SwirlBox

[Alef]

I had an idea to bring up swirls without cutouts, and using fast, continious and conformal transformations.
http://www.fractalforums.com/new-theories-and-research/succeess-with-3d-gravit-swirl/
At first, tested this in with exp smoothing and with orbit traps so that orbits are continious and then went to 3D. Modified abox fold, fish eye transform, scale and then 3d swirl transform + C. Pseudocode with parameters being Frequency, Spin, Scale, Fold and Fisheyeness:


//Do this once.
fishdivisor= 1/(Fisheyeness+1);

//seamless version of fold
zx = abs(zx-Fold) - abs(zx+Fold) + zx;
zy = abs(zy-Fold) - abs(zy+Fold) + zy;
zz = abs(zz-Fold) - abs(zz+Fold) + zz;

//fisheye is conformal transform and a scaling.
fisheye=(Fisheyeness+sqrt( sqr(zx)+ sqr(zy)+ sqr(zz) ) )*fishdivisor*Scale;
zx=fisheye*zx;
zy=fisheye*zy;
zz=fisheye*zz;

//3D Swirl. Probably at least quasi conformal, but iteration strong.
radius = sqrt( sqr(zx)+sqr(zy)+sqr(zz) );

angx=atan2(radius+1i*zx);
angy=atan2(zx+1i*zy);
angz=atan2(radius+1i*zz);

zx= cos(Frequency*radius+ Spin*angx)*radius+Cx;
zy= sin(Frequency*radius+ Spin*angy)*radius+Cy;
zz= sin(Frequency*radius+ Spin*angz)*radius+Cz;



In 2D. Looks pretty much like kalisets/ducks:




Julia set is very different, orderly and smooth.



UF parameter.

Code:

Swirlbrot_Rust_Flower {
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}

Lots of pics:
http://www.fractalforums.com/images-showcase-(rate-my-fractal)/swirlbox-10-and-zooms/

[Alef]

In 3D it's a jasper planet. This looks smoother than plain 3D swirl transform, probably becouse additional transforms softens orbits. Here are zooms out:




Closer:


This is strange


And this is even more strange


Julia set is cutted in z=-0.1


Throught no DE or GPU stuff. Chaos Pro parameter files are in attachement.

[kram1032]

Fisheyeness
...
fishdivisor

...
...
...
LOL!!

When I saw that pseudo-code, I really burst in laughter. That's awesome!
...

On a more relevant note: Those things look pretty pretty.
I wouldn't want to live on them - they tend to look rather sharp-edged and probably would cut me into slices upon first contact - but they do show some interesting shapes.

Why did you use zx+zy+zz as r² rather than xx+yy+zz?

[cKleinhuis]

never forget the "fishdivisor"

[Alef]

Fisheyeness was meant to be serious  

Hadn't tested r^2. But woudn't a r^2 instead of r ruin atan2 angles? Actualy I even aren't shure, do angles are correct, haven't put trigonometry on the paper. Just tryed different possibilities until result were what I expected. if you ment zx, that becouse x, y, and z is predefined variables I don't want to interfere with.
Probably fisheye's radius could be optimised with r^2, throught then it would match another apophysis transform a Bubble, who realy looks almoust as Fishaye. I took it from official fractal flame pdf.

Tweaked a strange fractal spot.

[kram1032]

what level of rotations are you using in those?

Also, it occured to me that, maybe, applying the swirl transform off-center could give interesting results.

E.g. do something like:

z=z-m
z=swirl(z)
z=z+m

to have some swirling happen around m rather than 0.

On top of that off-center-swirl, you then could choose any other transform like, for instance, the usual z=z^2 or z=z^8 or something alike.

[Alef]

Zoomed into second, even more strange fractal.

[Alef]

Julia set of  your suggestion were pretty cool:)

Z=Z - Fold
Z=Z*scale
Z=Swirl3D(Z)
Z=Z + Fold + C

Screen is centered on (0,0,0)


Something like this

zx= zx -Fold;
zy= zy -Fold;
zz= zz -Fold;

zx=scal*zx;
zy=scal*zy;
zz=scal*zz;

radius = sqrt( sqr(zx)+sqr(zy)+sqr(zz) );
angx=atan2(radius+1i*zx);
angy=atan2(zx+1i*zy);
angz=atan2(radius+1i*zz);
zx= cos(frequency*radius+ spin*angx)*radius;
zy= sin(frequency*radius+ spin*angy)*radius;
zz= sin(frequency*radius+ spin*angz)*radius;

zx= zx +Fold+Cx;
zy= zy +Fold+Cy;
zz= zz +Fold+Cz;


and mandelbrot set of this:
Z=Z - Fold
Z=Fisheye(Z)*scale
Z=Swirl3D(Zvector)
Z=Z + Fold + C

Screen is centered on (0,0,0) Its a bitt rugged, but that's becouse I did not spend enought time of tweaking  bailout, iterations, render. I shared this just becouse it have multi layered structures. But now it's displaced.



fisheye=(Fisheyeness+sqrt( sqr(zx)+ sqr(zy)+ sqr(zz) ) )/(Fisheyeness+1)  * scale;
zx=fisheye*zx;
zy=fisheye*zy;
zz=fisheye*zz;

[kram1032]

can you try more subtle swirl effects? Like, test on some formula, how sensitive it is to such a swirl transform.
Try very subtle values of turn and ripple frequencies to see how far away the values have to be from default to drastically reshape the fractal.

[Alef]

You can smoothly rotate mandelbrot with frequency *radius (ripple?), but swirl transform don't likes non integer spin *angx (turn?), and create non smooth fractal shape. At least with mandelbrot.
With box probably I would got nothing, becouse only swirls transform here is fractal generator.

[Alef]

Here turns (spin) = 8, frequency =4. Each turn is rising in power. It's actualy a zoom into second planet like 3D fractal + colours. In some looks spirals could be evident, but they eventualy turned into something more like knots. Spiral is least energeticaly dense shape, at least in some process.

[kram1032]

looks neatly cavey.

[Alef]

Some kaliset / ducks like 2D thing. Tweaked and tweaked, but still don't like it's colours.

Code:

Swirlbox_2D_Beta {
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}

[Alef]

Very fluid 2D, much becouse of anialiasing. Lines can be very thin, too thin without antialiasing. Kind of better than previous.

Code:

Swirlbox_2D_Gamma {
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}

<a href="http://www.youtube.com/v/Swm01YN_NRY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Swm01YN_NRY&rel=1&fs=1&hd=1</a>

[kram1032]

that last one looks great
I rank it somewhere between wood age rings, thick, oily liquid and heavy thunderstorm-winds