(This 885th Buffalo Sunday News column was first published on March 9, 2008.)
This coming Friday is designated Pi Day in many schools. The reason: it is the 14th day of the 3rd month which suggests 3.14, the three digit approximation to the mathematical constant, pi.
The number pi is in fact one of the most 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.
A bicycle wheel is rolled from left to right
until the point P reaches the ground again.
In this case ¼ = 88/28 = 3 1/7
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 will then 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 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 forty of those digits:
· High school students often learn the value 3.1416 for pi. There is a far 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 circle circumference is between the perimeters
of the inscribed and the circumscribed polygons.
Doubling the number of polygon sides to make
12, 24, 48 and 96 sides squeezes the circle circumference
between closer and closer values.
· 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 over 2200 years ago, long before modern numbering systems were available. (Decimals would not be introduced to western civilization for over 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, 6 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 Representative 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 on The Straight Dope website.
Thus we are left with our still indigestible pi.-- Gerry Rising
Note: I failed to include in this column the concerns about pi expressed to me some time ago by my friend, Mark Spahn. Mark has kindly summarized those concerns in the following (slightly edited) messages:
In Pi is Wrong, Bob Palais argues that the universal constant pi was chosen incorrectly.
At present, pi is defined to mean the ratio of the circumference of a circle to its diameter. It should have been defined, argues the author, as the ratio of the circumference of a circle to its radius. Because of the wrong choice, 2<pi> pops up everywhere, where a simple pi would be more elegant. As another example, if pi were defined properly, the formula for the area of a circle would be A = (1/2)<pi>r^2, which is analogous to the familiar formulas s = (1/2)gt^2 for the distance s fallen under gravitational acceleration g in time t, or E = (1/2)mv^2 for the kinetic energy E of a body of mass m moving at velocity v.
This wrong choice is like the wrong choice made by Benjamin Franklin. He assigned "+" to one kind of static electricity and "-" to the other kind. He could not have known it at the time, but the extra particles, which should have been associated with "+", turned out to have the charge he labeled "-". So now we have to make a distinction between "current" and "electron flow".
The wrong choice for pi has implications for our intergalactic reputation. If we send out "3.141592654..." to S.E.T.I. listeners elsewhere in the universe to show how smart we are, they will deride us for peculiarly broadcasting only half of the universal constant 6.283185307... . We'll be the laughingstock of the universe.
One fix that has been suggested is to use a three-legged pi for what we now call 2<pi>; formulas using the three-legged pi are given in the article.
As a follow-up to my earlier note about the misdefinition of pi, and how to fix the problem by using an new consonant defined as equal to 2<pi> and represented by a three-legged version of the lowercase Greek letter pi, we will need a distinctive name for this three-legged pi. I propose calling this consonant "poi". The allusion is not to the goopy taro-based Hawaiian food, but rather to a form of Maori juggling with balls swung at the end of cords held in the hands. For a picture and explanation, see Poi Juggling .
The association of circles with this "poi" is obvious, both because of the shape of the balls, and because of their circular trajectory when juggled. I note that the "o" in the middle of the word "poi" makes it even more circular.
Just my little contribution to the progress of mathematics. -- Mark Spahn (West Seneca, NY)