Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
In the 18th century, Compte de Buffon (1707-1788) posed and solved the very first geometric probability problem. 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 answer he discovered with the help of integral calculus is given by the simple formula [Beckmann, Dörrie, Eves, Kasner, Paulos, Stein]
With P approximated by the ratio of hits to the total number of tosses, the formula offers a way of evaluating p, 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 p. 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
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.
p is defined as the ratio of the circumference of a circle to its diameter, but it 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 p in (*) came (and stayed for a long while) as one of those surprises that mathematics supplies in abundance to its practitioners and fans.
But there is another surprise lurking behind (*) (see [Stein]). 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
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/p. 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 instead of 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 two needles are not dropped 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 we may apply it 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 must not necessarily have the same length. We may go even further. Let's experiment with a flexible wire instead of a rigid needle. 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 the 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.)
(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 p 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
So the circle does have a lot to do with Buffon's formula. But the formula still holds more surprises in store. The 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 like the circle. Thus, if the formula (*) holds, all such shapes have the same perimeter:
Another wonder: after all, Barbier's theorem has nothing to do with chance and probability.
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at firstname.lastname@example.org
Copyright © 1996-2001 Alexander Bogomolny