True 3D mandelbrot fractal (search for the holy grail continues)

[M Benesi]

@Mbenesi: can you control the reflections in the northerly directions? Maybe stop reflecting in the horizontal axis, so the bottom part can develop and be looked at in 3d?

 I suppose eliminating one of the variable assignations might work (any of the wvu pixel assignments), or at least altering a couple of them.  The variable assignation methods assign sign values to quadrants of the fractal.  Perhaps we can eliminate sign values, perhaps we can eliminate differences in calculated magnitude (instead of magnitude of sqrt | pixeli * |pixeli| + pixelj* |pixelj| | which results in more differences in the various quadrants).

 I figured out that the new type is definitely not reducible to a 2d Mandelbrot.

  A trigonometric 2d Mandelbrot's equations look like this:
r2=sqrt(bail);
theta= atan2 (x+flip(y));
a=cos(n*theta)*r2^n +pixelr;
b=sin(n*theta)*r2^n +pixeli;
bail=x^2+y^2;

  The new type reduces to this:

r2=sqrt(sx^2+sy^2);

phi=atan2(sw+flip(sx));
tango=atan2(sv+flip (sy));

sx=r2^n*sin(n*phi)+pixelr;
sy=r2^n*sin(n*tango)+pixeli;
      
sw=r2^n*cos(n*phi)+pixelw;
sv=r2^n*cos(n*tango)+pixelv;

Which actually combines 2 2d Mandelbrot calculations into one beast.  If we take:

r2=sqrt(sx^2+sw^2)   or  r2= sqrt(sy^2+sv^2)

we get a standard 2d Mandelbrot rotated 90 degrees between the 2 formulas.  We don't get anything great looking (at the 2d level) if we calculate the combined Mandelbrots at high iterations.  Some neat low iteration shapes emerge, but nothing of the higher iteration variety:  these aren't standard escapetime fractals with higher iterations leading to more replicated details emerging like in the 2d Mandelbrot or 3d Mandelbulb.  If we combine the 2 magnitudes by applying them to the specific variables they are associated with, we get a quasiMandelbrot reflected diagonally over a quasiMandelbrot.


  Extending the quasibrots up to 3d doesn't look promising.  Get rather boring images, more like quaternion fractals than some type of 3d Mandelbrot.  Perhaps using a combination of Mandelbulb and new type would give us something interesting.  

[M Benesi]

  So, I happened upon the definition of the word "fractal" in the wikipedia article, while trying to find out about the formula I've been using.

  Apparently it is not a fractal due to its lack of self replication at the smallest level.  I don't know the term for what the formula generates (specific term, it generates geometric objects, obviously). 

  Not even sure if it is appropriate for the fractal forums? 

  Anyways, a couple last pictures of it, hopefully Vector pursues his method and hits gold, I might try to mess around with this formula and tweek it until I can get actual fractals out of it. 

 

[jehovajah]

 @ mbenesi.  Fractal i define as being the product of an iterative procedure, so these are fractals no matter what. Would it help if you thought of this as an object sculpted out of a solid sphere reflected into four axes?   What we are trying to achieve is the most analogous form to the 2d mandelbrot that is as aesthetically interesting and fascinating at all levels of magnification. The formula for the procedure that sculpts this object is sought for a space we have no proper reference in, and so proceeds by analogy. To say that the solution is still out there is misleading. We have a solution, but it is not as aesthetically pleasing as some of us would like, However the many beautiful sculptures you and others have produced are not to be belittled.

While you reflect on what you have been producing remember that you have 2 strains of ideas here and masses of ways to explore. This can seem overwhelming, so narrow your researches down to explore each part of what you have produced. You have some fascinating stuff, and some novel approaches which are inspiring.

[M Benesi]

Thanks,

  Here is a low resolution animation I made with the new type I have been working on.  Hopefully I'll work with it, extend it to higher dimensions, clean it up, and perhaps combine it with the other approach and see what that gives us. 

  Here it is.  Definitely interesting variety of details.  If you can watch it in slow motion, it is interesting to see the changes that increasing angular rotation can cause.

  <a href="http://www.youtube.com/v/LIJIsc5oA9M&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/LIJIsc5oA9M&rel=1&fs=1&hd=1</a>

[kram1032]

beautiful animation
I'd also consider that a fractal in a way...
it self-organizes after repeating the same rules a multiple times... (If it would diverge or converge against a single point, it wouldn't be a fractal, I guess...)

Maybe you could do some box-counting... It most likely will have a fractal dimension

[M Benesi]

Thanks, gotta learn about box counting.  Don't know how easy it is to do with what I have, maybe do it in Maxima.

