Basically, I resurrected an old failed formula of mine and applied knowledge recently applied to another old formula: multiply x,y,z (vs. yz, xz, xy) values by negative rotations - 1 (for z^3 multiply by -4) and multiply the new x, y, and z values by 3 (magic) before adding in pixel values.
However, mess with the multipliers (based off rotation values) if you find that the system is not producing interesting images: for example, the z^17 fractal might not produce interesting images with a multiplier of -18, however using a multiplier of -3 produces a fractally, albeit spiky, object. I've found that using a multiplier of -1 gives good, but spiky fractals for z^n > ~9 (yeah... I know, should have checked to see if sign reversal worked first.. I did, so you know, but with the z^2 variety you don't get great fractallyness for multipliers that low: I prefer -2.5 to -3 for that fractal, until I find out something new). Anyways, I've yet to find a pleasing multiplier for z^7, although somewhere around -8 seems good enough. Need to figure out another method of multiplier assignment... or do abs(sx) *sx method of calculations.
Anyways, it seems that a value around 2.6-2.7 works best for the z^2 version (it gives far more interesting outcroppings on the one side) so.. further experimentation is needed. for z^3 ~ 3.7 . etc.. Here is the ChaosPro code (minus a few unnecessary, possibly confusing modifications that I play with):
//Copyright Matthew Benesi, per the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License
(http://creativecommons.org/licenses/by-nc-sa/3.0/)
r=(sqr(sx)+sqr(sy)+sqr(sz))^(n/2);
phi=atan2(sx*scalef+flip (sqrt(sqr(sy)+sqr(sz)))); // [s]scale of 3 makes it work nice[/s]
tango=atan2(sy*scalef+flip(sqrt(sqr(sx)+sqr(sz)))); // scale of - rotations + 1 seems best (for z^3 scalef=-4, z^9 scalef=-10)
whiskey=atan2(sz*scalef+flip(sqrt(sqr(sx)+sqr(sy)))); // - rotations doesn't alter evens (wondered why odds were wierd)
nx=r*cos(phi*v)*check; // check of 3 makes it work nice
ny=r*cos(tango*v)*check; //perhaps you noticed that I scavenged variables from another formula.... :D
nz=r*cos(whiskey*v)*check;
if (juliaMode) {
sx=nx+cr; // same as below: values for x y and z
sy=ny+ci;
sz=nz+cj;
} else {
sx=nx+pixelr; // pixelr is the x value of the pixel, pixeli the y, pixelj (guess... z?)
sy=ny+pixeli;
sz=nz+pixelj;
}
bail =sqr(sx)+sqr(sy)+sqr(sz); //this is my bailout check variable... the term bailout is already used (I think) so I use bail
Phenomenal results. The z^2 version is amazingly fractal. I skipped a few z^n and went to z^13 (good number: odd, plus lots of details.. or so I assumed and found out). There are interconnected areas within the fractals: pillars, loops, Mandelbulbesque pillars, hills, and places to explore. So first off are a few z^2:
Top view (there are 2 of these, then 2 back views, in other words, it can be rotated 90 degrees once and end up the same, then again and it is different, then again it is the same, once more you are back at the first/second (unless you rotated to the other image first... bleh))
Random zoom in from top (I turned 45 degrees to look at a structure after zooming between some things, then zoomed and iterated a bit (13?)):
this one I had a certain setting changed (set newxyz multiple to 4 instead of 3, not necessary to do so... so will have to find it again without screwed up settings):
this is a back shot. The back is pretty boring... at least where I looked. It is fractal... but... repetitive (unless maybe a different angle?)
That's it for the z^2s. Now the z^13s, 2 800x800, then a bunch of small ones (I was exploring the new formula, don't worry Trifox, they are in my picasa album):
So this one is an 800x800 zoom into 45 degrees off the squarish section of the z^13. I zoomed in on a couple of the bulbs (well, flat bulbs, there are regular bulbs, spiky bulbs, bulbs with a spiral of bulbs going up them, etc.):
This is an 800x800 zoom between the bulbs (color changed to green, but it's between the ones in the image above, perhaps the central and the one above it, then I rotated towards the one above it and zoomed into it a bit, added shadows to show a bit more details):
Here are a few small ones, first one of the spiky forest bulbs:
bad shot, zoomed between the bulbs into the loops and went upside down, increased iterations, not big enough, but here it is anyways:
This is just to show the variety of shapes of the main spikes/bulbs. Some flat, some round, some pointy. And there are other sections (not necessarily as awesome):
June 10, 2010, 09:24:42 AM
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