Is Pi a Constant in Non-Euclidean Geometry?

Date: 06/26/98 at 12:41:39
From: Sean Harper
Subject: Is pi a constant in non-Euclidean geometry?

My friends and I were arguing about whether or not Pi is a constant. 
I said that it is, and my friends said pi can't be a constant because 
it is irrational and it could be 3.1 or 3.14 or whatever, depending on 
how many digits you use. They were almost at the point of agreeing 
with me when they stumbled across a Web site that says Pi is not a 
constant because if a circle is stretched across a curved surface, 
then pi will be smaller, and it will grow larger if the circle has a 
negative curvature. I countered that for everyday purposes and for pi 
as we know it we need to assume that the circle is on a flat surface. 
My question is, who is right?   

Perhaps all of us are right in different cases. In which cases am I 
right and in which cases am I wrong, and why?

Date: 06/26/98 at 16:25:15
From: Doctor Barrus
Subject: Re: Is pi a constant in non-Euclidean geometry?

Hi, Sean!

Well, you're both right in different cases. Whenever you're talking 
about perfect circles on flat (Euclidean) surfaces, then pi, which 
represents the ratio of the circumference of the circle to its 
diameter, is a constant. (By the way, this number is irrational but it 
IS a constant, just as the square root of 2 is a single irrational 
constant whose decimal expansion goes on forever. Take 1, 1.4, 1.41, 
1.414, or any other approximation of the square root of 2 and square 
it, and you won't get 2. Only the irrational square root of 2 squares 
to 2. Similarly, there is only one number, pi, in all its irrational 
glory, that represents the circle's ratio.)

However, if you go into NON-Euclidean geometry, where you deal with 
curved surfaces, then the ratio of a circle's circumference to its 
diameter does not remain constant. For example, say you stretched a 
piece of rubber over a circular hoop. When the rubber lies flat, 
you've got a flat (Euclidean) circle, and the ratio of the 
circumference to the diameter is pi = 3.14159265358... But say you 
poked your finger through the center of the circle and stretched the 
rubber a bit. Then the diameter of the circle would grow, but the 
circumference would be the same. The ratio would change - it wouldn't 
be constant. Consequently, if you called this ratio pi, then pi 
wouldn't be constant.

It gets a whole lot more complicated in non-Euclidan geometry than 
this. For example, suppose you traced a circle (and you have to be 
careful of your definition of circles in non-Euclidean geometry) on a 
horse's saddle. This constitutes a surface with negative curvature. 
The ratio of circumference to diameter is not easy to compute exactly 
without advanced mathematics, let alone to describe!

So you see, you're right that pi is a constant when you're dealing 
with Euclidean, or flat surfaces. However, your friends are right in 
that the ratio of the circumference of a circle to its diameter 
changes and is not constant in non-Euclidean spaces.

Whew!  Long answer, but I hope it's helped. :)

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