Can't Square the Circle
You may have heard the saying, "you can't square the circle" - meaning "you can't do the impossible." We'll talk with a mathematician about the resolution of a problem with no solution - after this on Earth and Sky:
JB: This is Earth and Sky. You may have heard the saying, "you can't square the circle" - meaning "you can't do the impossible." But people didn't always know that squaring the circle - in other words, constructing a square that has the same area as a circle - was indeed impossible. We spoke with Dr. David Harbater, a mathematician at the University of Pennsylvania, about this ancient problem:
(Tape 0:03:31-0:03:44) The ancient Greeks had posed various problems, namely to do various constructions with a straight edge and compass, and there were all sorts of constructions they could do, but one of the things they could never figure out how to do was to square the circle. (Tape 0:04:14-0:04:45) And the realization in modern times, really in the 1800's, was that it's impossible. That no matter what construction you do with a straight edge and compass, no matter how complicated it is, you will never be able to square the circle. You will never be able to find a square with the same area as the circle. So that was somehow a real shift in realizing that it doesn't make sense to keep trying to do this. In fact the answer is, it can't be done.
JB: So the conclusion that there was no solution was the resolution of the problem. And this was a revolution in mathematical thought. It's something mathematicians today have to consider.
(Tape 0:08:30-0:08:46) And that means that whenever you pose a problem, you have to ask yourself a question, namely, is it, could it be that the answer to what you're saying, namely ‘do such and such' is ‘it can't be done.'
JB: Thanks to Dr. David Harbater for speaking with us. And with thanks to the National Science Foundation, I'm Joel Block, for Deborah Byrd, for Earth and Sky.
Author: Beverly Wachtel
Thanks to the following individual for aiding in the preparation of this script:
Dr. David Harbater
Department of Mathematics
University of Pennsylvania
Basically, the reason you can't construct a square with the same area as a circle using a straight edge and compass is that since the area of a circle contains the number pi, which is transcendental (not the root of any polynomial with rational coefficients,) you could not construct a square such that it has a side of the length of the square root of pi. But for more details, check out these links:
Squaring the Circle:
Math Forum: Squaring the Circle:
(for a slightly different perspective)