More Holy Grail

[David Makin]

Someone's mentioned this already but I think for the 3D holy grail what we actually need is the X-Y and X-Z slices of the 3D z^2+c Mandy to be the 2D complex z^2+c Mandy and the Y-Z slice to be the 2D z^5+c Mandy

[Alef]

I think, I mentioned this in thread IMHO reason why there are no 3D mandelbrot;)
So complex number formula for z^5 and z^2  Don't sounds too hard. Only thing, if you'll have polynomial for z axis, polynomial for y axis, and to polynomials for x axis, so they somehow must be putt together.

[David Makin]

I think, I mentioned this in thread IMHO reason why there are no 3D mandelbrot;)
So complex number formula for z^5 and z^2  Don't sounds too hard. Only thing, if you'll have polynomial for z axis, polynomial for y axis, and to polynomials for x axis, so they somehow must be putt together.

Yep - put together so that when X is zero we get z^5+c where z is made up of Y and Z and of course if Y or Z are zero then we get z^2+c where z is made up of X and Z or X and Y respectively.
Ideally this should be done such that unwanted terms are removed by cancellation rather than using the obvious scale by other dimension but I haven't looked into it and it's probably not possible to do that.

[Alef]

Idea seems to be right, at least becouse resulting fractal would be mathematically meaningfull, what is expected for real 3D mandelbrot.

To see how different power shapes would correspond to mandelbrot I put together layers of mandelbrot and multibrots.

This would iterated simulatineusly, so probably z^2 with z^5 would loose mandelbrots head, throught side bulbs would be placed perfectly.
So I think, even power would go with even powers, and odd with odd powers, becouse they behave differently -1^(2n)=+1 and -1^(2n+1) = -1 and then you add c=-1 and get or 0 or -2. It seems, that power 6 would fit the best, power 4 shape would have side bulbs would be too close, and power 8 shape stands too much outside of mandelbrot outline.

If y=0 and z=0 x would be x=x^2+x^6 +c what seems main problem.

[Alef]

I got something, IMHO result of mixing powers 2 and 5 were pretty visual fractal. X value from xz plane unused, but it seems that stretchedness depends coefficient 0.

Code:

C=pixel;
compXY = complex (real(z), imag(z));
compXZ = complex (real(z), part_j(z));
	compXY = compXY^2;
	compXZ = compXZ^5;
zy=imag(compXY);
zz=imag(compXZ);
zx=real(compXY) + (real(compXZ))*0;
z= quaternion(zx, zy, zz, 0)+C;

This was easy solution, mixed power bicomplex and far from holy grail. Maybe putting two x= y= z= polynomials together and moving around and changeing coeffittients of x could lead to something. Or maybe not;) But this stump have small buds and looking closer there are more buds and tinny stalks, feature expected but lacking in many other versions, just stretched shape.

[Alef]

Budds are interesting, each having smaller buds in two height levels. But in the middle are extruded mandelbrot. Throught render is not very smooth even for chaos pro. Not quite an Holly Grail, but:

[LhoghoNurbs]

I'm also hunting for the Holy Grail. I got some new ideas (at least I hope they are new) ... which look promising. But I have a problem: I do prefer to program this idea (instead of feeding some third-party software with the formula and waiting for it to cook an image). And the problem is that I will get a collection of pairs [3D point]-[color]. I have no idea how to convert this cloud of points into a solid object or a surface or a mesh.

Maybe there is some simple but effective algorithm for this?

Ah, as for the formula -- I prefer to keep it secret until I see what image does it generate. To be honest, the probability to get something ugly and unholy is extremely high (>99%).

[kjknohw]

You-know what theorem makes grails impossible, but what if we can get arbitrarily close? Kind of like Heizenburg's uncertainty principle not preventing macroscopic objects from having well-devined positions at rest.

[M Benesi]

Someone's mentioned this already but I think for the 3D holy grail what we actually need is the X-Y and X-Z slices of the 3D z^2+c Mandy to be the 2D complex z^2+c Mandy and the Y-Z slice to be the 2D z^5+c Mandy

 I toyed with a similar idea a little bit ago (maybe this thread subconsciously prompted it??...). 

1.  I took the 2d xy Mandelbrot, did edge checks for each point on the x-axis 
      --- is point on the edge of the fractal and inside fractal?  if so....
      --- check for next point... if y>1.5 go to next x point

2.  used the y value of this point to divide the y[/b] and z[/b] plane magnitude, then calculated the z^5 (tried others as well) Mandelbrot escapes for the yz fractal. 

