The first studied case of geometric probability – not discrete, therefore – was proposed and solved by Georges Louis Leclerc (1707-1788), Count of Buffon, better known as a naturalist and botanist, although he also had an interest in mathematics. The problem consists of calculating the probability that a needle of length \(l\) thrown on a surface where there are marked parallel lines separated by a distance \(d\) will cut one of the lines – let’s say for simplicity that \(d>l\). The surprising solution is \(\frac{2l}{d\pi}\) – the calculations of this probability can be found at the end of the post.
The Marquis de Laplace later studied the same problem by changing parallel lines for a lattice of perpendicular lines. What I am interested in pointing out here is that Laplace also pointed out that Buffon’s result allowed one to get experimental approximations of the number \(\pi\). Indeed, if we repeated a large number \(P\) of times the experiment of throwing a needle of length \(l\) on a surface where there are marked parallel lines separated by a distance \(d\), and we counted the number of times \(F\) in which the needle cut some of the lines, we would obtain an approximate probability for Buffon’s problem equal to \(\frac{F}{P}\) – following the principle that the probability of an event is equal to the number of favourable cases divided by the number of possible cases. If we equate with the probability calculated by Buffon, we would obtain for the number \(\pi\) an experimental approximation of
$$ \pi\sim \frac{2lP}{dF} $$
After Laplace observed this fact, many experiments have been carried out to calculate approximations of the number \(\pi\) by throwing a needle over and over again on a sheet marked with parallel lines. Undoubtedly, the most famous of these experiments was the one carried out by the Italian Mario Lazzarini in 1901. Almost all that is known about Lazzarini is that he was the author of this famous experiment and must have been a high school teacher.
It seems that Lazzarini designed a fascinating machine that threw the needle over a grid of electric wires – which made the effect of parallel lines – and automatically marked on a continuous piece of paper every time the needle touched the wires. In this way, he was able to perform thousands of experiments in a very short time. As his aims were pedagogical rather than scientific, Lazzarini published his results in an educational journal: Periodico di matematica per l’insegnamento secondario.
These results are undoubtedly surprising, because Lazzarini achieved the feat of getting six exact figures of the number \(\pi\). For this he used a needle of \(l=2.5\) cm with a separation between the parallel lines of \(d=3\) cm; he made 3,408 throws with it and on 1,808 occasions the needle cut one of the parallel lines. As we calculated before, this gives for \(\pi\) a statistical value of
$$\pi\sim \frac{2lP}{dF}=\frac{2\times 2’5\times 3408}{3\times 1808} =3’141592…$$
Surprising… perhaps too surprising: either Lazzarini was very lucky or, more likely, he cheated. Indeed, it is enough to increase or decrease by one unit the number of times the needle touches the parallel lines to obtain very poor values of \(\pi\): 3’1398 and 3’1433, respectively.
Now we shall see that, as Lee Badger showed almost a century later, Lazzarini’s alleged trap is most elegant. Indeed, if one does the math carefully, it turns out that the statistical approximation that Lazzarini unwittingly found is exactly the fraction \(\frac{355}{113}\). And this is not just any fraction; on the contrary, it is a unique fraction because it has the property of being a very good approximation of \(\pi\): of all the irreducible fractions with denominator less than or equal to 113, \(\frac{355}{113}\) is the one that best approximates the number \(\pi\).
This property comes to the fraction \(\frac{355}{113}\) because it is an approximant of the following continued fraction
$$\pi =3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{242+\cdots}}}}.$$
This continued fraction was considered by Euler in his magnificent Introductio in analysin infinitorum of 1748. The first terms he obtained by applying Euclid’s algorithm to the rational number
$$ \frac{31415926535}{10000000000}.$$
Euler noted that the second approximant of the continued fraction, that is,
$$3+\frac{1}{7}=\frac{22}{7}$$
is the famous Archimedean approximation for the number \(\pi\). The fourth approximant, i.e,
$$ 3+\frac{1}{7+\frac{1}{15+\frac{1}{1 }}}=\frac{355}{113},$$
is precisely the approximation found by Lazzarini. Euler attributed it to Adriaen Metius, who published it in 1611, although it had been known in the West since the 16th century. However, the first person to find it was the Chinese mathematician Tsu Ch’ung-Chih in the 5th century.
Every real number \(x\) has a continued fraction of the previous type, and it turns out that every approximant \(p/q\), generated by truncating the continued fraction, has the property of being a better approximation of the number \(x\), in the sense that of all the irreducible fractions with denominator less than or equal to \(q\), \(p/q\) is the one that best approximates the number \(x\).
This is what happens to the approximation \(\frac{355}{113}\), which the clever Lazzarini found for the number \(\pi\): given the uniqueness of \(\frac{355}{113}\), it is very difficult to believe that he found it by throwing needles at random.
References
Lee Badger, Lazzarini’s lucky approximation of \(\pi\), Mathematics Magazine, 67, 83-91, 1994.
Antonio J. Durán, Crónicas Matemáticas, Crítica, Barcelona, 2018.
Solution to the Buffon’s problem
We suppose that the parallel lines are oriented in the direction of the OX axis. Once the needle has been pulled, we look at its northernmost end or, if the needle is parallel to the OX axis, its rightmost end. We place the OX axis on the parallel line closest to that end of the needle from the south and the OY axis so that it passes through that end. We can then identify the needle by a pair of coordinates \((\rho,\theta)\). The coordinate \(\rho\) measures the distance from the fixed end of the needle to the OX axis; it is clear that this coordinate satisfies \(0\le \rho <d\). The coordinate \(\theta\) measures the angle that the straight line on which the needle is placed forms with the OX axis, measured counterclockwise; we therefore have \(0\le \theta < \pi\). Thus the total number of positions for the needle corresponds to the square: \(0\le x< d\) and \(0\le \theta < \pi\), whose area is \(d\pi\). Of all these positions, those in which the needle cuts one of the parallel lines – by the previous conditions it would necessarily be to the line situated on the OX axis (I remember that \(l<d\)) – are given by the set whose coordinates satisfy \(0\le x\le l\mathrm {sen} \theta\) and \(0\le \theta < \pi\); a simple calculation with integrals shows that the area of this set is \(2l\). If the probability of an event is the quotient of the number of favourable cases by the number of possible cases, and we agree that, in this geometric problem, the number of favourable cases corresponds to the area of the set of positions in which the needle cuts the lines, while the number of possible cases corresponds to the area of all the possible positions of the needle, we deduce that the solution to Buffon’s problem is given by the quotient of the previous areas, that is: \(\frac{2l}{d\pi}\).
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