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

[Timeroot]

Just a thought about the speed of these programmes: Does any one use  flags in the register for each iterated point to indicate if it is part of a sequence that  escapes or if it orbits? This would mean when the programme steps to that point it can skip the calculation calls as it already has determined this point is in the set 0r not. I know you add a flag look up call,but maybe overall this might save time?

I think some people - rather than using DE - use an orbit trap at the center of the Mandelbulb, because this indicates a periodic cycle. Like David Makin said, this doesn't work for Mandelbrot Sets well, only for Julia Sets. This can be used to help points bail out faster if the point goes in any point outside the Antibuddhabrot/bulb. I don't know if these more complicated bailout mechanisms actually are worth the potential speed ups, but it can be relatively easy in some places. For example, if the 2D Mandelbrot, if Re(z)<-1.4 and abs(imag(z))>0.06 then you know it will bailout. Here's some code to plug in and see the difference: "bailout: (|z| < 4)&&(Real(z)>-1.4||abs(Imag(z))<0.06)".

[jehovajah]

Thanks guys. I think this may be because of scaling issues in the mandelbrot. It seems each mandelbrot is at a different scale and so smaller and smaller mandelbrots are being sculpted, but they are magnified up to look the same apart from the new detail sculpted out.

[jehovajah]

Re: True 3D mandlebrot type fractal
« Reply #38 on: November 19, 2008, 04:19:12 PM »   

Oh, my deeply sleeping thread!
I revive Thee from the dead... 


Ok, seriously, after staying away from FractalForums for a while because of RealLife(TM), I came back and (obviously) found this thread. I had a little exchange of mails with twinbee about the formula involved and he has asked me to post the results here. So, here it comes:


twinbee defined (a "few" posts back)

double r    = sqrt(x*x + y*y + z*z );   
double yang = atan2(sqrt(x*x + y*y) , z  ) // that would be theta in std polar coordinates
double zang = atan2(y , x);                      // that would be phi  in std polar coordinates

so I would suppose he has implicitly

x = r*sin(yang)*cos(zang)
y = r*sin(yang)*sin(zang)
z = r*cos(yang)

and I would have expected (doubling the angles!)

newx = (r*r) * sin(yang*2)*cos(zang*2)
newy = (r*r) * sin(yang*2)*sin(zang*2)
newz = (r*r) * cos(yang*2)

but he defines

newx = (r*r) * sin( yang*2 + 0.5*pi ) * cos(zang*2 +pi);
newy = (r*r) * sin( yang*2 + 0.5*pi ) * sin(zang*2 +pi);
newz = (r*r) * cos( yang*2 + 0.5*pi );

which can be simplified by taking into account the symmetries of sin() and cos() to

newx = - (r*r) * cos(yang*2) * cos(zang*2)
newy = - (r*r) * cos(yang*2) * sin(zang*2)
newz = - (r*r) * sin(yang*2)

which is not exactly equal to doubling the angles.
... but it leads to interesting pictures.
One DOES NOT need atan2(), sin() and cos() to implement these formulae,
because they can be simplified a lot by using the following identities for any
angle a :
cos(a)*cos(a)+sin(a)*sin(a)=1  (well known, I suppose)
cos(2*a) = cos(a)*cos(a)-sin(a)*sin(a)   (less well known, it seems )
sin(2*a) = 2*cos(a)*sin(a)

I'll spare you the details, but you end on:

newx = ( x*x + y*y - z*z )*( x*x - y*y) / ( x*x + y*y )
newy = 2 * ( x*x + y*y - z*z )*x*y / ( x*x + y*y )
newz = - 2 * z * sqrt( x*x + y*y )

no trigonometric functions at all, just additions, multiplications, divisions and a squareroot.
There is NO pole on the z-axis, BUT there might be numerical problems because of the
denominator, solvable e.g. by taking

if( abs(y) < really_small_value )
newx = x*x-z*z
newy = 0
newz = -2*z*sqrt(x*x)
else (view above)

(end of simplification)

