Expanding the Mandelbrot Set into Higher Dimensions (2010) [pdf] | Hacker News Reader

Expanding the Mandelbrot Set into Higher Dimensions (2010) [pdf]

archive.bridgesmathart.org

The Mandelbrot set already has that rather pretty interpretation as the logistic map [0, 1] that is enough to serve as a third dimension. While the mathematicians can have their fun with alternative, more technically satisfying, approaches when people talk about "3D Mandelbrot" it is probably more interesting if that references to the logistic connection.

[0] https://en.wikipedia.org/wiki/File:Verhulst-Mandelbrot-Bifur...

[1] https://en.wikipedia.org/wiki/File:Logistic_Map_Bifurcations...

A while back I wanted to render some of these myself, the results can be seen at https://renders.datenvorr.at/. I have a few 760 megapixel versions of these that you can easily print at 1x1 meter with full detail if anyone is interested. I have a couple of these on my wall...

I used the project to familiarize myself with the C programming language, so the code quality is pretty bad. A while back I rewrote most parts of it, but never merged the finished result back into master...

Mandelbulber [0] is a visualiser that does a fairly decent job of exploring these sets, and a few more.

[0] https://mandelbulber.org/

Look up "mandelbox" on youtube. Amazing stuff!

> The main problem by introducing the fourth dimension in the calculation is the enormous increase of processing needs and the difficulties of selecting an adequate projecting surface.

This article was written over 10 years ago and this is no longer true. Recently I wrote a simple program which can explore the CxC space of Mandelbrot and Julia sets. I don’t know why there aren’t better visualizer which do this.

Can you share a link?

I actually just dusted this code off, and unfortunately the code is way too crappy to share as a fun learning and POC using TypeScript and React. but here is the basic idea:

start with the standard mandelbrot equation: Zn+1^2 = Zn^2 + c where Zn and c are complex numbers. You can think of these as 4 real dims: (Z0_re, Z0_im, c_re, c_im)

Now associate the X dimension of the display window to one of these and the Y dim to another. The other dims can be set to arbitrary constants. This can get you all orthogonal slices of the 4D space.

For example the standard mandelbrot set would just be mapping: Z0_re->0, Z0_im->0, c_re->X, c_im->Y

But you can also generate every combination of mandelbrot and julia sets in this space.

As the paper mentioned though, you want to choose any arbitrary 2d slice, not just orthogonal ones, but really this shouldnt be much harder to implement.

If I could do a POC of this in TS and React this should be easy to do for real.

Reminds me of romanesco broccoli, one of the coolest looking foods you can eat.

https://en.m.wikipedia.org/wiki/Romanesco_broccoli

What about a continous version? You can replace that Z[n+1] with a Z[n+dn]...

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