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.)

I'm thinking that this boils down to coming up with an algorithm for how to place all the 8's.

First consider each real number as its decimal representation.  There will be an integer part and a fractional part(but in decimal form).   Assume we can come up with a way to do all the integers.   So we have a given '8' for the integer part.  Now draw nested '8's inside to represent the fractional part.

Inside the 8 draw 5 '8's in the top half and 5 in the lower half.  Each one represents the digit right after the decimal point.  Inside each one of those is 10 smaller '8's to represent the next decimal point.  On to the smaller '8's for the smaller numbers.

If we can do that for the fractions, then we should be able to have larger and larger '8's for the digits of the 1's column, the 10's column etc.

Posted by Larry on 2005-02-15 16:52:06