Interesting side effect: when I first read about "doubling both angles" I wanted
to try that for myself. When doing geometry instead of math, I prefer measuring
the angle theta not against the z-axis, but against the x-y-plane, so in that case

phi = atan2( y, x ) // just like before
theta = atan2( z, sqrt(x*x + y*y) ) // exchange of arguments, angle is positive above, negative below x-y-Plane

then
x = r*cos(theta)*cos(phi)
y = r*cos(theta)*sin(phi)
z = r*sin(theta)

and simply doubling the angles

newx = r*r*cos(2*theta)*cos(2*phi)
newy = r*r*cos(2*theta)*sin(2*phi)
newz = r*r*sin(2*theta)

simplifying this analogous to above gives

newx = ( x*x + y*y - z*z )*( x*x - y*y) / ( x*x + y*y )
newy = 2 * ( x*x + y*y - z*z )*x*y / ( x*x + y*y )
newz = 2 * z * sqrt( x*x + y*y )

IDENTICAL to twinbee's stuff except for the sign in z !

Since I'm sitting at an OLD Mac (350MHz) and try to use POV-Ray to produce pictures I can't show any yet; that takes TIME! But the first little "thumbnails" I produced show that this change of sign dramatically changes the resulting set!

Forgive my ranting - I hope somebody might find these "simplifications" useful - twinbee thought so, at least!

Happy iterating...
Karl

[jehovajah]

So adding to Karls original analysis:
Power-Reducing/Half Angle Formulas




Product-to-Sum Formulas



For the mandelbrot in 2d

z= x+ iy=rcosø+irsinø

z²=r²(cos²ø-sin²ø+2icosøsinø)

                  =r²cos2ø+ir²sin2ø


For the 3d mandelbrot


z= x+ iy+jz=rsinΩcosø+irsinΩsinø+jrcosΩ

z²=r²sin²Ωcos²ø-r²sin²Ωsin²ø-r²cos²Ω+i2r²sin²Ωsinøcosø+j2r²sinΩcosΩcosø+ijr²sinΩcosΩsinø+jir²sinΩcosΩsinø
                  =r²(+i+j+ij+ji)

So doubling the angles is only part of the transformation if this analysis is right. Please check and point out my mistakes as it is easy to rectify.

[M Benesi]

   All right.  Removing the formulas for now.  Found out that the (old) radius formula might be incorrect, have to update it and really busy right now.

  If you mess with it:

r= sqrt( (sx^2+sy^2+sz^2)^n + (sw^2+sv^2+su^2)^n);    /*  the planar and linear magnitudes are separate: better math, better fractal  */
  Although the old formula makes cool images as well.  Must experiment.

[jehovajah]

From analysis of the trig methods i wonder  three things:
Hairy ball effect does it depend on stepping ? That is to say the finer the steps the less hairy the renders look. If so the hairy effect i due to the great cicle nature of the transformation and sculpting;
Fuzziness does this depend on what accuracy π is set to? In which case double or quadruple covers may mean pixel identification is not "spot on":
Whipped cream is this due to the fact that only only 2 degrees of freedom are used for rotations when in fact there is no such limitation in geometrical space, unlike flatland 2d world. If an axis is determined and a plane to which this axis is a norm then rotation about that axis in that plane can be defined. A norm to a plane is not necessary as a spin axis but it makes the math easier to describe and les relativistic: more vector math less tensor math.

Although i have not explored them yet clifford algebras and rotors are claimed to have  this property.

For the time being i am thinking of freeing up the degrees of freedom by adding a third angle of rotation to spherical coords so

is extended to where rotates around a normal to the radius which is in the plane of the great circle that the radius is in as determined by . This means that at certain times this nomal will coincide with the established orthogonal axes, principally when and are multiples of π/2.

[M Benesi]

  Your comment on degrees of freedom reminds me of something I just read on wikipedia, either yesterday or the day before.

  Gimbal lock. 

