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

[jehovajah]

Just trying to code twinbee formula in 150 characters or less lead to this!

Found here http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/

and the simplification here http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/30/

Whoops my mistake i see it is karl131058 who did the simplifying sorry.
Anyway because the variables in quasz are quad variables and i have used them like real variables i do not know if the intended outcome has been realised. More likely some unintended consequences of quad algebra have been introduced. This is the formula used.
s=imaj(z),g=x#*x#,h=y#*y#,m=s*s,f=g+h,e=g-h,d=f-m,if(abs(y#)<.001),z=g-m+0*i
> -2*s*sqrt(g)*j+c
> else z=d*e/f+i*2*d*x#*y#/f-2-s-sqrt(f)*j+c endif

You have to visualise that this is all on one line!!
so the s,d.e.f.g,h and c are all quaternions.In addition they have not been properly initialised apart from c that is. Terry has posted some updates to the windows platform yesterday but not the mac yet. The trial version has limitations obviously but it is so intuitive and sweet to handle. The grid structure appeared in a region that looked like spots. See
http://www.fractalforums.com/mystic-fractal-programs-gallery/mandelbrotin-3d/ 

Thanks for that info on the spirals.

[jehovajah]

Ok, so now i am getting somewhere in understanding how this mandelbrot works.
.

This is the mapping of the mandelbrot  m set in the set A of the tensors mapped by the following tranformation onto a set B of the tensors by the following procedure. B needs to contain a set |B| which is all those elements b of B such that |b|<2 .Elements of m map exactly into |B|. The restriction on |B| can be altered but this alters the set m.

I have described the mapping, the second necessary part is the transform. The transform is an iteration mod(n) on a function of z,c in A in which z is a fixed value in A which is squared and c is a variable in A which is varied at each iteration period (mod(n)) and added at each iteration. During each iteration period (mod(n)) the result of each manipulation becomes the new z so z is determined by the result of each iteration, Thus it is a feedback variable determined by the function form and the iteration rules. The rules state that at the end of each iteration period (mod(n)) z returns to its fixed value while c can represent any element of A, however during each iteration period c is fixed while z is determined by the outcome of the function manipulation. So to write z=z^2+c is hardly adequate and essentially misleading.

The mapping and the transform together allow one to comprehend what is happening in this cybernetic feed forward feed backward system.

As artists will appreciate the set m is sculpted from the set A by the transform And the Restriction on set B . By making |B| the mapping of A under the transform and requiring it to be one to one we obtain set m as the residue or the sculpted form of the whole process.

[jehovajah]

De nombreux lecteurs connaissent probablement l’image de gauche qui est l’ensemble Mandelbrot en 2D. Pour chaque nombre complexe c, on considère la transformation du plan complexe définie par z z^2+c. Partant de z0=0, on itère cette transformation, c’est-à-dire qu’on construit la suite zn avec zn+1=zn^2+c. L’ensemble de Mandelbrot est l’ensemble des c qui sont tels que la suite zn ne tend pas vers l’infini quand l’entier n tend vers l’infini.

So the transformation can be written in french which is marginally better than this



Thinking about the transformation sculpting, this is controlled by the restriction on B . With the restriction in place we can think of the rotation as positioning the sculpting tool and the expansion and translation as chipping off a bit of a sphere if the iterated z breaks the restriction.  Also the iterated z traces a path which may be an orbit or some other path. If we map A to A then the paths mapped out will trace the path and depth of the sculpting tool's cut if those points lie on a path that breaks the restriction.

[jehovajah]

The current mathematical construction of geometrical space uses two mod(4) unary operators defined on a plane. It is possible to define a unary operator for geometrical space using one mod(6) unary operator. I have not done so yet but i have plans to investigate this. The fact that we use an even mod(2n) set of unary operators is so that we can preserve the unary operator sign on each of the axes in the plane and this essentially is the same as requiring orthoganality in the axes.

We combine the two mod(4) unary operators  so that one pair of axes coincides with each other while the other pair are orthogonal to each other. So if the pair of axes in the x direction coincide then the orthogonal pair are arranged in the z and y directions respectively. This construction however is a restricted set for a so called 3d space, as we only have two planes of rotation these being xy and xz. That is to say that the unary operators mod(4) are rotating about the z axis and the y axis respectively.

While this is sufficient to describe the position of any point in 3d space it is not sufficient to describe the rotation of any point in so called 3d space. It is clear to me that i need a third unary mod(4) operator to describe the rotations in the yz plane about the x axis. This means that since the axes of the third operator will coincide with both the y axis and the z axis i need only to describe how the unary operator flips between the other 2 operators in these directions bearing in mind sign .

For the moment using v for this operator,

             .

The question now is how to represent this in a way that does not confuse it with so called 4d space or quaternions.

One thought occurs and that is to use a similar construction to my definition of vx but with the unary operator for the yz plane rotation being v.

