Very simple formula for fractal patterns

[Kali]

Ok, I know it's pretty vain... but I'm gonna call this "Kaliset" 

Or maybe plural "Kalisets", because the interesting things are the sets of Julias.

[Jesse]

Please add scale and minscale for the ballfold, I'm playing with them in 2D and works nice! Thanks!!

Man, you are killing me  

For who want to try in the 2D formula I posted, replace:

x=x/m+cx
y=y/m+cy

with:

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

Then play with s and r values as you do with mandelbox

So r stands for minscale?

One good thing about the mandelbox is the limiting of sphere folding above 1, so there are no big changes of scales and rendering is smoother.
The sphereinv without limitations brings much noise into the render, i guess because the scalings change between iterations and maybe 1 iteration is skipped for next bailout condition... or something.  Maybe one can improve normals calculation by keeping the same iterationcount, like in the 2 or 4 point DE?

Nevertheless, i had already the simpler version tested, and the results differ from the 'handmade' formula with addons, because the bailou condition is only tested after all transforms.
The bailout value itself is also very crucial for the result, but i have not tested the new method yet.
Values of 2 downto 1 have a grfeat effect on the image, at least for 3d.

ok, to much talking...

ps:
the mandelbox compares m (=x*x+y*y+z*z) with sqr(minscale), should it be if m < sqr(minscale)?

[Aexion]

I have made a little twist to four formula and ported it to the Quadray dimension..here how it looks at julia point c=(-0.8,-0.1,-0.8,-0.1)

The render is ugly, but it was fast made..
As always, just bare escapetime algorithm..

[Kali]

@Jesse: Many thanks you are working on this. To be honest, I'm doing the 2D ballfold as I read in one post I don't recall now, but feel free to try what you think it works best!

@Aexion: Just wonderful... Thanks! The Quadray system is really nice!

[Kali]

I've just realized that when I was doing a combination of what I call Mandelbrot on real numbers (wich is just x^2+cx and y^2+cy), with a ballfold (sphere/circle inversion), I was actuallly doing something like this, because instead of abs function I was turning negatives to positives with the exponentiation. The results of this are pretty distorted but nice anyway.

This one uses that method, colored with exponential smoothing:

[Kali]

Wait a minute... look at this:

z=abs(1/z)+c

Using complex values, Julia mode, with c.real = -0.2 and c.imag = -0.35, the result is...



  (well, not so surprised, I did it because I was expecting this to happen)

z=abs(z^-1)+c

So we are dealing with kind of an "absoluted negative power mandelbrot" <- good name! 

[Jesse]

@Jesse: Many thanks you are working on this. To be honest, I'm doing the 2D ballfold as I read in one post I don't recall now, but feel free to try what you think it works best!

Yet only the one i already made:
http://www.fractalforums.com/index.php?topic=6061.msg28944#msg28944

To get rid of some noise you could try the 'Normals on Zbuffer' postprocessing.

The complete box spherical folding looks to much boxlike, i think.  The benefit would be a fast analytic render method, maybe there is another way to get something more new.

[yv3]

I integrated the Formula into my rendering tool. Called it "Kali DuckKaliset"  

[Jesse]

Wait a minute... look at this:

Hmm, lost a bit the track, but 3d version of this...

Mandelbulb3Dv16{
O.....Y/...33...S....26....oN3vOMZmzzuS1D/Vv0U/EkqdgPVeRJwnzan4M.7MmzmqPs76KncyD
................................nL6o6m2AuzXaNaNaNaNyz...................y.2.....
................/c.0/....6EA3...Z/....E2.....kGLB165peoD/c...............w1/BnAn
z.EnAncD1...02EnAnAnAnArzeNaNaNaN4yj........Ez9..................y1...sD...../..
.w1...sDX.HnySNEax1..........YLlWvwRv6djbh47MXeC0uHR6INU9OoMzwIsNela.NpD/SWBc6AZ
DuXzyS55P0YJzay3rzrW4BqD......ow8...U0................sD..E.0...................
..............................MRb7.4qRa.kOrN0UoRb7.srRa.sVrN0.FSb7..............
.....................w7e21EM.s3.c....E4....D....60...M4...E3....2/...M1...k/r.22
...U.eaKtwdOOZ1eD2XG21.3U.6.1c..zrhe.AWAZRBEi7yjX6HNL1YPWz1..........Ek.8..Xo/2.
4352Zs9U4y1CJrLV1AqJzU1qXIyGntwj36.3xzzzz1EFOhytTU7vzMyTadEdO/xDcXUU3BPr2zP/lcEz
uUXA.27bqQf.DuyD5fIDN9w5Fz1FK16adRghzKEA8oDU.06.uaKaZ6Aeiz1x58nQJ6TnzcSRpBK.iAwj
/6V0z5FAZ............oklrlkCx9yjW9wTJHz2Vz1..I6Vo/.DuI1.bxER7C7.G6/3.Yg5VBrU.kbU
70UOhQtVS0E4KY/.1aHWt75.a3qL.QRF7abQ.MKMT/k6G74VY/kN2W4.Pt3Rf/7.9ck0.gbO7StN.gl4
J.UqqxrXb/.Of/4...EMg75.eeWRAW2.IJJMg75.wybRAW2.................................
E....2....E.....I....6....kGVlKOnJ4Rn31.rJaQ....................................
..........UaNaNaNaNwzMWdge8/qarA................................................
................................................................................
........................}

[Kali]

@Jesse:
Thanks for the formula for M3D, I will try it later cause I'm not at home...
And also the params you posted... Is that the complex formula I mentioned translated to triplex numbers?

@yv3:

I integrated the Formula into my rendering tool. Called it "Kali DuckKaliset"  
<Quoted Image Removed>

No problem with the name - It shares the Ducks concept anyway, the main difference is that mine has a direct 3D analog.

[Jesse]

@Jesse:
Thanks for the formula for M3D, I will try it later cause I'm not at home...
And also the params you posted... Is that the complex formula I mentioned translated to triplex numbers?

It is the "Kalisets1" formula i made, of course 

[Kali]

It is the "Kalisets1" formula i made, of course 

Ah, ok... looks nice!

I get it wrong, because you quoted my post about the "z=abs(1/z)+c" version... maybe using triplex math this could be good also

Also I asked Sam for permission on naming it KaliDucks, not really important but change it if you can 

I don't want to even try installing M3D in this netbook, so I can't wait to get home and give it a try

[Softology]

z=abs(z^-1)+c

So we are dealing with kind of an "absoluted negative power mandelbrot" <- good name!  

That makes it much simpler for other people to code who are used to complex number maths.

A variation on that is z=log(abs(z^-1)+c) which also leads to interesting structures.





Jason.

[Softology]

Another variation z=log(sin(abs(z^-1))+c)





Jason.

[Kali]

@Softology: Very good variations... they are more close to Ducks, because of the log function, but the neg power makes the difference. Nice images.


@Jesse: tried your formula and works better than the custom I made and also faster, nice!

Still think about r parameter... I mean m<r part, and not using m<1...

Just this way:

if m<r
---
---
else
---
---
endif

Did you try somthing like this and it looks such the same as Mandelbox with all julias?
Asking this cause I get different stuff in 2D