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I'm a graphic designer and my client has asked me to use this fractal in a design that I'm working on. As you can see, it's not a very good copy, so I'm trying to see if I can generate a high-resolution version for the project. I've found all sorts of fractal generators that work great and many of them will allow me to enter a specific equation. The trouble is that I'm a designer and not a mathematician! So the question is this: is it plausible to think that this could be reverse engineered, or is that an insane idea? If it's plausible, how would I achieve it?

Distorted Julia Set Fractal

EDIT:

I found the original from the book (below). The values @Carlo Beenakker posted got me very close to the right area. I was able to render and warp the output and achieve what we were looking for! It's not exact, but very close.

Julia Set Fractal from War in the Age of Intelligent Machines

enter image description here Warped and Colorized Julia Set Fractal

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    $\begingroup$ At first I was doubting this question was on-topic, but I think there's a very interesting problem here, and it seems like an applied maths kind of thing: how to model observed behaviour, especially the kind of multi-scale self-similarity we have here? $\endgroup$ – David Roberts Sep 7 '20 at 0:09
  • $\begingroup$ Are we sure this is not some high-order point in the Mandelbrot set? $\endgroup$ – Gerald Edgar Sep 7 '20 at 11:07
  • $\begingroup$ So, I've found out a little more info on the fractal, my client found it in a book titled War in the Age of Intelligent Machines... It seems like a fellow named Robert Devaney is who generated it. $\endgroup$ – Circle B Sep 7 '20 at 15:37
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The source info (War in the Age of Intelligent Machines) identifies the fractal as a Julia set, iterates of $z\mapsto z^2+z_0$. It has evidently been distorted (warped) to give it a 3D appearance. I played around with a spiral-shaped Julia set I found at pixels.com and could create an image such as this.

The Mathematica command
Manipulate[JuliaSetPlot[a- b*I,MaxIterations->n], {a,0,1},{b,0,1},{n,100,1000}] provides a way to find suitable values for $z_0=a-bi$ (and the iteration count $n$). I find a similar spiral for $z_0=0.202-0.55\,i$, $n=400$.

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If the requirements are:

(a) fractal (image is repeated at small scales); and

(b) clockwise spiral

then the following may be of use:

enter image description here

This image is generated by the complex exponential function $\exp(z)+1+2.9i$. The resolution can be improved to whatever level you desire (given enough computing power).

Alternatively, if you had a high resolution image of one of the "islands" in your image, then you could make a lattice out of many copies of that, and then apply some sort of a spiral filter which I have seen in some graphics programs like GIMP and maybe Photoshop.

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  • $\begingroup$ I think the TS asking about the exact reconstruction of the given fractal. It is obviously, machine-generated, so there should be a formula... But finding it may be enormous mathematical challenge. $\endgroup$ – Anixx Sep 7 '20 at 10:34
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It looks like a tiling of some type of julia fractals, then post-distorted with a 'spiral' map. Hence, it seems to fall in the "flame fractal" category. The downside is that there are a lot of parameters to play around with. Also, a reverse image search gave me nothing, so this is not a picture grabbed from the internet.

There is a huge fractal art community at DeviantArt.com - if your client really want something like this, perhaps one can ask one of the artists for a commission?

Side note: I have developed some software to make this type of fractals, so I am rather familiar with the process. Also, in all fractals I generate, I hide the underlying equations as metadata in the image file, so that I can always recreate the fractal (this can be of interest for other mathematicians out there).

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