2D Mandelbox variations

[Kali]

I was testing a 2D version of the formula I suggested to Jesse for Mandelbulb3D.

The formula is, for each iteration:

Code:

x=abs(x)*s+cx
y=abs(y)*s+cy
m=x*x+y*y
if m<r
   x=x/r
   y=y/r
elseif m<1
   x=x/m
   y=y/m
endif

So there are only two parameters (other than julia values): s = scaling | r = min.radius

Formula is not exactly the same, because there is no additional "translation", just "adding C" before "ballfold".

Original is: abs+translation+ballfold+c

This is: abs+scaling+c+ballfold (I also tested this version in 3D and works fine)

To use in mandelbrot mode (kind of, not good), swap cx and cy values before the iteration part, otherwise just pick negative julia values.

A quick example: s=1.50 | r=0.30 | cx=-3 | cy=-1




For coloring I use "exponential smoothing", but best results are achieved with the trick of skipping one iteration when doing the coloring calculation. Skipping  odd or even iterations gives some different results, as you will see below.

even:




odd:




I will post some more formulas and images later...

[Kali]

More images...

[Fractal Ken]

This formula is a really nice variation on the 2D mandelbox. The pictures have an elegant look.

[Kali]

This formula is a really nice variation on the 2D mandelbox. The pictures have an elegant look.

Thank you Ken!


This is the same formula but taking away the "abs" on the y axis:



As you can see in my last upload to the gallery, some tweaks of this params and proper coloring could end up doing naughty things...  

[Kali]

Had to change the subject because I found new interesting formulas based on Mandelbox, not directly related to AboxModKali, and I don't want to start a new thread for each variation I found. I'm exploring them in 2D first, because it's easier to try out things and I'm interested in 2D and the "inner patterns"... but I will see later if some of this ideas can work in 3D, maybe using Traffassel's Gestaltlupe (can't use M3D custom transforms for this).

Ok, this is one of the formulas (UF5 code as I wrote it for the "loop" part):

Code:

x=abs(x+f)-abs(x-f)-x
y=abs(y+f)-abs(y-f)-y
m=x*x+y*y
if m>r
   x=x/m
   y=y/m
else
   x=x/r
   y=y/r
endif
x=x*s+cx
y=y*s+cy

So, we must provide this parameters: f (folding size), s (scaling), and r (conditional radius).

This is a sample using julia value 1.00+0.50i and params: s=3.00, r=0.50, f=1.00 - only 15 iterations
Coloring method is "exponential smoothing", with one iteration skipping (not always required, but skipping gives best results sometimes)



I will post some more images, but there is so much to explore with many different parameters so I invite you to give it a try and see.
Also Mandelbrot modes have some good, non-stretched (or not very stretched) areas to explore.

Don't have much time now, will continue later...

[Kali]

More images using the same formula.

| s=3.00 | r=1.50 | f=1.00 | cx=0.00 | cy=2.00 | 20 iterations |



| s=4.00 | r=0.50 | f=0.50 | cx=0.50 | cy=-0.30 | 20 iterations |



As I said before there are many combinations of parameters to explore with this alone, but I also have more formula variations that I will post later, when I decide wich ones are worth showing here

[Kali]

This is the same formula but changing the inversion condition to m<r instead of m>r

| s=-2.00 | r=2.00 | f=1.00 | Mandelbrot mode | 20 iterations |

[kr0mat1k]

whoaaa that looks awesome, especially the ones in your reply #5
I'd love to know how to use it as a texture for those piles of bubbles

[Kali]

whoaaa that looks awesome, especially the ones in your reply #5
I'd love to know how to use it as a texture for those piles of bubbles

@Krom: Thanks! Aexion proposed Jesse a method for adding predefined textures or bump mapping into surfaces of 3D fractals. I guess applying the patterns from a 2D formula could also work for that.

---

Well, I'll continue with some variations of ballfold (conditional inversion). This one produces discontinuities with some values, something that for aesthetical reasons is usually avoided. But here can make nice patterns with the right number of iterations, and viewed from the right zoom distance. I really like the effect!

ballfold part is changed to:

Code:

if m>r
   x=x/m
   y=y/m
endif

so, with r=0 is an unconditional inversion (like Kaliset), and with r=1 is a reversed ballfold (original is m<1, this is m>1 and works good in 2D). Any other values leads to discontinuities, but can produce interesting results.

Below are some examples, remember that the coloring method is exponential smoothing, and in some cases I use iteration skipping for it's calculation, so I'm mentioning in the params either if I'm skipping iterations or not.


| s=-3.00 | r=0.10 | f=1.00 | Mandelbrot mode | 16 iterations | no skipping |




| s=-2.00 | r=0.20 | f=1.00 | cx=0.42 | cy=0.42 | 16 iterations | odd iters. skipping |



Sorry about the penis-like shape, I don't know why things like this happens with my fractals lately...    


| s=2.00 | r=0.20 | f=0.50 | cx=0.00 | cy=1.00 | 20 iterations | even iters. skipping |




| s=4.00 | r=3.00 | f=1.00 | cx=1.50 | cy=1.50 | 20 iterations | no skipping |




BTW, I couldn't find a good 3D analog for the last formulas, even when I didn't try it carefully, the way they behaves makes me think they are mainly useful for 2D patterns, so I changed subject to "2D Mandelbox variations". I'll open a new thread if I find any good 3D variation.

[eiffie]

I have been using your "abs fold" for 3d formulas like this:
p.zyx=abs(p+f)-abs(p-f)-p;  //p is the 3d vector, notice the swizzle swapping x and z
if(p.y>p.x)p.xy=p.yx;          //this conditional swap along with the swizzle above makes it "behave" for me

<a href="http://www.youtube.com/v/Axgl0XYkONY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Axgl0XYkONY&rel=1&fs=1&hd=1</a>

Thanks for all the great formulas. Sorry about the glsl code but its all I can think in now.