Yummy. Buffon’s Pi.

um piAs I write this, pi day is happening. On our local (Mountain Savings Time) clocks, it will soon be 3.14.15 (month-day-year) then 9:26:53. We use the American month-day-year for this event (rather than day-month-year as the rest of the world uses, or the computer-friendly year-month-day used to organize file/folder names). Without the peculiar system from the USA, there would be no pi day. Finding π – the actual number (3.141592653…) – has been a source of great fun for humanity.

One of the oldest known declarations of pi is given in the Bible. I Kings 7:23:

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

Yes, the Bible tells us the value of pi. It is equal to 30 divided by 10. Three, the trinity number. Am I the first to notice that there might be a mistake in the Bible? That could change everything – though I suppose there are people who will insist that it is the mathematicians who have got it wrong.

Archimedes inscribes pi.

Archimedes inscribes pi.

Over the years, others refined the divine, adding a dreadful decimal point and an apparently infinite line of integers after it. About 2,000 years ago, Archimedes calculated π to 3.146 by inscribing an endless series of triangles inside a circle and doing the arithmetic, making him the first to get close to the right value. Darn close.

Until Archimedes, the value used to calculate the volume of spheres and the brims of molten seas was either measured by rolling something round (like an antique can of coke) and then comparing the length revolved against the can’s diameter, or it was found by reading the Bible. With Archimedes, and then calculus a quick sixteen centuries later, this changed.

Louis-George LeClerke, the Count of Buffon

Georges-Louis Leclerc, the Count of Buffon

The most enjoyable calculation of pi came from Count Buffon (Georges-Louis Leclerc) around the year 1800. Buffon was a real count. He bought his title when he inherited 20 million dollars (today’s value) from his uncle. (Uncle Georges had been a French tax-farmer who robbed the island of Sicily into poverty. If he didn’t lose his head in revolutionary France’s guillotine, he should have.)

Buffon was a rich spoiled brat until mathematics entered his life and saved him from ruinous debauchery. He also became a noted geologist and nature writer, composing a huge science encyclopedia. He wrote well. For example, volcanoes were all the rage back in the early 19th century and Buffon had this to say about them:

“a volcano is an immense cannon, from its wide mouth are vomited torrents of smoke and flames, sulphur, and melted metals, clouds of cinders and stones, the conflagration is so terrible, and the quantity of burnt and melted matters so great that they destroy cities and forests.”

Buffon continues spewing his volcanic description, then attributes it all to an act of nature, even though the volcano’s throat was mistaken by “ignorant people for the mouth of Hell. Astonishment produces fear, and fear is the mother of superstition. The natives of Iceland imagine the roarings of the volcano are the cries of the damned, and its eruptions the rage of devils and the despair of the wretched.” Buffon would have none of this superstition: “all its effects, however, arise from fire and smoke.”

"Needles" on the floor

“Needles” on the floor

Buffon was an especially gifted mathematician. Quite by accident, he brilliantly stumbled upon a calculation of pi using probability.  A popular pub-night game in 1800 involved dropping needles on the floor after guessing how many would intersect the floor’s wooden splices.

Although he was already fabulously wealthy, Buffon thought he could rig the game using probability. Intuitively, we know the answer has something to do with the length of the needle and the distance between floor splices. Buffon invented a whole system to tie together those various lengths and the infinite range of angles the needle might encounter. He solved this, and other problems, by merging calculus with probability theory.

Buffon’s solution for pi is fairly easy to follow, but is best shown using a blackboard – as I did last year when I led my son’s 6th grade class through the calculation. (Not every kid suffers through having his dad in the classroom for a day, teaching fun stuff like calculus.) Buffon’s needle can be shown with the simplest of integral calculus, which the kids followed fairly well. Since I don’t have a blackboard attached to this computer, here is a professional presentation –  a great little video by the folks at Numberphile.

The actual test of Buffon was the fun part for the 12-year-olds. We simply tossed toothpicks on a lined sheet of paper, counted the number that intersects (yielding the probability of intersection) and solved for pi. We dropped 100 needles and got 53 intersects. The ‘needles’ were 2.5 inches long and the spacing of the lines on the paper was 3 inches. The formula is π = (2*needle length/spacing) divided by the rate of success. (See this link for a derivation.) Using Buffon’s, our probability (actual rate) of success (53/100), and the lengths we measured, the final calculation was 3.144. Easy as pi!

About Ron Miksha

Ron Miksha is a bee ecologist working at the University of Calgary. He is also a geophysicist and does a bit of science writing and blogging. Ron has worked as a radio broadcaster, a beekeeper, and Earth scientist. (Ask him about seismic waves.) He's based in Calgary, Alberta, Canada.
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