How to create a 3D fractal flower

Question

There is another post asking about the steps of creating a Barnsley's Fern, and it has very good answers. Now the reason I am asking this, is because I stumbled upon a post on MathOverflow which mentions Barnsley's Fern. But adds something surprising:

If you allow for a larger class of functions (stochastic, $\mathbb R^3\to\mathbb R^3$, and introduce a log-density plot and color each point according to orbit history, the possibilities are endless (image created by Silvia C.):

I am interested in creating something like that. I first visited the artist's page, Silvia Cordedda, to get some ideas about the math behind this kind of images. It seems they are created using the software Apophysis 7x and Chaotica and then polished. The former is open source and the latter is commercial. There wasn't any explanation about the generating equations for this.

I tried changing the equations of Barnsley's Fern and expanding them to 3D, but the results are horribly hopeless, and I have no idea what to do. Any help would be appreciated.

Answer 1

Well, I do not have code in Mathematica that does this, but I have an implementation in Java.

It is capable of making such pictures, see the attached one for example (which is related to my current research actually).

I should perhaps add the latest version to GitHub at some point, the Sourceforge version is a bit outdated.

For the mathematical history/background, I suggest you read Scott Draves paper in this (he basically invented the family of fractals and drawing algorithm you are interested in).