Saturday, October 14, 2006

Non-Existent Circles

I was teaching a physics class on Uniform Circular motion the other day which made me think about the concept of a circle.

A circle is defined as the locus of all points equidistant from a center. As any school kid can tell you a circle has both a circumference and a diameter and the relationship between the two is:

C = Pi*d
where C = Circumference
d = diamater (or 2 *radius)

Now Pi is an irrational number that does not terminate. So that we cannot say with 100% surety what its value is.
Therefore if we know exactly what the circles diameter is there will always be some uncertainty with respect to its circumference and if we know its circumference then its diameter is in doubt.
So we cannot know both its diameter and its circumference. Since a circle has to have both a defineable circumference and diameter. We run into a mathematical paradox.

A number of resolutions to this paradox are:
a. Circles don't exist. Neither do other conics dependent on Pi for dimensions.
b. Pi is in reality a rational number.
c. Higher Dimensions account for the uncertainty in circle geometry (ie. 2-D Euclidean Geometry is a simplification even in the flattest worlds) - A String Theory type of Approach.
d. Pi is not a constant but fluctuates somehow creating an illusion of existence.

Another point to consider is whether circle uncertainty is the mechanism behind Heisenberg Indeterminancy? Does it also effect observation and if so how? Its worth thinking about.
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