  You end up with a z^5 yz cross section that follows the z^2 Mandelbrot xy.  It's a pain in the ass to implement in ChaosPro, and I didn't bother streamlining/debugging the code as it wasn't all that pretty. 

  Basically you get a relatively smooth pattern of transitions- exactly what you would expect.

  The problem being, at least for ChaosPro, there isn't a really easy way to implement something that checks the x axis first, then the yz plane vs. points on that axis.  Instead, you check points in whatever order CP desires, gotta recalculate xy plane radius for every yz pixel... TAKES A LONG TIME!@#!@#

  You'll notice that I didn't debug the script- has completely filled in sections "outside" of the x- radius because I didn't bother writing in a part of the script for that part (which would bailout...).

  Here is a z^2  x  z^5, straight on... only 333x333 pixels because it takes forever to calculate on my old ass computer:



  Here is the inside of a z^2 by z^2... gives more of a feeling of whats going on:


  The reason it is jaggedy (has those sudden cut offs) is pretty simple: it's determined by the resolution I check the xy radius at.  That resolution is the thing that slows the whole thing down in ChaosPro.  Now, for a directly written routine?  It wouldn't be a problem.  The thing is, I can't do that in CP....  basically have to achieve bootstrap levitation in order to do any of this stuff.

   I've got a busy weekend coming up... but maybe I'll help out with a flowchart so someone can implement this formula in MB or some other thing.  You guys will figure it out.   

[M Benesi]

  It seems to me that this formula is the best bet.  Unfortunately, it consumes more compute cycles than the standard mandelbulb formula.  One must calculate all edge radii for each x, then check y and z values against EACH radii- more on this after the image.

  Older images below these.  This is the newest formula- I realize the flat parts are because I have to start from the farthest point out when doing the radius check (so I don't hit the inside parts first).  The flat parts are where it would be curling over the inside... as you would expect.

   Will rewrite to get rid of flat parts in a few hours.. way harder then I thought.  Gonna think about it...



  It took ~12 minutes to calculate this low resolution (low radius resolution!!) 444x444 z^2 by z^5 mandy:



  Here are a few with slightly improved code:



  So, we have to calculate the radius (off of the x axis) of the spun (really, a simple 2d) Mandy at each x point.

  x= x  axis point       r= radius off of x axis (start at 0)

  if (x,0) is within the (2d) Mandy, we set our IN variable to true, otherwise we set it to false.

  if IN is true, we keep on increasing the radius by the radius resolution until we hit a point outside the 2d Mandy (use the number of iterations you use to check the whole fractal).  radius=radius+radius resolution until we change from inside to outside

  Once we hit a point outside the 2d Mandy, we know that the point back one radius resolution is at the edge of the Mandy.  This will be ONE of our divisors (I don't divide, instead use radius^-1 so I don't get divide by zero errors) for our y and z pixel values.

  if IN is false, wait till you hit a point within the fractal to set your radius.

  y_pixel_new=y_pixel_actual * radius^-1

  z_pixel_new=z_pixel_actual * radius^-1

  Now we run a simple 2d Mandy check of our y and z pixel values.

  (y+ iz)^n + y and iz new components, check for bailout, iterate number of times you set...

  If it's inside, it's a hit.  If it's outside, increment your radius by your radius resolution and check your y/z values again.

 
  UPDATE:  The flat sections require a bit more work, and it might be impossible with ChaosPro.  It looks like we need to take the farthest out hits on the yz plane, and calculate inwards.  This will give us the gaps we are missing in the flat areas.  

  By this I mean we will divide a new radius (1.26) by the magnitude of the yz pixel line sqrt(y_pixel^2+z_pixel^2)- in other words how far out the y and z components are.  We will then calculate a z^2 mandy from the farthest point out using the x pixel component of the plane and the new radius based off of the yz magnitude, checking values along the line to see if they are in the mandy or outside.

  lol... we might even need to then check the new details against the yz plane again.  The true mandy might be computationally intensive. 

  Basically, we can't raytrace to pixels- we have to take each slice of the x axis and calculate what lies on it.  This makes the standard check (moving towards the fractal) a pain, and without being able to control which way pixels are checked... blehh