I recently read Mandelbrot’s book, The Misbehavior of Markets, and in it he describes multifractals, but I still don’t understand what these are or what makes a multifractal. In part of the explanation he says that if you look at points in a fractal set as black and outside as white, then multifractals introduce a “shade of grey”. (I’m familiar with plotting points of a Julia set and coloring them, assuming this is what he is referring to).
I have no idea what this means – can someone help me understand what multifractals are with a clearer explanation? Are they fractals that have partial self similarity, i.e. the shapes as you scale down only partially match the whole? or am I way off?
On my quest to find a clear explanation of multifractals I did find this Google tech talk (missing sound of course) that has a table comparing fractals to multifractals. [https://youtu.be/SgwWyq9j3-k?t=572](https://youtu.be/SgwWyq9j3-k?t=572)
As far as I can assess apparently a multifractal is multiple sets of a fractal with different dimensions, but I don’t really understand what that means. I know what a set is, and have a vague idea/read about dimension in fractals (though still don’t have an intuitive understanding of that either).
In: Mathematics
The shading of sets in the famous plots of Mandelbrot and Julia sets are something else.
The points *in* the fractal set are the points = complex numbers for which a particular infinite sequence that starts with itself remains bounded. The rest are points for which the sequence eventually diverges.
Intuitively, if you start far from the set it will be immediately obvious that it diverges but if you start somewhere close to the set, it may require calculating many steps of the sequence before you see that it diverges. The plots you have likely seen are a visualization of this proximity: You choose a threshold such that once a sequence crosses it, it is guaranteed to diverge. Then you plot each point *outside* the fractal set using a color that corresponds to the number of steps needed to cross the threshold.
Multifractals are essentially what you get by distorting fractals by continuously changing their fractal dimension.
Say you want to model the shore of a country. Shores are known to be fractal-like, but when you compare a real shore to say a Koch curve, it’s nowhere near as regular: different regions will have somewhat different properties, different bays within the same region will differ, and so on. You can still model a fractal-like curve to fit the actual shore well, but the curve will not be a pure, exactly self-similar fractal.
I don’t know what exactly Mandelbrot meant by the “shades of grey” but my assumption is that this represents something you would get if you plotted not just a single fractal curve but a collection of similar ones one on top of the other, each with its own local distortions of their fractal dimension.
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