  I tend to think that the "hairyness" is due to the fact that the objects are 3 dimensional.  There are spaces behind the front pixel that are out of the set, and if the z-axis resolution is not set fine enough, we can miss the pixel in front and hit one behind.

  So we end up with little shaded pixels (set into the object) because we missed the actual pixel (even though we checked (bisected) between the first pixel hit and last pixel not hit, when we ran our check, we hit towards the first pixel hit FIRST, although if we had done 2 iterations backwards we would have hit a pixel towards the last pixel not hit). 

  2d demo:
   1,2,3,4.....C,D,E = pixel in set  * = check point not in set     ( )= not in set  <-- second one is a space...   

  Checking in this direction ---->        *   123  45      678   9A  BCDE

   So we have something like this, miss the first group of pixels and hit pixel 9.  We do 1/2 back to previous no hit checkpoint:

  Checking in this direction ---->        *   123  45 *    678   9A  BCDE

  We hit nothing, move 1/2 towards 9:

  Checking in this direction ---->        *   123  45 *    6*8   9A  BCDE

  We hit 7, now we check ~1/2 towards previous miss:

  Checking in this direction ---->        *   123  45 *   *6*8   9A  BCDE

  hit nothing.  Now we check 1/2 towards previous hit on 7:

  Checking in this direction ---->        *   123  45 *   ***8   9A  BCDE

  And hit 6.  We've totally missed the groups of pixels in the front due to our bisection algorithm and end up with an indented pixel.  Perhaps if we get a value deeper than the 9 pixels around the pixel, we should just bring it to the level of the pixels around, or at least have it as an option (to do so).  Otherwise, we have fuzz, some of which I'm running into now with these Julia's, as I prefer the higher details of one more iteration, but don't have the patience to set z axis resolution high enough to hit every pixel (at least for now, until I find a set of amazing images that I want to make intensely detailed and correctly rendered):

  1) 9th order (z^9)       c= -.25, .1, -.4
  2) 15th order (z^15)    c= -.29, -.29, -.29
  3) 15th order (z^15)    c= -.29, -.29, -.29    (rotated)
  4) 15th order (z^15)    c= -.31, -.31, -.31
  5) 15th order (z^15)    c= -.37, -.23, -.31

[jehovajah]

Thanks for that insight. I have no clue of the rendering algorithms necessary to visualise these escapes and orbits. I have only recently realised that the visualisations are every bit as computational intensive as the iterations if what is happening in the sculpting is going to make sense.

i am essentially talking about missed points and planes due to the calculation schemes referencing points in the 'tracks' ie orbits or escapes in different ways. The trig formulae will sculpt in the planes of great circles thus leaving a noticeable combing effect which i think is the basis of hairy ball. If so this effect should diminish with finer and finer stepping. But now add on top the rendering difficulties again in a great circle plane and you end up with a multiple effect.

Your explanation give insight into the development of spirals in some of the images that twinbee posted, where sharp images suddenly appear to exude cream before becoming sharp again .

Gimbal lock. Thanks  

[M Benesi]

Here is yet another z^2 fractal of the new type (although this one has the magnitude set by the new rules).

  I've heard that coastlines are fractals, but did you ever see fractals that are the edge of an ocean?



  This is just a zoom in of a quadrant of the fractal.

  There is plenty of variety within it, but I followed the quadrant line, and zoomed into the lakes that exist in the z^2 version (there are lakes in the others as well, but I thought we needed some true 3d z^2 fractals, so...).

[jehovajah]

So here are a few different  mandys at different iterations.

[jehovajah]

Now a mandy found in nature/yh~w.

[jehovajah]

This is a few sculpts from acorn mandy and lathed mandy. These are not slices per se but true lathed or "rotated" sets. These were produced by terry gintz mac version 1.02 of Quasz.

[jehovajah]

these two images show variations of the lathed mandy which indicate that the lathed effect can be controlled sufficiently to reveal the details of the mandy.

[jehovajah]

Although not as intricate as the trig formulae it is still a good sketch of the structure at power 8

[jehovajah]

Dragons Teeth! The rise of Godzilla mandy!