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I'm playing around with a couple of different fractals, and a recurring theme I'm seeing is the presence of baby Mandelbrots (small versions of the Mandelbrot set) in nearly all of these.

enter image description here

Here's one between -0.183346 + 0.36017685i and -0.18229 + 0.36072355i.

As you can see, it's got these cool structures around it. The equation used to generate this fractal is:

$$ z \to (z^2+2c)^4+c $$

So why does the Mandelbrot seem to appear here? And in other fractals generated with similar equations.

Vivaan Singhvi
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  • I'm pretty sure that this question has been answered before on this site (recently, even... and by me). – Xander Henderson Oct 22 '23 at 17:24
  • Yeah... similar: https://math.stackexchange.com/a/4750798/ . The question is different, but the answer is about the same. The map "looks" basically the same near any point which is attracted to zero. – Xander Henderson Oct 22 '23 at 17:26
  • It's similar for sure, but the answer is regarding the Taylor approximation for the cosine function, while this seems like it would have a different answer. – Vivaan Singhvi Oct 22 '23 at 18:02
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    In as sense, thought, it is really about the behaviour of the map near zeros. In the case of the cosine function, there is a little bit of extra work to see that $\cos(x) + c \approx kx^2 + c$, but the fundamental idea is the same: if a point is attracted to zero, then the behviour of the map near that point will be approximately the same. – Xander Henderson Oct 22 '23 at 18:34
  • But note that I did not close your question as a duplicate---I just pointed out that there is a similar question. – Xander Henderson Oct 22 '23 at 18:35
  • $c_0 = -0.18291053\ldots + i 0.36049589\ldots$ appears to be a solution for $c$ of $f_c^{10}(0) = 0$ where $f_c(z) = (z^2+2c)^4+c$ and $f_c^{n+1}(z) = f_c^n(f_c(z))$. Then for $c$ near $c_0$, and $z$ near $0$, the Taylor series for $f_c^{10}(z)$ presumably "looks like" the Taylor series for $w \to w^2+d$ with $d$ near some $d_0$, and $w$ near $0$, whence a baby Mandelbrot set occurs. – Claude Oct 23 '23 at 12:45
  • The answer is probably similar to the answer to the question, "why are there baby Mandelbrots in the Mandelbrot fractal?" – Claude Oct 27 '23 at 10:31

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