each little segment is basically a tiny needle, and its expected number of crossings follows the [music] same law set by its little length.

[00:11:23]Crossings simply add up, so the expected [music] number of crossings for the whole curve is set by the whole length. Bend the wire however you like, and as long as the total length stays the same, the average number of times it crosses the lines [music] stays the same.

[00:11:39]That is a striking statement. The expected number of crossings of a curve with a grid of random lines is proportional to the length of the curve and to nothing else about its shape.

[00:11:50]This is the heart of a subject called integral geometry.

[00:11:53]>> [music] >> And the precise version is known as the Cauchy-Crofton formula. It says length itself can be measured by counting crossings with random lines. In fact, you can flip it around and use crossing counts to define what the length of a complicated curve even means. Buffon's little needle, dropped on a ruled page, is the very first and simplest case of that whole idea. So, step back and look at what a needle on the floor actually did.

[00:12:19]There was no circle in the setup, only straight lines and a straight rod, and a lot of chance. Yet, the fraction that crossed was [music] two over pi. And by counting crossings, you can measure pi to as many digits as your patience [music] allows, growing like the square root of the number of throws. The circle constant appeared because averaging the needle over every angle it could take is secretly averaging over a quarter of a circle. And the average of a sign across that quarter turn is two over pi. The whole effect only depends on length, which is why a bent wire obeys the same rule, and why this [music] humble puzzle opened the door to integral geometry.

[00:12:59]Georges Louis Leclerc, the Count of Buffon, asked this question in 1733 and published it in 1777, [music] near the very beginning of geometric probability. He could not have run a million throws, but the number he was reaching for is the same one a computer confirms today, hidden in plain sight in the simple act of dropping a needle and counting how often it lands on a line.