  For some reason, I don't think these objects are "fractal" over  standard space, rather I think they might have fractal properties (of self similarity) over rotation "velocity" (what you multiply the angle by). 

  matt

  Ohh, here is an "old" one, rendered with anti-aliasing enabled.  Looks better, I think:

[M Benesi]

  Here is an additional animation of the new type.  It's low resolution, so a lot of details are hard to pick out, but you can see some of the variety that can be found with rotation & magnitude variation.  Also, for some reason, it kept losing color when I started the animation, and I was in the middle of watching a movie with the woman so I just let it go after the third try.

  In other words, it's gray, in addition to being low resolution. 

  <a href="http://www.youtube.com/v/T_Ws7f5kPkc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/T_Ws7f5kPkc&rel=1&fs=1&hd=1</a>

  Here is a second one, same exact formula, a few starting parameters different, which makes for a different base pattern.  More like a flower.  In this one, colors worked AND it's a slightly higher resolution.  Maybe a little less clipping issues as well.

  <a href="http://www.youtube.com/v/7QH0cQIvK4I&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/7QH0cQIvK4I&rel=1&fs=1&hd=1</a>

[jehovajah]

Absolutely fascinting! The animations alone have a purpose to illustrate electron probability shells! One could even illustrate gravitational collapse. Anothwe could be a form of big bang explosion and the expansion of the universe! big ideas i know for what you have been tinkering with,but i mean to inspire you! 

[vector]

Hi Benesi,
the animations really look very well and have a certain beauty,
i sometimes(as most of You also will have) got Julia objects, with a very caleidoscopical aspect.
Indeed, each circle(or little bulb) on the periphery of a Julia set is a picture of the circle, which is defined by
the circle with radius Bailout(or it can at least be interpreted as such a picture). So, the Julia sets consist(as far as i understood with limited mathematical base) of lots of mirrored pictures of a starting circle. So in nhyperbolic geometry, there are Julia sets, got only by consecutive mirroring at the hyperbolic lines(the orthogonal circles representing those straight lines).
Maybe, a 3d M-set could be constructed by those mirrors, which, very sadly, would be no planar mirrors, but f.i. spheres as in the case of the hyperbolic geometry-pictures. I think it has been done to some extents at the very beautiful, well known, video of Quasi-Fuchsian Kleinian groups(i hope, it is named correctly by me). Have You got an idea, what Your formula is doing? Which way a point will be projected by it?
 

[M Benesi]

 Hi all,

  I suppose I could find out which way a point will be projected since they are only going through a few iterations to reach the complexity I desire (4-6 for the most part, some are higher).  

  That's only 4-6 rotations and additions of the original value, so I could write out an algebraic equivalent (for each number of iterations/ bailout settings).  From there we could map the direction the points will travel and perhaps find their acceleration; which is related to the angular velocity (change in angle multiplication) of the point in space.  Of course, it's a little more complex than simple accelerations.  

  Working on an old formula trying to find out about it (did others work on this?  I suppose I should read the whole original thread):

r2=sqrt(bail);   // I set the "original" bail before this loop, avoiding an additional variable for bailout
                    // and I also set the original sx,sy,sz as pixelr,pixeli, and pixelj respectively before starting the loop
                    // pixelr is the x component, pixeli the y, pixelj the z, it's easy to make a julia of this...

theta=atan2(sx+flip(sqrt(sz^2+sy^2)));   //flip means multiply the value by i
phi=atan2(sy+flip(sqrt(sx^2+sz^2)));
tango=atan2(sz+flip(sqrt (sx^2+sy^2)));
         
         
sx= r2^n * cos (n*theta) +pixelr;  // n is the number of rotations and magnitude exponent... just like 2d complex equivalent
sy= r2^n * cos (n*phi)    +pixeli;
sz= r2^n * cos (n*tango) +pixelj;
         
bail=sx^2+sy^2+sz^2;

  You only get regular "bulbs" for odd powers of n (or mixing values....).  Powers 5, 9, 13 .... seem to be the best for whatever reason, although I was playing with 11 recently.  No great images.  Basically lots of fractal type replication, but more like chasing a 2d mandelbrot set (you have to seek out the 3d equivalent of a "seahorse valley" instead of being able to zoom in anywhere in the fractal).  There is fractally goodness along certain areas, but I haven't been able to go to deep with ChaosPro, get lots of clipping issues, etc.

  I'm thinking the fractal is more symmetric at powers of 1+n*4 because these are the powers of the 2d mandelbrot set that are most symmetric.  The spikes seem to be in areas that correspond to the bulbs breaking off the equivalent 2d mandelbrot, and the structured parts are in the corresponding valleys between the 2d mandelbrot bulbs.

  I did notice that you can travel off too the side of the main pattern after zooming in a bit and find slightly different side patterns: keep in mind this is only at ~19-23 iterations, not 100-1500 iterations like the 2d mandelbrot when you start finding crazy new patterns in the valleys.

