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.
https://www. deviantart.com/theli-at/art/Kleinian-drops-192676501
This is a special Mandelbox hybrid, where the transformations of the Mandelbox are executed in this case 12 times without scaling and without adding c, followed by a second fractal transformation ‒ also several times ‒ with adding c at the end of the iteration cycle.
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.
http://blog. hvidtfeldts.net/index.php/2012/05/distance-estimated-3d-fractals-part-viii-epilogue/
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 “something” to fill the space, which leads to spheres as intermediate figure. In the hybrid display you can fill the space with a second fractal like shown in this gallery.
Again from a geometric point of view in the hybrid description c remains as a linear movement (something is added) at the end of iteration cycle. So the clear graphical correlation between M-set and J-sets work in this hybrid as well as the Mandelbot set with Julia sets in complex plane and as known from the Mandelbox itself, which has the same correlation.
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.
You find parameters for Mandelbulber for most images in this gallery.
carambollage
The essence, I saw in Theli-at’s combo is (simplified)
transform(boxfold|ballfold)+transform(f2)+cAll following examples work with that pattern.
A more precise notation would be (as example)
12·(boxfold|ballfold)+repeat(f2+c)
repeat means, that after the boxfold|ballfold sequence f2+c is executed until one of the termination conditions is reached.
AmazingSurface two times c=(0 0 0)
