In the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so?

The Mandelbrot set

From Wikimedia Commons Mandelbrot set

Part of the Collatz map fractal

From Wikimedia Commons Collatz map fractal

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    $\begingroup$ The Mandelbrot set appears in many other fractals, and there's some discussion of that here. As to why the Mandelbrot-like regions appear in the Collatz fractal, I suspect they mark regions where the dynamics got by iterating the Collatz map (varying the starting point of the iteration) are similar to the dynamics got by iterating $z\to z^2 +c$, always starting at zero and varying $c$, but I'm not familiar enough with the Collatz fractal to give a detailed answer. $\endgroup$ – Aaron Golden Nov 7 '15 at 9:55

This resemblance, of which I see only a bit of, is likely due to the fact that both are fractals generated on the complex plane by convergence of non-trivial equations (to distinguish them form fractals like Sierpinski's Triangle/Sponge/Carpet or Koch's Snowflake that are generated geometrically); they both share similarities with Julia Sets. Since the fractals are self-similar, the nodules of similarity are likely to show up, and the human brain picks up patterns and highlights the similarities between patterns, making instances where the dynamics of the two fractals are vaguely similar stand out, even if the actual similarities are weak.


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