![]() ![]() ![]() Specifically, a set is a fractal if its Hausdorff dimension is strictly greater than its topological dimension. The first clear definition of fractal was written in 1975 by Benoit Mandelbrot himself. Under the characteristics section of the Wikipedia page on fractals, we see a couple of relevant points. If you want to "prove" that the Mandelbrot set is a "fractal", then you'll need to work with some specific definition. How do we deal with yet more types of construction, such as The issue arises with algebraic constructions, such as that of the Mandelbrot, Julia sets etc. Obviously, as I noted above, there isn't much of a problem in the case of geometric constructions such as the Koch snowflake, the Sierpinski triangle, etc. some sort of test we can apply, in the case of iterative maps like the Mandelbrot set? (NB: proving such a test in the general case could be more difficult than proving the Mandelbrot set to be a fractal, but then it would be easier to apply). Then, armed with this newfound (hopefully!) knowledge about the fractalness of the Mandelbrot set, the slightly more general question: is there a way to know which maps/processes produce fractals? E.g. ![]() Yet it's not obvious to me where you could even start to prove that the map which produces the Mandelbrot set creates this self-similarity, or even infinite detail. The second leads on from the first.Ĭan we prove the Mandelbrot set is a fractal? It is very easy to see that something like the Sierpinski triangle is fractal by design. So, as you probably noticed, I have two questions. ![]()
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