Buffon's needle

The Buffon needle problem asks for the probability that a randomly tossed needle intersects a grid of parallel lines. It allows, among other things, the circle of pi to determine experimentally. The problem belongs to the field of integral geometry and was one of the first problems in this area. Georges- Louis Leclerc de Buffon, treated the problem for the first time in 1733 before the Paris Academy of Sciences, and again in more detail in the Supplement to his Histoire Naturelle 1777.

Implementation

Are needed as many identical rods. On a flat surface parallel construction lines to be constructed at a distance.

All existing sticks are randomly scattered on the surface. Finally, you count how many sticks intersect one of the lines.

If the spacing of the lines is equal to the length of the rods, we obtain an approximation for by multiplied by 2 the total number of used chopsticks and. By the number of rods, which divides cross a line

From the picture on the right to cross 11 of 17 sticks a line; It therefore follows

Valid ( case of short rods), then the formula

Applying, with the total number of fingers refers to the length. is the total number of rods which cross one of the lines and the spacing of two adjacent lines.

In the case ( the case of long rods), the relationship is more complicated.

Mathematical Background

It is the formula derived for the simpler case where the distance between two adjacent lines of the length of a rod corresponds to, ie. However, the same idea can also be used for the derivation of the formula for the case of long needles.

Given a surface with equidistant parallel lines. The angle in radians by which a rod is located on the surface is defined as.

A rod that is covered by a given angle to the surface, is given by the probability density function of a steady uniform distribution. The probability that the rod, a line is given by. The two extreme cases are covered: Is the stick so falls perpendicular to the lines on the surface, so it cuts out still; Is it, however, parallel to the lines, there can be no intersection ( the stick is next to lines) or infinitely many points of intersection ( the stick is on a line ).

The probability that a needle hits a line is thus defined by the ratio of surface to be given ( see picture).

If we calculate the integrals over the two probability density functions, we obtain the probabilities:

Therefore. According to the law of large numbers, the relative frequency approximates in the practice for a large number of rods of the above calculated probability. Therefore, one can say.

Proof of needles of any length

For case ( short needle ), consider only a vertical strip of width d and only the right edge (the other regions of the surface are obtained by translation). A needle of length touches the edge if the left end point of the needle is within a strip of width from the right edge. The probability is. Integration over gives the probability that the needle touches a line:

In case ( a long needle ) one has to take into account the possibility of multiple contact lines, so that the formulas are more complex:

You can use instead of the Arkussekans in the last formula, too. In the case of both yield formulas. In the case of long needle grows monotonically with the needle length and goes on to 1

The proof of barber for small needles

Barbier gave in 1860 for the case of small needles a proof which does not require integration. First, it is shown that the sought expectation E is a linear function of the length of the needle for the number of hits ( of length l ), which also applies if one approximates an arbitrary curve by a polygonal line. To determine the proportionality constant c. To this end, the case of circles with diameter d is considered, who always exactly 2 results on the parallel lines at a distance d. It approaches the circle by one and rewritten polygons P, Q ( with perimeter p, q) and obtain:

And at border crossing of the page number of polygons to infinity:

Thus, and thus the desired result.

Because of its elegance of the proof of Barber of Aigner and Ziegler was recorded in the book of evidence.

Generalizations

You can ask for other figures as routes that are randomly thrown on a plane, eg, polygons. From the formula for the Buffonproblem follows for polygons with diameter small and extent of the hit probability

When approaching a closed curve through such a polygon, also arises in this case, the probability of a hit, when you replaced the scope of the traverse through the length of the curve (again, the diameter must be smaller ). These are the starting points to methods of integral geometry to obtain from appropriate hit probabilities formulas, for example, for the arc length of curves.

In the so-called Buffon -Laplace needle problem one asks for the probability of a hit for a square lattice with side lengths and. For small needles ( and ) results

The problem can also be extended to the union of other bodies, as in the well of Buffon in 1733 addressed Franc Carreau problem: you throw a coin ( circular disc, diameter ) on a square lattice (side length ), and it should be. What is the probability that the coin on the edge comes to rest? Since the coin can only touch the edge, if its center is at a distance from the edge, can be specified by simple geometrical considerations, the hit probability as area ratio:

For a fair game must be.

Historical

About the needle problem and similar problems reported Buffon in 1733 before the French Academy of Sciences, as Fontenelle reported. Of interest, it was in connection with a then popular with noblemen game: you throw a coin on a tiled pattern and bet on the position of the coin, if it touches one of the cracks or not ( Franc- Carreau problem). Buffon went on a detail in 1777, where he already guessed the correct answer in the case of the needle problem. In particular, he derived from the formula for short needles that for a fair bets ( ie strength for betting with 50 % winning percentage ) at the needle problem must be the money. For the design on a square lattice, however, he gave an incorrect formula. The correct formula ( in the case of a rectangular lattice) gave only Pierre Simon de Laplace in 1812, without mentioning Buffon. Citing the example of the application of the theory of probability to the determination of the curve length, and surface areas. This was then carried out, among others, by Isaac Todhunter 1865. From the study of the Buffon's needle problem also Croftons formula revealed (see Morgan Crofton ) for the arc length of a curve in the integral geometry (1868 ).

A certain Mario Lazzarini said to have 1901 performed the most extensive experimental test of the formula ( with a specially built machine ), with 3408 litters of needles with a length ratio. His result was 1808 hits, so that a value of from revealed to six decimal places exactly. However, the figures given by him are suspicious ( other authors interpreted this as a fluke. ) And seem tailored to mark the well-known approximation for Pi to obtain. The astronomer Wolf introduced with the experiment in 1850 and received in 5000 litters 2532 hit, according to an estimate for Pi Further experiments led Ambrose Smith 1855, with 3204 litters () at 1218 hits, which yields a value of for Pi. Hans -J. Bentz led the experiment with 2000 litters and received for Pi

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