Buffon's Needle

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====Generalization of the problem====
====Generalization of the problem====
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The Buffon’s needle problem has been generalized so that the probability of a cross can be calculated for a needle of any length and paper with any spacing. It has been proven that for a needle shorter than the distance between the lines, the probability of a cross is <math> \frac {2*l}{\pi*d} </math>. This equation makes sense when we consider the normal case, where <math> l=1 </math> and <math> d=1 </math>, so these variables disappear and the probability is <math> \frac {2}{\pi} </math>.
The Buffon’s needle problem has been generalized so that the probability of a cross can be calculated for a needle of any length and paper with any spacing. It has been proven that for a needle shorter than the distance between the lines, the probability of a cross is <math> \frac {2*l}{\pi*d} </math>. This equation makes sense when we consider the normal case, where <math> l=1 </math> and <math> d=1 </math>, so these variables disappear and the probability is <math> \frac {2}{\pi} </math>.

Revision as of 10:25, 28 May 2010

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Buffon's Needle
The Buffon's Needle problem is a mathematical method of approximating the value of pi (\pi = 3.1415...) involving repeatedly dropping needles on a sheet of lined paper and observing how often the needle crosses a line.

Contents

Basic Description

The method was first used to approximate pi by Georges-Louis Leclerc, the Comte de Buffon, in 1777. Buffon apparently first tried throwing bread sticks over his shoulder and counting how often they crossed a line on his tile floor. He was intrigued by the relationship he observed between the probability of a cross and pi.


Subsequent mathematicians have used the method with needles instead of bread sticks, or with computer simulations. In the case where the distance between the lines is equal the length of the needle, an approximation of pi can be calculated using the equation


\pi \approx {2*\mbox{number of drops} \over \mbox{number of hits}}

A More Mathematical Explanation

Will the Needle Cross a Line?

[[Image:willtheneedlecross ps [...]

Will the Needle Cross a Line?

To prove that the Buffon's Needle experiment will give an approximation of pi, we can consider which positions of the needle will cause an intersection. The variable θ is the acute angle between 0 and  \tfrac {\pi}{2} made by the needle and an imaginary line parallel to the ones on the paper. Finally, D is the distance between the center of the needle and the nearest line.


Since the needle drops are random, there is no reason why the needle should be more likely to intersect one line than another. As a result, we can simplify our proof by focusing on a particular strip of the paper bounded by two lines.


We can extend line segments from the center and tip of the needle to meet at a right angle. A needle will cut a line if the green arrow, D, is shorter than the leg opposite θ. More precisely, it will cross when

 D \leq \left( \frac{1}{2} \right) \sin(\theta). \

See case 1, where the needle falls at a relatively small angle with respect to the lines. Visually we can see that a needle lying at such a small angle will only intersect if the center falls close to one of the parallel lines. In case 2, the needle crosses even though the center of the needle is far from both lines because the angle is so large.


The Probability of an Intersection

To approximate pi using Buffon's method, we need to know the probability of a cross. If we graph the outcomes of θ along the X axis and D along the Y, we have the sample space for the trials. In the diagram below, the sample space is contained by the dashed lines.


The sample space is useful in this type of simulation because it gives a visual representation of all the possible ways the needle can fall. Each point on the graph represents some combination of an angle and distance that a needle might occupy. We divide the area that contains combinations that will cause an intersection by the total possible positions to calculate the probability.

image:Minigraph.jpg

There will be an intersection if  D \leq \left ( \frac{1}{2} \right ) \sin(\theta) \  , which is represented by the blue region. The area under this curve represents all the combinations of distances and angles that will cause the needle to cross a line. The area under the blue curve, which is equal to  \tfrac {2}{\pi} in this case, can found by evaluating the integral

\int_0^{\frac {\pi}{2}} \frac{1}{2} \sin(\theta) d\theta

Then, the area of the sample space can be found by multiplying the length of the rectangle by the height.

 \frac {1}{2} * \frac {\pi}{2} = \frac {\pi}{4}

The probability of a hit can be calculated by taking the number of total ways a cross can occur over the total number possible outcomes (the number of trials). For needle drops, the probability is proportional to the ratio of the two areas in this case because each possible value of θ and D is equally probable. The probability of a cross is

P_{hit} = \cfrac{ \frac{1}{2} }{\frac{\pi}{4}} = \frac {2}{\pi} = .6366197...


Using Random Samples to Approximate Pi

The original goal of the Buffon's needle method, approximating pi, can be achieved by using probability to solve for pi. If a large number of trials is conducted, the proportion of times a needle intersects a line will be close to the probability of an intersection. That is, the number of line hits divided by the number of drops will equal approximately the probability of hitting the line.

