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/
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