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 and 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)
ported principle
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.
edit 11/2023: a parameter which includes a bunch more formulas for slot 2 is at the end of the gallery.
A Mandelbulb3d portation with Abox in slot one and two.
janus
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
aurelia
The standard Mandelbulb works in that system, but not well. So here a Mandelbulb quat.
menger pseudo
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.
This hybrid was first documented by knighty and Mikael Hvidtfeldt Christensen in Syntopia Blog
basic M-set
A Mandelbox* (mset) overlaid with the Pseudo Kleinian grid.
* Mandelbox means the same fractal as AmazingBox, Abox or sometimes Mbox
static view
A Kaleidoscopic IFS put in the PseudoKleinian order. You can see this KIFS bottom right in single formula use.
2.5d Pseudo
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.
desert copper
This is an elaboration of that 2.5d Pseudo.
The figure is cut out by a surrounding sphere.
equilibrium
A sphere inversion as pre transform helps to grasp the object, creating a limited entity. At this point I had the feeling for the first time that a circle was closing – or a sphere…
owl
Finally there is a (simple) way to integrate basic transforms like rectangular grids, tri- and hexgrids and some more complex, dedicated pure transformations into the hybrid Pseudo concept.
Proof of concept
I lately changed the approach of implementing formula 2, making it possible to use some more formulas also in mandelbulb 3d. Most KIFS like formulas – like the Menger Sponge – should do now as well. I share the parameter directly as copy/paste parameter. Maybe I will sort this to single files later.
Mandelbulb3Dv18{
g….EZ/…C2…u1…26…EbNJF/ht4mzeB6JHlV720Ex.p7SMhHKyXX8PD69BZ..f.FgM01K4xD
…………………………..48FkhKkVU.2……..Y./………………c.2…wD
…UzA0…Eq…./Q.0/….2k.3…z2….Ej…..694HVU/aHmD/Q.TVdxD..Ae0dkpXin/…K
./EnAnQD12..02kAnAnAnAn1.1………k…………………..3/….y1…sD…../..
.z1…sDgjYVUJ6umw1SLmiX4856zo2hyrU6ygkDD3rfsZ/cEwnY54DTqMKwyS4KHo7knFnjDMZ.FmvS
Tw9kAi3s0umAzYXTqLoaoHkjU…..IT….B/……..sD.6….sD..G…………E……..
6.A……..3.pAnAt1…sD….zw1………………………………TVdxDW….k1.
…..93iyz1…..5.kz.wzzz1.U..6.P….o5…EB….m….c3…EE….4/…In4…UgJ54
kE2U.qFG9yzb2zzzRYoWzz7lz16.3c..zXCc…………6………………..2.28.kFrA0.
.Ub96aAIVz9.1se7Umvxz0………./EU0.wzzz1………..s/……………….E.2c..
zzzz……………………………2./8.kzzzD…………8………………..
/EU0.wzzz1……………………………..MkO9/EUs5hvVtmg7w..uXdabvoJzfiQ.wzw
kuR1yNkO9/EUs5hvPtb/fh2./WToi9fT4gqG.26yFvy1zNkO9/EUs5hvWxb/fh2./WToijuT4gqG.26y
FvyzzNkO9/EUs5hv..EsUa3feeWCNqGQIJ36wk8EwyLsUa3f…………………………..
E….AU2V2.3….I….A….kL/pKMuZaPb7oPsBpI371…………………..k/9…….
……………….wz………..3iSIsuFVjz1……………………………….
……………………………………………………………………..
…………………2…..3….8….oINiRKNmB1.4x4PY/..r/.PZB5……………..
…..MU/4MU/4…………….U.E……..kz1……..wz………zD…………….
……………………………………………………………………..
…………………………../….E/…k…..T34NYB2…………………….
………………………………….kz1……..wz………zD…………….
……………………………………………………………………..
……………………………………..}
{Titel: abox_pseudo_menger~}
