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Boris Gourévitch
The world of Pi - V2.57
modif. 13/04/2013



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Georges-Louis Leclerc Comte de Buffon
(1707 - 1788)



An elegant result

If a needle of length 2a is dropped on a parquet formed of floorboard of width 2b, the probability that the needle cuts one of the lines of this parquet is
Reference from the Encyclopedie of Integer Sequences : A060294

Slices of life

Georges Louis Leclerc was born in 1707. Naturally, his main work is that of a naturalist. So, general and particular natural History (15 volumes!), the natural History of birds (9 vol.), Supplement in the natural history (7 vol.) and other natural History of minerals and treaty of the magnet (5 vol.) will be worth him the greatest celebrity... Especially since his style is very pleasant... But, philosopher also , this excellent administrator was member of all the great European academies and was interested in mathematics... Oh, oh!.. In the Attempt of moral arithmetic published in 1777, an entitled volume Memory on the game of the "franc carreau" presents indeed the famous problem of the needle...

About

This problem is one of the first to bring in probability and Pi. Another proof of the omnipresence of this number in mathematics!!! Obviously, this presence is not foreign to the original geometrical definition of Pi as perimeter of the circle of diameter 1... But the subtle mixture with the analysis is very pleasant! So pleasant moreover that two proofs are told below! Naturally, the floor is supposed flat (let us stay in euclidian spaces!).
In fact, it is not necessary to hope to obtain a good estimation of Pi by going only to buy a package of needles on the corner of the street!!! A preciseness of 10 -3 is obtained with a probability of 95 % from 888 697 throws!

Proofs

1) A little of geometry...

With hypotheses above (length a needle 2a, width of floorboard 2b) :
Let us appoint by y the distance of the middle of the needle () to the line of bottom and by F the angle between the needle and the line .
If or then respectively, the needle cuts the line of the height or the one of the bottom...
So, there will be intersection if the point P defined by its coordinates (ß, y) belongs to the zone hatched of the graph opposite:
Now, the distribution of y on [0,2b] and ß on [0,/2] is uniform and the looked probability represent the ratio of the area of the surface hatched to the area of the rectangle [0,2b]*[0,/2] (which contains all the possible cases). Now the area of the rectangle is A=/2*2b=b.
and the area of the hatched surface is B=2 (the 2 hatched surfaces have the same area because they are symmetric with regard to the line of equation y=b)
So the looked probability is p===.
If we have a great patience, it is enough to throw a great number of time n the needle on the ground and to count the number of intersections. By setting b=2 and a=1, the law of the great numbers allows us to conclude that .

2) Emile Borel also found a crafty and fast demonstration of this Buffon's result... Whatever is the form, the number of intersections of a needle with the edge of slats is proportional to its length 2a and inversely proportional to the slats of width 2b. So he can write in the form of Ka/b. Remain to find constant K...
And there, craftiness! Let us take a circular needle of diameter 2b. The perimeter, so the length, is obviously 2b.Whatever the way of which it falls, it cuts exactly twice lines so we deduct from it 2=Ka/b=Kb/b=K, where from K=2/ and the looked probability is !

Trials

Some patient individuals tempted their chance in the trow of needles... Notably Wolf in 1850 which arms itself with 5000 needles with a=0,8b and observes 2532 intersections that brings him to the estimation =3,1596.
For my part, I have not yet tried. In my opinion, principle is especially interesting, but experiment is long and little effective, that limits the interest. But nothing prevents from automating process!


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