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> Now, a fractal is a set with a fractional Hausdorff dimension.

Is there an authoritative definition of a fractal? The one you use rules out structures like Hilbert curves, which are generally considered fractals.

> Is there an authoritative definition of a fractal?

Sort of. I believe Mandelbrot's original definition required only that the Hausdorff dimension exceeds the topological dimension. And that kinda-sorta includes Hilbert curves, if you count them as topologically 1-dimensional.

But I've seen other definitions, including a rather hazy one that was not a definition in the formal sense, but just talked about properties that certain interesting sets tend to have: self-similarity, etc.

In any case, I have yet to find a situation in which the formal definition of "fractal" actually mattered significantly. (If someone knows of one, I'd be interested.)

No, it's pretty vague. http://en.wikipedia.org/wiki/Fractal#Characteristics Also, the Hausdorff dimension of a Hilbert curve is 2.

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