Why is Pi a Constant?

Date: 08/01/2000 at 20:44:32
From: Donielle
Subject: Why is pi the same?

Hi,

My question is, why is pi the same number for any circle? If one 
circle is 5 feet across and another circle is 1,000 feet across, how 
can they both have the same circumference divided by diameter, or pi? 
It just doesn't make sense to me.

Thanks for your time.
Donielle

Date: 08/02/2000 at 13:00:43
From: Doctor Ian
Subject: Re: Why is pi the same?

Hi Donielle,

On the one hand, that _does_ seem pretty amazing, doesn't it? And it 
leads to some pretty bizarre conclusions.

For example, if you tie a string around a beach ball, and you want to 
add enough string to make it one inch away from the ball all the way 
around, you have to add about 6 inches (pi times the increase in the 
diameter, two inches) of string.

If you tied a string around the entire earth, and you wanted to add 
enough string to lift it one inch off the ground everywhere, you would 
have to add the same amount of string -- about 6 inches!

On the other hand, does it surprise you that you can construct two 
triangles of vastly different size, e.g.,

             |                        |
       5   / |                      / |
         /   |  4                 /   |
       /     |         75,000   /     |
     /_______|                /       |  60,000
                            /         |
         3                /           |
                        /             |
                      /_______________|

                           45,000

that contain exactly the same angles? One answer to your question is 
just that scaling doesn't always work the way you expect it to.

But you don't have to stop there. Your question is actually a very 
deep one, and in trying to answer it, people have been driven to 
create entirely new fields of mathematics.  

In a sense, the reason you're asking your question is that you've made 
certain unexamined assumptions about the world. If you were to make 
different assumptions, you'd come up with different questions.

For example, it turns out that the statement 'The circumference 
divided by the diameter is the same for every circle' is only true if 
you're talking about a 'flat' space. But here's an experiment you can 
do to get a feel for what it would be like to live in a 'curved' 
space:

Look at a globe, like this one:

     

Call the horizontal circle that goes through the top of South America 
(i.e., the equator) 'Circle 1', and the horizontal circle that goes 
through Chicago 'Circle 2'. If you measure the 'radius' of each circle 
along the surface of the earth from the north pole (which is the 
'center' of each circle), then dividing circumference by diameter 
gives you two different values of pi, doesn't it? 

In fact, if the earth were perfectly spherical, the 'radius' of the 
equator (measured along the surface) would be 1/4 of its circumference 
(do you see why?), so the ratio of circumference to diameter - that 
is, 'the value of pi' - would be exactly equal to 2, rather than 
3.14...

Does it seem like cheating to measure radius that way? Well, the 
radius of any circle is the length of a straight line segment that 
runs from the center of a circle to the circle itself. And a 
'straight' line segment, in the most general sense, is the shortest 
one that you can draw between any two points.

To someone who thinks that he can only measure things along the 
surface of the earth - because he lives in two dimensions, not three 
- the definition of 'radius' that we're using above would be 
perfectly reasonable, even obvious. If he looked at smaller and 
smaller circles, he might notice that as the diameter of a circle gets 
close to zero, the circumference gets closer and closer to the number 
3.14..., and he might call that number 'pi'.

And since pi shows up in so many other places in mathematics, this guy 
might find himself asking: "Why is the ratio of circumference to 
diameter close to pi for small circles, but not for large ones?" And 
if he investigated that question deeply enough, he might eventually be 
discover that there are other ways of measuring distances than always 
going along the surface of the earth. That is, he might learn to think 
in terms of three (or more) dimensions, even if he can't directly 
experience those dimensions.

The field of mathematics that deals with this kind of thing is called 
'non-Euclidean geometry' because Euclid made some assumptions about 
the shape of space, which, it turns out, aren't _necessarily_ true.

If you find this kind of thing interesting, you might want to check 
out the book _Flatland_, by Edwin Abbott, which is a novel about 
creatures who live in worlds with different numbers of dimensions. 
It's pretty easy to read, and will give you quite a lot to think 
about!

You can read the entire book online at

   http://www.alcyone.com/max/lit/flatland/   

Also, there is a series of lectures by the late physicist Richard 
Feynman called "Six Not So Easy Pieces" that you can probably find at 
your local library. The last lecture is called "Curved Space," and is 
worth listening to even if you don't feel as if you won't know enough 
about math or physics to understand it. He doesn't expect you to know 
anything, and it's one of the most entertaining and understandable 
explanations of this subject to be found anywhere.

Finally, one of the best science fiction books ever written - 
_Contact_, by Carl Sagan - considers a twist on your question, and 
provides an answer that is simply astonishing. I won't spoil it for 
you, though. The book is well worth reading for a number of reasons.

I hope this helps. Be sure to write back if you're still confused 
about pi, or if you have any other questions.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/