Suppose that I drew an infinite number of disjoint closed curves in the plane (such as circles, squares, etc.). Suppose that I then tell you that there is one curve for each positive real number.

You would not have too much trouble believing my assertions at this point. For example, I could have drawn all circles with center at the origin. They are all disjoint, and for each positive real number x, there is a corresponding circle - namely, the circle of radius x.

But suppose that I also tell you that all the curves I drew were figure eights. Can you believe my assertions now?

(A figure eight is a curve in the plane obtained from the basic "8" shape by any combination of translation, rotation, expansion, or shrinking.)

Figure eight's can be represented by polar equations. Graph r^2=cos(2*theta) if you do not believe me. In fact, any equation of the form r^2=x*cos(2*theta) will yield a figure eight.

We can easily see that each curve has no points in common other than the origin. I'm not sure if this is acceptable or not as they're not technically disjoint.

FYI, we can also square the cosine will ensure that x can only be positive. It would still be a figure eight.

Posted by np_rt on 2005-02-16 08:38:55