Comment by yarg

This doesn't seem to fly with the inductive fact that 1/2 of a power of two is always one over a power of two no matter how many times you perform the iteration.

There are a countably infinite number of rationals between any two rationals, you can even keep splitting up those rational infinitesimal gaps into countably many rationals that are infinitesimal even relative to the earlier infinitesimals.

And you still only end up with a countably infinite set of expressible locations and not the real continuum.

Either x, y, or both are guaranteed to be a number of that form for all values on the curve.