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Shotgun Pi

May 5, 2014

I've had an interest in the mathematical constant, pi, since I first learned of it in elementary school. Since I've published a book about pi this year, it's apparent that this interest still holds. Although just a few digits of pi are required in any scientific or practical calculation, pi has been calculated to ten trillion (1013) digits. I reviewed some simple analytical and physical methods of estimating the value of pi in a previous article (Another Piece of Pi, July 28, 2010).

The observable universe is a sphere with a diameter of about 9 x 1026 meters. This means that only 27 digits of pi are needed to calculate its circumference to better than a meter. Since the size of a hydrogen atom is about 100 picometer (10-10 meter), only 37 digits of pi are needed to calculate the circumference of the universe to the size of an atom.

The fraction, 22/7, is a close approximation to the value of pi, as Archimedes discovered by circumscribing polygons around a circle, noting their perimeter P, and taking a ratio to the diameter D of the circle to get π ≈ P/D. Archimedes obtained his estimate of 22/7 (3.142857...) for pi using a circumscribed 96-sided polygon. Ptolemy improved Archimedes' estimate with a 360-sided polygon, obtaining 3,141666..., a value that's correct to three decimal places.

Sculpture of Archimedes by Gerhard Thieme, Güstrow, Germany, 1977-8Sculpture of Archimedes by Gerhard Thieme, Güstrow, Germany, 1977-1978.

(2012 photograph by "Niteshift," via Wikimedia Commons.)

Certain species of millet have small, 2 mm diameter seeds that approximate a physical point. While the mustard seed takes center stage when talking about something small, the millet seed figured into a paradox of the Greek philosopher, Zeno. The paradox, which has been interpreted in various ways, involves the fact that a single millet seed doesn't appear to make a sound when it falls, but a bushel of millet seed does.

Somewhere in my reading of the history of mathematics, I remember that millet seeds were employed as a method for estimating the value of pi. Alas, a previous article of mine is the only source for this that Google finds. The method is easy, as the figure shows. You toss a handful of millet seed onto a drawing a circle inscribed in a square, then you count how many seeds have landed on the circle, and how many there are in total. The ratio of these numbers is an estimate of the ratio of the area of the circle to the area of square, from which pi may be calculated.

A square and an inscribed circle populated with random points to estimate piA square and an inscribed circle populated with random points to estimate pi.

I programmed this using Visual Basic more than twenty years ago.

The figure above was constructed using a Visual Basic program. If a graphical interface is not required, the C programming language, or many other languages, will give better estimates, since you can "throw" many thousands of seeds many thousands of times. You can review my C source code for this calculation, here. As shown by the histogram of results, 10,000 trials of 100,000 points/each gives us a mean value of 3.1415697 with a standard deviation of 0.00517947. This means that we're 95% certain that pi is between 3.1414587 and 3.1416613.


Histogram of the 'millet seed' method of estimating pi.
Histogram of the 'millet seed' method of estimating pi. (Graphed using Gnumeric.)

Vincent Dumoulin of the Département d'informatique et de recherche opérationnelle of the Université de Montréal and Félix Thouin of the Département de physique, also of the Université de Montréal, have traded millet seeds for shotgun pellets in a similar method for estimating pi.[1-2] Using a Mossberg 500 pump-action shotgun, they determined an approximate value of 3.131 for pi, corresponding to a 0.33% error.

Perhaps tongue-in-cheek, they observe that they're "...opening up new perspectives towards computing mathematical constants using everyday tools." They also write, "We feel confident that ballistic Monte Carlo methods constitute reliable ways of computing mathematical constants should a tremendous civilization collapse occur."

Their "calculation" involved firing the shotgun at a sheet of aluminum foil at a twenty meter range. Careful scientists as they are, they specify that the shotgun cartridges had 3 dram-equivalents of gunpowder and 32 grams of #8 lead pellets. The pellets are estimated to have an average velocity of about 366 meters per second as they exit the gun. Firing 200 times resulted in 30,857 perforations. Multiple foils were used, and their results combined. The foils were photographed and the hole locations were derived by image analysis.[1-2]

Various physical factors such as the aim of the gun, wind speed and direction, etc., will diminish the randomness of the shot distribution. To counter this, Dumoulin and Thouin used a random selection of the points (20,000) to normalize the distribution, and they used the rest (10,000) to estimate pi.[1-2]

References:

  1. Vincent Dumoulin and Félix Thouin, "A Ballistic Monte Carlo Approximation of π," arXiv Preprint Server, April 8, 2014.
  2. How Mathematicians Used A Pump-Action Shotgun to Estimate Pi, The Physics arXiv Blog.

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