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Sunday, March 9, 2008

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Nature Watch / By Gerry Rising

No matter how you slice it, Pi fascinates


Updated: 03/09/08 7:51 AM

P d=28 P P C = 88 Roll pi for an estimate: Divide one cycle of point “p” by wheel’s diameter. P P

This coming Friday is designated Pi Day in many schools. The reason: it is the 14th day of the third month which suggests 3.14, the three-digit approximation to the mathematical constant, or pi.

The number pi is in fact one of the central concepts of all nature. It is defined as the ratio of the circumference (C) to the diameter (d) of a circle. In equation form this is pi = C/d.

This simple relationship gives us a means of estimating the value of pi. Make a chalk mark on the tire of a bicycle wheel where it touches the ground, roll it forward until that mark once again touches the ground. Measure the distance between marks and divide that distance by the diameter of the wheel. For a 28-inch bike, for example, you should find that it rolls about 88 inches. Division then will give you a pi estimate, quite a good one for my example.

There are many features of this remarkable constant that I hope you will join me in finding interesting:

• In elementary school, students first learn the pi estimate, 22/7 or 3 1/7. Then a few grades later they learn the decimal approximation, 3.14. Because decimals are taught later, most of them (we too) retain the idea that 3.14 is a better estimate. It is not: 3 1/7 is a closer approximation.

• The ratio pi = C/d immediately leads to the formula, C = pi times d for the circumference of a circle. And some geometry leads to the formula for the area of a circle which states in words (with r for the circle radius): “A equals pi r squared.” It is said that a teacher tried unsuccessfully to have her students learn this formula. The reason, one student finally explained: “You got it wrong: pie are round; cake are square.”

• Remarkably, given how simply it is defined, pi “doesn’t come out even” no matter how many decimal places are calculated. Here, for example, are 40 of those digits: 3.1415926535897932 384626433832795028841971 …

• High school students often learn the value 3.1416 for pi. There is a better estimate that can be remembered as a fraction. Write the first three odd numbers each twice to give 113355. Now divide the last three digits by the first three: 355/113. This is pi accurate to six decimal places, 3.141593.

• There are dozens of mnemonics (memory helpers) that supply many of those pi digits. Here is a poem giving pi to 21 digits when you replace each word with the number of letters in that word (How = 3, for example): How I wish I could recollect pi. “Eureka,” cried the great inventor. Christmas Pudding; Christmas Pie Is the problem’s very center.

• The great inventor of that poem, the mathematician who cried “Eureka” when he solved a different problem, is Archimedes, who was the first to use math to calculate a pi approximation. What makes his work so extraordinary is the fact that he did so more than 2,200 years ago, long before modern numbering systems were available. (Decimals would not be introduced to western civilization for more than 850 years.) Archimedes knew how to calculate the perimeter of polygons so he determined those perimeters for polygons inscribed and circumscribed about a circle.

Beginning with hexagons, six-sided figures, he continued with 12, 24, 48 and 96 sides. Even with all this work, however, his estimate only squeezed pi to between 223/71 and 22/7, or to two of those decimal places. That is only about as accurate as the one we can obtain with the bicycle experiment.

• Perhaps the most entertaining story about pi relates to an attempt to change its value. In 1897 Rep. T.I. Record introduced Bill 246 in the Indiana House of Representatives suggesting not one but three candidates for pi, which reduced to 3.2, about 3.23 and 4, because the present value “should be discarded as wholly wanting and misleading in the practical applications.” The bill passed the Indiana House unanimously, but fortuitously a Purdue mathematician, learning about the bill, stopped its progress before it could be passed by the Senate. More on this story is found at a Web site: The Straight Dope: "Did a state legislature once pass a law saying pi equals 3?". Thus we are left with our still indigestible pi.

insrisg@buffalo.edu


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