Fold a strip of paper in half two times and unfold. Put the strip on its side and adjust each angle to 90 degrees. It will look something like this:

___   
   |  |
   |__|

If you could fold the paper in half an infinite number of times and adjust every angle to 90 degrees what appears to be a fractal would be formed.

Is it a fractal? If so, what is its fractal dimension?

Tristan is right about the magnification factor: The size of the fractal (distance between endpoints of the curve) remains constant if one scales it up by a factor of sqrt(2) at every step.

The fractal dimension is the logarithm of the number of straight lines at each step divided by the log of the corresponding scaling factor.  That gives log(2^n)/log(sqrt(2)^n) = n log2 / (n/2 log(2)) = 2.  This means it's a plane-filling curve, that is it gets arbitrarily close, with increasing n, to any point within a region with non-zero area (did I get this right from memory? hope so).

Posted by vswitchs on 2006-08-20 16:51:23