[jehovajah]

Here's a polynomial mandlebrot type image. z=z+c+c^2+c^3+c^4+c^5+c^6. I am going to be referring to polynomials soon.

[M Benesi]

  Guys, found it:

r2=sqrt(sx^2+sy^2+sz^2);
theta=atan2(sqrt(sx^2+sy^2)+flip(sz));
phi=atan2(sy+flip(sx));

r3=r2^2;

nx=r3*cos(2*theta)*sin(2*phi);
ny=r3*cos(2*theta)*cos(2*phi);
nz=r3*sin(1.5*theta);

  Apparently the Z axis rotations should be .5 less than the x/y rotation.  Must keep same xy rotations, or maybe not (perhaps other thetas should be changed as well, don't know yet, for now, here are some pics of the current format (2 theta for x and y, 1.5 for z, also I used same magnitude for z, so perhaps it should be set to 1.5 as well... we will check, won't we?):


  First images attached.

[M Benesi]

If we set:

nx=r3*cos(mag*theta)*sin(v*phi);
ny=r3*cos(mag*theta)*cos(v*phi);
nz=r2^mag*sin(mag*theta);

with mag=1.5, we don't seem to get the foam problem (much).  Of course, increasing details a bunch from far away will still create foam, but it doesn't look like we get way too much.  Apparently not.  It just seems to shrink the fractal, which seems logical considering that we get smaller fractals when we increase iterations in 2d Mandelbrots.  Still get a bit of a mess at too high number of iterations when zooming in (lots of details means lots of stuff), and I've got a bunch of clipping problems with the renderer I'm using, which may be due to exceeding its floating point limits (don't know its extended capabilities, and don't really want to find out until I have a DX11 card to play with).    

  I'll have to wait for those of you with better computers (DX11 rendering capability) to really zoom in.  


  So, once again, to be clear:
r2 =       sqrt (sx^2 + sy^2 + sz^2)
theta =  atan2 (sqrt (sx^2 + sy^2) + flip(sz))  // flip (sz) means multiply sz by i
phi=       atan2 (sy + flip(sx))

magnitude= r2 ^ 2

nx= magnitude * cos (1.5 * theta) * sin (2 * phi)
ny= magnitude * cos (1.5 * theta) * cos (2 * phi)
nz= r2^1.5      * sin (1.5 * theta)


  So far, the pictures look clearer with all 3 theta values + z magnitude set to 1.5, leaving phi values at 2.

  The pictures below are an example.  The clearer one is with all set to 1.5, although there are interesting details in the 2 at lower iterations (probably due to the distortions it introduces, and I like the way it (2 value back side of Mandelbrot) looks a bit more organic, but you can see for yourself).  

  The 8th order Mandelbulb, and the rest seem to be clearer with z value of exactly .5 less like the 2nd order, although we need further tests with better rendering engines.  They are all (2nd, 3rd... 8th) a LOT more symmetric.  

  Makes me wonder about the time dimensions I've been adding, if they should be 1/3 for 1st time dimension, 1/4th for second, or 1/4th for first, 1/16th for second..... but it's late and I need food.  

[M Benesi]

Alright, here are the TRUE 3d mandelbrots with 1.5 for all values (1.5 * theta, radius^1.5 for z).

[jehovajah]

Looks cool! But it is clearly sculpted flatter than previous versions and has sacrificed a lot to have the Mandelbrot border.

[M Benesi]

Yeah, 

  I was also told that it would have 4x90 degree symmetry over the real axis, although I don't entirely agree with that assessment.

  I was doing something similar to what you did earlier in this thread (adding an additional angular increment to the one axis) and thought why not try out +/-.5 angle differences. 

  Looked good at 2 in the morning.  Now... looks similar to a bunch of other bulbs. 

[David Makin]

  I was also told that it would have 4x90 degree symmetry over the real axis, although I don't entirely agree with that assessment.

Somewhere online there's a sketch of what the "true 3D" Mandelbrot *should* look like - it was linked I think somewhere in the original discussion thread but I just looked and couldn't find it.
When I say "should look like" I just mean it's exactly what most folks would expect/want it to look like based on the 2D version i.e. it's purely aesthetically based

[M Benesi]

  I was also told that it would have 4x90 degree symmetry over the real axis, although I don't entirely agree with that assessment.

Somewhere online there's a sketch of what the "true 3D" Mandelbrot *should* look like - it was linked I think somewhere in the original discussion thread but I just looked and couldn't find it.
When I say "should look like" I just mean it's exactly what most folks would expect/want it to look like based on the 2D version i.e. it's purely aesthetically based

  I think I saw the sketch you mention somewhere around here, perhaps in the original thread (someone mentioned it was similar to the tetrabrot, if I recall correctly).  

  I guess I could see someone expecting something along those lines, assuming 1 axis= 0 we should get a 2d Mandelbrot on the other 2 axes plane, but I don't imagine there being something that simple when we add an additional dimension.  I'd think that we wouldn't be able to expect to use the same angular rotation that we use with the x and y components, apply the same rotation to the sqrt(x^2+y^2) and z components and get anything like a regular 2d Mandelbrot on the z/x plane.

