Hybrid Pseudo Kleinian
a pragmatic approach of a 3d fractal hybrid, which combines the formal properties of two fractals in one system.
PseudoKleinians are a type of fractal, graphically simillar to Kleinian fractals, but of a completely different formal math origin. The principle was discovered by a user called Theli-at back in 2011 in a “hybrid” of two fractals.
Soon after this post a coder called knighty ‒ well known in the genre of fractals ‒ defined it as Mandelbox + something and created a stand alone formula of it. A great achievement, that is standard presentation for PseudoKleinian fractals since that time. Further more knighty first documented hybrid stuctres with the Menger Sponge.
Unfortunately I wasn’t aware of that collection, so I needed to go back to the original hybrid to work out the differences to knighty’s standalone formula.
The difference is in short, that the standalone version adds spheres to fill the space. In the hybrid display you can fill the space with a second fractal like shown in this gallery.
My aim here was to work out the details of the hybrid construction in a generalized way. Further more I could show, that AmazingSurface can be configured, that it “generates” a PseudoKleinian grid, just like AmazingBox. I once called this AsurfPseudo
And in fact building a PseudoKleinian only from AmazingSurface, was the starting point for my exploration of that system.
The shown principles are implemented in Mandelbulber, one of the common programmes for that task. In Mandelbulb3d – the other commom software – this can only partly be reproduced with restrictions in choice of formula 2.
A Mandelbulb3d portation with Abox in slot one and two.
Initially my aim was to implement “box-like” fractals – AmazingBox (Mandelbox), AmazingSurface and others. This worked pretty well and it soon turned out, that formal properties of fractal 2 can be projected almost undisturbed in a “Pseudo Kleinian grid”. And even more. All standard formulas work in the given pattern with full preservation of formal properties in formula 2.
Asurf with Abox in slot two
A classic fractal – a Menger Sponge – in slot two. This shows nicely, how the formal properties of fractal two shows up in the Pseudo Kleinian order. The global constant c in hybrid Pseudo Kleinians gives the opportunity to control the density and shape of that appearance.
A specialty of AmazingSurface. Without rotation a Pseudo Kleinian in extended 2d can be created. I once called this a 2.5d Pseudo Kleinian. A rather metaphoric description. This is different to all other Pseudo Kleinians, also the dedicated Formulas.