# Why does the Mandelbrot shape show up in other fractals?

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

## Part of the Collatz map fractal

• 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. – Aaron Golden Nov 7 '15 at 9:55