  Here is a triple Mandelbrot, bailing out for complex z = x+iy, y+iz, x+iz, iterations z=z^n + relevant pixel components.  Looks weird and picassoesque.  I am sure somebody has tried this before, however.  I suppose someone should try bailing out between z= sqrt(x^2+y^2) + i*sqrt(y^2+z^2).... as well.


 

[M Benesi]

  Yet another 3d type of fractal.  Doesn't look like a Mandelbrot, but it looks like we could maybe slice it at a certain point and get a Mandelbrot 2d cross section.  It's reflected everywhere, so... weirdness.  More reflection than expected.  Maybe hybridize it with mandelbulb type formulas later.  Seems to be reflected over every axis, probably because of the way I did it.  <-- that looks right

  I like to set bailout really high (1e30), to make bulbous stuff, but I find a bailout of around 64 seems to be a good sweetspot, 128 is nice too.  

Some pictures at the bottom, here is an animation through certain angular increments (and magnitude as well, not really sure actually, I did change the iterations at some point, and you can see the pop):
<a href="http://www.youtube.com/v/iPAnK8I0-lU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/iPAnK8I0-lU&rel=1&fs=1&hd=1</a>

In addition to the x,y,z pixel components I created these:
pixelw=sqrt(pixelj^2+pixeli^2);
pixelv=sqrt(pixelr^2+pixelj^2);
pixelu=sqrt(pixelr^2+pixeli^2);

 in the above pixelr= x axis component, pixeli= y axis component...

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

phi=atan2(sw+flip(sx));     //  flip means change from real to imaginary
tango=atan2(sv+flip (sy));
theta=atan2(su+flip(sz));     // I used theta for the z component instead of the x

nx=r2^n*sin(v*phi);      //  I like to be able to adjust the magnitude
ny=r2^n*sin(v*tango);  //  and angular rotation of these guys
nz=r2^n*sin(v*theta);  //  next step is to do it for each individual
nw=r2^mag2*cos(mag*phi);   //  component, as altering any should give you
nv=r2^mag2*cos(mag*tango);  // a different fractal
nu=r2^mag2*cos(mag*theta);

sx=nx+pixelr;
sy=ny+pixeli;
sz=nz+pixelj;
sw=nw+pixelw;
sv=nv+pixelv;
su=nu+pixelu;


I do bail<bailout check.  So is set bail in the iteration component:

bail=abs(sw)^bail1*abs(sx)^bail2+abs(sv)^bail1*abs(sy)^bail2+abs(su)^bail1*abs(sz)^bail2;

  abs(x) = absolute value of x  (you probably know that)
  bail1 is the wvu component exponent, bail2 the xyz components...

  It doesn't seem to matter how you add/subtract bail between bail1 and bail2 if the xyz and wvu rotation (and maybe mags) are equal.  However, when the rotation/mags are different, picking different values for the bail changes things.
  The first 2 pics are examples of this: the one has bail1=4 bail2=-6, the other is bail1=3 bail2=-3 and every other fractal component is the same: I'll explain my picture name format in a later post, I like to include iterations, bailout, and all the other components with the pics.

  I haven't done julias of this new type yet.  Never got too familiar with julias, but you guys know the coding difference: static values instead of the pixel values added every iteration.

  Sorry about the sloppy writing.  Brain tired.

[jehovajah]

i know, it gets you like that. Anyway welcome to the thread. Your palette reminds me of those rocket ice lollies you used to be able to get.yummy!
Reflection and the mandelbrot might be a useful avenue to pursue especially as you are getting the rotation and extension through the v and n variables, but can you control the axes that express reflection? You may need to do this to approach a sculpting of a 3d mandelbrot.

[M Benesi]

  Thanks jehovajah,

  I'm thinking we can accomplish that by taking a hybrid version between the mandalball (from mandala and mandelbrot/bulb and its spherical tendencies in higher rotations) and the mandelbulb/mandelbrot x/y.  

  Do the standard x,y things from mandelbulb/mandelbrot, but do the mandalball z axis equations (which generate reflections).  Gonna try it in a few hours after this current animation is done computing and I've done all that I have to get done.  Maybe after dinner I will have a few minutes.

  Can't get caught up now, so I'll try and just leave this message, in case I forget what I was going to try.. I'll probably see this again.  


  Hahaha, mid work out I thought: maybe we should differentiate pixelw, pixelu, pixelv components:
for pixelw, if pixeli and pixelj are the same sign, leave as is.  If they are the opposite:
pixelw=sqrt(abs(pixeli^2-pixelj^2))

  So basically, if pixeli*pixelj<0 ....   (do the same for other components)
  Of course, the later equations might need to be modified as well.  Back to the work out.  Try not to think about fractals.