Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Math Surprises: An Example

August 2001

Compte de Buffon (1707-1788) in the 18th century posed and solved the very first problem of geometric probability. A needle of a given length L is tossed on a wooden floor with evenly spaced cracks, distance D apart. What is the probability of the needle hitting a crack? (The problem is nowadays known as Buffon's Needle problem.) The answer he discovered with the help of integral calculus is given by the simple formula [Beckmann, Dörrie, Eves, Kasner, Paulos, Stein]

(*)P = 2L/πD

With P approximated by the ratio of hits to the total number of tosses, the formula offers a way of evaluating π, an observation that eventually led Pierre Simon Laplace (1749-1827) to propose a method, known today as the Monte Carlo Method, for numerical evaluation of various quantities by realizing appropriate random events.

History records several names of people who applied the method manually to approximate π. A Captain Fox [Beckmann, p. 163] indulged himself while recuperating from wounds incurred in the Civil War. H. Dörrie [Dörrie, p. 77] mentions Wolf from Zurich (1850) who obtained π = 3.1596 after 5000 throws, and the Englishmen Smith (1855) and Fox (1864) who obtained π = 3.1553 and π = 3.1419 after 3200 and 1100 throws, respectively. The record is held [Eves, 290 and Kasner, p. 247] by an Italian mathematician Lazzerini (1901), who attained the accuracy of six decimal (3.1415929) digits after just 3408 throws. As there does not appear any definite relationship between the number of throws and the accuracy of the approximation, Lazzerini's success should probably be attributed to chance or luck as the case may be. (The best Theory of Probability can do is to assure accuracy with a certain probability. The expected accuracy may or may not be achieved in any particular set of experiments.)

Several present day simulations are available on the Web that permit one to toss the needle thousands of times without leaving one's armchair. Below I wish to approach Buffon's result from a different perspective.

π, that is defined as the ratio of the circumference of a circle to its diameter, has proved time and again to be one of the most fundamental mathematical constants by showing up unexpectedly in many important formulas apparently unrelated to circle. (*) appears to be one such instance. One drops a needle on a wooden floor and counts the number of times the needle crosses the cracks between the slats. What does that have to do with the circle?

Indeed, appearance of π in (*) came (and stayed for a long while) as one of those surprises that mathematics supplies in abundance to its practitioners and mere fans.

But there is another surprise lurking behind (*), see [Stein, Engel]. Even more unexpectedly, circle has a vital role to play in deriving and, in fact, explaining Buffon's formula. The explanation emerges from reverse engineering of (*).

The formula has been derived under the assumption L≤ D, which is in fact not necessary. For L > D, P should be treated as the average number of crossings per a toss.

What the formula says in particular is that, for two needles, the probabilities of their hitting the cracks are in proportion to their lengths. In other words, the average number of crossings per 1 unit of relative length L/D is constant and is equal to 2/π. Color one half of a needle blue and the other red. The probability of the blue half hitting a crack is exactly half of that for the entire needle. The same of course holds for the red half. Obviously, we'll have on average twice as many crossings tossing two needles than one, but the above argument claims more. There are going to be twice (on average) as many crossings with two needles than with one even when the needles are attached to each other as to form twice as long a needle.

This is a crucial point that bears repetition. Even if the two needles are not tossed independently, but constitute two halves of the same needle, their contributions to the total number of crossings remain pretty much independent. The point is important because it may apply recursively to smaller pieces of the needle.

The next step is to recognize that the same will be true for two needles forming an angle, and that they need not necessarily have the same length. We may go even further. Let's experiment with a flexible wire instead of a rigid needle. (Should the approach be now referred to as Buffon's Noodle?) Such a wire may cross a crack several times. When this happens all the crossings are counted in the total. By a generalization of the foregoing argument, on average the total number of crossings only depends on the length of the wire and not on its shape.

This is a bold generalization that begs a good definition of length of a curve and the validity of the Law of Large Numbers for the average of dependent random variables [Feller, Exercises to Ch. X], although dependencies can be folded into a single random variable associated with the given shape. The applet below may help one to get accustomed to the idea. (With the Draw box checked, you can draw broken lines by intermittently dragging and clicking the mouse. When you close the popup window by pressing the Save button, the shape you drew is resized to the length equal to the distance between two parallel lines.)


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Buffon's Noodle


(In the lower right corner the applet shows the number of crossings and the total number of throws. The third number is an approximation to π obtained when the simulation is considered as a run of the Monte Carlo Method.)

Now back to the formula (*) and the claim that it is valid for curves other than straight line segments. Assume that for a wire of length L, the average number of crossings is proportional to L/D independent of the wire's shape. The coefficient of proportionality is the same for all curves. What is it?

Let's take a circle of diameter D, with the circumference equal to L = πD. This shape has a nice property that however tossed on the floor with cracks distance D apart, the number of crossings produced is always 2. So is the average number of crossings. For the circle, therefore, the latter is 2 and L/D = π, wherefrom the coefficient of proportionality is exactly 2/π, as in (*).

So circle does have a lot to do with Buffon's formula. But the formula still holds more surprises in store. Circle is not the only curve that has the same number of crossings with parallel lines irrespective of how it has been tossed. There are other shapes that share this property. These are the so called shapes of constant width. For the width equal to the distance D between the cracks, any such shape crosses the cracks twice exactly as the circle. Thus, if the formula (*) holds, all such shapes have the same perimeter: L = πD. Which is the theorem due to Joseph Emile Barbier (1839-1889): all shapes of constant width D have the same perimeter, L = πD.

Another wonder: after all, Barbier's theorem has nothing to do with chance and probability.

References

  1. P. Beckmann, (The History of) π, St. Martin's Press, 1971
  2. H. Dörrie, 100 Great Problems Of Elementary Mathematics, Dover Publications, NY, 1965.
  3. A. Engel, Exploring Mathematics With Your Computer, MAA, 1993
  4. H. Eves, In Mathematical Circles, Prindle, Weber & Schmidt, Inc., 1969
  5. W. Feller, An Introduction to Probability Theory and Its Applications, v. 1, John Wiley & Sons, 1968
  6. J. Havil, Nonplussed!, Princeton University Press, 2007
  7. E. Kasner and J. Newman, Mathematics and the Imagination, Simon and Schuster, 1958
  8. J. A. Paulos, Beyond Numeracy, Vintage Books, 1992
  9. S. Stein, How The Other Half Thinks, McGraw-Hill, 2001

Buffon's Needle

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