Pi sticks

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Georges Louis Leclerc, Comte de Buffon, is best remembered as one of the founders of the science of natural history and the author of a massive 44 volume work on the subject.  But earlier in life, in his twenties, he had an interest in mathematics.  In 1733 he set out to gain membership of the Royal Academy Of Sciences in Paris by proposing a solution to a fascinating mathematical problem.

Buffon began his paper with an exploration of a simple popular game by the name of franc-carreaux, where players would throw a coin on to the tiled floor, with wagers placed on whether the coin would touch one of the cracks between the tiles.  Buffon’s solution to the probability is contained in this diagram:

buffon-franc-carreaux

The diagram shows that for the coin to touch the crack on a particular tile the centre of the coin must fall outside the smaller shaded square in the middle.  The small square has sides equal to the sides of the larger square minus the radius of the coin, so while the largest square has an area of L2 squared, the smaller one has an area of (L -2r)2.  The probability that the centre of the coin will fall within the smaller square is simply the area of the smaller square divided by the area of the largest square.  For the game to be a fair one, with equal chances of the coin touching a crack or not:

pi-sticks-equation-01

If you follow that equation through you find that in order to have equal chances, the side of the tile must be 
pi-sticks-equation-02times the radius of the coin.

Buffon knew that such a simple solution would not suffice to gain the approval of the Academy but he knew where he was going next.  What, he asked, if instead of a coin the object to the thrown were to be of a different shape, like a triangle or a needle.  In fact it was the problem of the needle that Buffon decided to tackle.  Rather than square tiles he imagined a floor made up of boards with equally spaced cracks and a needle which was exactly as long as the boards were wide.

The problem is more complicated because, as opposed to the coin, which is entirely symmetrical, the needle will cover a variable amount of space between the cracks according to the angle at which it falls.  If it falls parallel to the cracks then it covers no horizontal space at all; if it falls at right angles to the cracks then it will cover all the space between two cracks.  When parallel it has almost no chance of touching a crack (given that these are theoretical cracks and needles, with length but no width) while if it falls at right angles it is always going to touch a crack.  In between parallel and a right angle the amount of horizontal space taken up will depend upon the angle of the needle and, more precisely, upon the cosine of the angle.

buffon-needle-positions

It took some simple calculus for Buffon to arrive at the answer and it would be tiresome to follow it because the real interest lies in the sheer simplicity of the answer.  The probability of the needle touching a crack turns out to be 2/π , about 64 per cent.  It was a neat solution to an apparently difficult problem, and it was enough to gain Buffon his membership of the Academy.

Maybe Buffon lost interest at this point because he was very close to something even more interesting.  Apparently he did not investigate the connection between the two equations above, so he never realized that if he had continued his thought experiment using the coin as well as the needle he would immediately have seen that if the coin’s diameter was equal to the width of the gaps (i.e. the length of the needle) the probability of touching a crack with the circumference of the coin would always be two.

buffon-coin-positions

Now Buffon knew as well as we do that the circumference of a circle is π times the diameter and he would surely have seen that whereas the probability of needle of length L touching a crack was 2/π  the circumference of the circle, with a length of π × L had a probability of 2, which is π × (2/π) .  It doesn’t take a genius (which he was) to see that the probability of touching a crack seems to depend directly on the length of the line.

In fact, that’s the whole of the answer but it took 100 years before another mathematician by the name of Barbier proved it.  Whether it’s a straight needle, some regular or irregular geometric shape or even a piece of string, the probability of touching a crack depends simply on the length! Double the length and you double the chance. It’s an example of what is known in mathematics as additive expectation, where if you know what to expect from one event you can easily translate that into what to expect from multiple events.

But there’s something else fascinating about Buffon’s discovery.  If you look at his simple formula for the probability of the needle touching a crack you can see that it involves the value of .  Assuming the formula to be correct, which it is, every time you throw a needle on to the floor you are being given information about the value of .  To take a silly example that might cast light, supposing you didn’t know the value of ‘half’ you just knew the label ‘half’.  Now someone tells you that coin tossing can help you find what ‘half’ means. If you were a to toss a coin a few times, each toss would be giving you information about the value of ‘half’ because the probability of heads or tails is 1/2.  Randomness being what it is, a small number of tosses would probably be very misleading but if you continue to generate a large number of tosses it’s highly likely that you would arrive at quite an accurate estimate of what ‘half’ means.

Buffon’s needle does the same thing for .  If you throw the needle a large number of times you can use the simple formula to get an estimate of : take the total number of throws and divide that number by half the number of times the needle crosses a crack. In fact an Italian mathematician by the name of Mario Lazzarini claimed to have done just that at the turn of the 20th century.  He had, he said, thrown the needle 3408 times and it had provided an estimate for  of 355/113.  This is a fraction well known to mathematicians as a reasonable approximation of the value of  – in fact it is accurate to six figures.  Some people doubt whether the experiment was ever carried out but there’s no reason in principle, apart from arm ache, why it wouldn’t have worked.

In other words, if you set up the right simple experiment, you can persuade the universe to calculate for you a number that great minds have struggled with over the ages. In what possible sense is that not awesome?

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Where mathematics becomes an object of desire.