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?

I believe that with every fold, the shape becomes smaller and smaller (since it becomes more dense).  So as the number of folds approaches infinity, the fractal approaches nothingness.

We can correct this by making the paper longer by a certain factor at every step.  If we pick just the right value for the factor (sqrt(2)?) I think we should get the fractal we are supposed to be thinking about.  If the factor is higher, the fractal should spiral outwards forever, covering a quarter of the plane.

Posted by Tristan on 2006-08-15 23:51:26