 \frac {\mbox{number of drops}}{\mbox{number of hits}} \approx P_{hit} = \frac {2}{\pi}

So

 \frac {\mbox{number of drops}}{\mbox{number of hits}} \approx \frac {2}{\pi}

Therefore, we can solve for pi:

 \pi \approx \frac {2 * {\mbox{number of drops}}}{\mbox{number of hits}}


Watch a Simulation

http://mste.illinois.edu/reese/buffon/bufjava.html


Why It's Interesting

Monte Carlo Methods

The Buffon's needle problem was the first recorded use of a Monte Carlo method. These methods employ repeated random sampling to approximate a probability, instead of computing the probability directly. Monte Carlo calculations are especially useful when the nature of the problem makes a direct calculation impossible or unfeasible, and they have become more common as the introduction of computers makes randomization less laborious.

Pi is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space. Pi is an irrational number, which means that its value cannot be expressed exactly as a fraction a/b, where a and b are integers. Since π is irrational, mathematicians have been challenged with trying to determine increasingly accurate approximations. In the past 50 years especially, improvements in computer capability allow mathematicians to determine more decimal places. Nonetheless, better methods of approximation are still desired.


A recent study conducted the Buffon's Needle experiment to approximate pi using computer software. The researchers administered 30 trials for each number of drops, and averaged their estimates for pi. They noted the improvement in accuracy as more trials were conducted.

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These results show that Buffon's Needle approximation is a relatively tedious process because the estimates converge towards pi rather slowly compared to other computer generated techniques. Nonetheless, the intriguing relationship between the probability of a needle's intersection and the value of pi has attracted mathematicians to study the Buffon's Needle method since its introduction in the 18th century. Buffon's approach has since been used in studies ranging from the behavior of ants to the Manhattan Project of developing atomic bombs in WWII. The problem remains relevant as variations can make the its accessible to a wide range of mathematical abilities. The method can be found everywhere from a simple demonstration in a middle school classroom to an exam for graduate students.


Generalization of the problem

Image:Picture 1.jpg


The Buffon’s needle problem has been generalized so that the probability of a cross can be calculated for a needle of any length and paper with any spacing. It has been proven that for a needle shorter than the distance between the lines, the probability of a cross is  \frac {2*l}{\pi*d} . This equation makes sense when we consider the normal case, where  l=1 and  d=1 , so these variables disappear and the probability is  \frac {2}{\pi} .

The generalization of the problem is useful because it allows us to examine the relationship between length of the needle, distance between the lines, and probability of an intersection. The variable for length is in the numerator, so a longer needle will have a greater probability of a cross. The variable for distance is in the denominator, so greater space between lines will decrease the probability.

To see how a longer needle will affect probability, follow this link: http://whistleralley.com/java/buffon_graph.htm


Needles in Nature

Applications of the Buffon's Needle method are even found naturally in nature. The Centre for Mathematical Biology at the University of Bath found uses of the Buffon's Needle algorithm in a recent study of ant colonies. The researchers found that an ant can estimate the size of an anthill by visiting the hill twice and noting how often it recrosses its first path.

Ants generally nest in groups of about 50 or 100, and the size of their nest preference is determined by the size of the colony. When a nest is destroyed, the colony must find a suitable replacement, so they send out scouts to find new potential homes.

In the study, scout ants were provided with "nest cavities of different sizes, shapes, and configurations in order to examine preferences" [2]. From their observations, researchers were able to draw the conclusion that scout ants must have a method of measuring areas.

A scout initially begins exploration of a nest by walking around the site to leave tracks. Then, the ant will return later and walk a new path that repeatedly intersects the first tracks. The first track will be laced with a chemical that causes the ant to note each time it crosses the original path. The researchers believe that these scout ants can calculate an estimate for the nest's area using the number of intersections between its two visits.

"In effect, an ant scout applies a variant of Buffon's needle theorem: The estimated area of a flat surface is inversely proportional to the number of intersections between the set of lines randomly scattered across the surface." [7]

This idea can be related back to the generalization of the problem by imagining if the parallel lines were much further apart. A larger distance between the two lines would mean a much smaller probability of intersection. We can see in case 3 that when the distance between the lines is greater than the length of the needle, even very large angle won’t necessarily cause an intersection.

The ants can measure the size of their hill using a related and fairly intuitive method: If they are constantly intersecting their first path, the area must be small. If they rarely recross the first track, the area of the hill must be much larger so there is plenty of space for a non-intersecting second path.

This natural method of random motion in nature allows the ants to gauge the size of their potential new hill regardless of its shape. Scout ants are even able to asses the area of a hill in complete darkness. The animals show that algorithms can be used to make decisions where an array of restrictions may prevent other methods from being effective.


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References

[1] http://www.maa.org/mathland/mathtrek_5_15_00.html

[2] http://mste.illinois.edu/reese/buffon/bufjava.html

[3] http://www.absoluteastronomy.com/topics/Monte_Carlo_method

[4] The Number Pi. Eymard, Lafon, and Wilson.

[5] Monte Carlo Methods Volume I: Basics. Kalos and Whitlock.

[6] Heart of Mathematics. Burger and Starbird

[7] http://math.tntech.edu/techreports/TR_2001_4.pdf





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