Fractal of any dimension Define a fractal in the following way:

1. First step is a line, say of length 1.
2. Attach new lines of length 0&lt;a&lt;1 by their middle point at each endpoint, perpendicular to the first line.
3. Rinse and repeat.

Maybe this is not a fractal, and that is why I get the result I am getting, but I calculate that the Hausdorff dimension of this is -log(2)/log(a). This means that if I choose my a correctly, I can create a fractal of any particular dimension by this process.

This does not sound right at all - where is the flaw in my logic?