**Pi Day**

(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:

·
3.1415926535897932384626433832795028841971...

· 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)