Is the golden ratio or are spirals in general fractals? If not, why?

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Mandelbrot wrote: A fractal is a shape whose “Hausdorff dimension” is greater than its “topological dimension.”

In simple (and less precise) terms: Fractals are shapes with a non-integer dimension. Shapes that are rough, and that stay rough as you zoom in. For a purely geometric shape to be a genuine fractal, it has to keep looking rough, even as you zoom in infinitely far.

Please correct me if I'm wrong.

Do Fibonacci spirals qualify as fractals? If not, why? How do you measure the Hausdorff dimension or 'roughness' of a spiral?

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3 Answers

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No, a Fibonacci spiral is not a fractal - it's a smooth curve and has dimension one

Here is a fractal spiral:

The distinction is that this set is self-similar - it consists of two copies of itself one blue copy, scaled by the factor 0.98, and one orange copy, scaled by the factor 0.08. The dimension can be computed using the theory of self-similar sets; it's approximately 1.4135.

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A fractal does not need to have a non-integer Hausdorff dimension Check Peano's curve, Hilbert's curve, Julia set, and many other fractals who have Hausdorff dimension of exactly 2, which is integer.

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See also spirals near Misiurewicz points and external rays that land on it:

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