  I also have lots of troubles with clipping (hit the "adjust" button a bunch of times and wait for the image to recalibrate, hoping that it hits the right range... patience is required) when I start getting up into "high" iterations (23 or so) as the adjust button apparently hits a maximum (both numbers are "0" after iteration 19 or 20 instead of being .00000001 and .0000000001 or whatever).  

  If you do explore the version: below is a z^9 and z^13 version, including an extreme zoom of the z13.  You can see what I call escape (it's the opposite, I know) spikes, where the fractal's area is actually larger (so the spikes are the part of the set that DOESNT escape, but...) , and you can see the axes where fractally stuff happens.  Apparently the 'escape' spikes are attractors that distort the fractally areas, although you probably need to zoom into the fractally areas a bit before going towards the escape spikes or you will simply enter an escape zone, which is pretty boring.  

  I also included a maximum zoom (for my software, you will see the weird lines that start appearing in the graphics and notice some clipping issues) z^13 (only 16 iterations for the high z^n value reaches max zoom detail).  You can keep on zooming into the detailed areas (not the escape spikes: the escape spikes appear between the details, even when you keep on zooming in) that look a bit like the eye thingies that pop up in mandelbulbs.  I assume you can zoom into the tentacle thingy coming out from under the escape spikes as well, but my software is maxed out and will introduce crazy-ass artifacts if I zoom more.  

  If you stay on the central point of one of the bulbs from the main figure, you can zoom (until my software maxes out: perhaps forever) and keep on finding the same bulb as a little bulb within the bulbs around it.  I did this for the extreme zoom z^13 one, but traveled off to the side of the central bud and started selecting deviations from the main.  If you deviate you can find replicas as well, just like in normal 2d mandelbrots.

  I also included a couple of zooms of the batcave in z^9 in the post below.

[M Benesi]

Batcave zooms.  For this z^9 run I haven't run into my max resolution yet, even though I'm at higher iterations (I think I went up to 24, maybe higher) than when I maxed out for the z^13 run (16 iterations). 

  You can keep zooming into the thing, or pix another nexus of activity and zoom into that for a slight variation.

[kram1032]

well, that one's clearly fractal

[M Benesi]

Yeah.   Yet another 3d Mandelbrot variety.  At least this formula is fractal, instead of just a pretty pattern generator.

  Wonder how it would look in someone's better raytracer?

[M Benesi]

  Allright, yet another new formula type:

r=sqrt(bail);
theta=atan2(sk+flip(sz));
phi=atan2(sy+flip(sx));

  Set bail as x^2+y^2+z^2 to initialize.  Initialized sx,sy,sz,sk as corresponding pixel values.  pixelk= sqrt( pixelr^2+pixeli^2), as usual pixelr is the real or x component, pixeli is imaginary or y component, pixelj is the z component of the pixel...

sx=2^n*sin(n*phi)*cos(n*theta) + pixelr;
sy=r^n*cos(n*phi)*cos(n*theta) +pixeli;
sz=r2^n*sin(n*theta)                +pixej;
sk=r3*cos(zfold*theta)              +pixelk;   //  I did make an if statement in my formula so I can switch this to -pixelk instead of +pixelk

bail=sx^2+sy^2+sz^2;   //also check if bail < bailout value, if so, bail...

Here is an incremental zoom out from a deep zoom little knob with lots of details, first one is higher res/iterations but same as the second one.



here comes the zooming, zoomed out a bunch between images, usually with a little spot towards the center as the old image.






Had to reset because I was in a bad location.  The blue nob at the top is where I zoomed into (or out from).



  Anyways, the new fractal type has some potential.  The other new type is really fractally as well, and needs more exploration.  But I keep on playing with the formulas... ....  

[M Benesi]

Ok, the second order (z^2) thingy of this is nuts.  It's got all these twisted patterns, flowers, etc.  

  Just set k=-k to see it (I have a thing in my formula to change scale and change k's sign, setting it to -k is wierd, ultra detailed, and weird also).
Zooms generally have increased iterations, just to bring out more details (and I'm not trying to render tons of details until I find something anyways)...


Zoomed into notch at bottom, you might be able to make out exactly where:
 
This is some neat bud thing somewhere in it (I got lost when I found it):


Twisty buds zoomed in from left hand pic to right:


  Gonna add a top and side view in a bit.  Here's the top, side is calculating:



side:


rear:



  The fractal details are weird, I think it doesn't have the patterns that the higher orders z^3++ do.  You can zoom in on them, but they are pretty repetitive, even if they are slightly different.  Perhaps I'm just not looking at the right places.

  Here is a z^4 (working on a z^3, wonder how it will look... I'll post it):


Update: or not post it.  Pretty boring.