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Author Topic: True 3D Mandelbrot ?  (Read 6885 times)
Description: Yet another try at the "true 3D" Mandelbrot
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David Makin
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« Reply #15 on: July 18, 2014, 10:55:31 PM »

Ah - no the disconnected bits (which definitely exist on the z^2) are the bits which I think are an artefact of the DE - the DE detecting points close but in fact outside the 3D slice of 4D space - I've had the same issue before when rendering true 4D fractals with the 4th term being non-zero.

The whipped cream is again I think possibly an issue with the image being a 3D slice of 4D space - the flat non-fractal bits could be missing fractalness because we simply can't view the rest that's outside the 3D slice - we're viewing the 2D (ignoring fractalness) surface of a 3D slice of a 4D object - this is a bit like viewing the 1D (ignoring fractalness) outline of a 2D slice of a 3D object.
Edit: Even simpler consider taking straight line segments through the standard complex Mandelbrot - assuming the line at least crosses some inside and some outside the amount of fractalness of the points on the line will vary wildly - for instance if the cutting line is the real axis then there's going to be no fractal detail at all with respect to the boundary points of the Set.
« Last Edit: July 19, 2014, 12:56:31 PM by David Makin, Reason: Clarification » Logged

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« Reply #16 on: July 18, 2014, 11:29:18 PM »

I love it when a mistake works - when rendering that last pic for using 1-sin(w)^2 for the double-angle cosine I messed up and corrupted w.

Anyway this is the result - "The Griffin (an ambulatory plant)"

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David Makin
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« Reply #17 on: July 19, 2014, 01:20:55 PM »

I don't know about anyone else but I was wondering about the 3 different values calculated for cos(2a) using either 2*cos^2-1 or 1-2*sin^2 or cos^2-sin^2 even though complex numbers are supposed to work exactly like their real counterparts - the answer is simply because I used the magnitude of the hypotenuse rather than the complex value of sqrt(opposite^2+adjacent^2) which of course is a complex number itself.
I also realised the reason we have 3 possible values for cos(2a) (and in fact sin(2a) as well) is because we are in 4D and therefore have 3 independent angles (the 4th being fixed by the other 3).
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« Reply #18 on: July 19, 2014, 03:32:46 PM »

The whipped cream is again I think possibly an issue with the image being a 3D slice of 4D space.

It probably depends on a fractallness 4 dimensions. Quaternions are fractals in just 2 dimensions. Maybe try a bitt zoom around small bulbs and antennas in pow2. It seems quite a promising. There looks as could be a different structures not so visible from the outside.
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« Reply #19 on: July 20, 2014, 09:03:10 AM »

I love it when a mistake works - when rendering that last pic for using 1-sin(w)^2 for the double-angle cosine I messed up and corrupted w.

Anyway this is the result - "The Griffin (an ambulatory plant)"

<Quoted Image Removed>
ALEF have right.
The graphic is not like 3D MANDELBROT set because is too stretched. As well Mandelbrot set 2D is pure fractal and not have neither stretched convergent point (like HENON attractor). So this function is yet a failure attempt to render a 3D MANDEL style, as other many mistakes.
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David Makin
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« Reply #20 on: July 20, 2014, 10:07:40 PM »

IMO it's a fail for triplex, but I challenge anyone to prove it's a fail as a 4D version since it's impossible to see if the whipped cream exists when viewing the entire object (since we can't - unless someone's worked out a way).

I'm now working on multiplication, division and hopefully log and exp.

The whipped cream (as far as I can tell) disappears if you zoom in far enough increasing the "solid" level appropriately - for instance the whole area in the one below looks like a pipe in an area of stretched pipes until you zoom in this far (*125650):



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16429
« Last Edit: July 20, 2014, 10:13:29 PM by David Makin » Logged

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David Makin
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« Reply #21 on: July 20, 2014, 10:10:05 PM »

ALEF have right.
The graphic is not like 3D MANDELBROT set because is too stretched. As well Mandelbrot set 2D is pure fractal and not have neither stretched convergent point (like HENON attractor). So this function is yet a failure attempt to render a 3D MANDEL style, as other many mistakes.

Please read the post with that one again - that was a bug in the formula that I just rather liked - not a 3D or 4D Mandy !!
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David Makin
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« Reply #22 on: July 20, 2014, 10:32:10 PM »

Here's a simplified square (and power cool:

Code:
    float s
    float t
    float m = (s=|z|)+(t=|w|)
    if m!=0.0
    w = 2.0*z*w
    z = sqr(z)*(1.0 - t/s)
if true ; power 8
    m = (s=|z|)+(t=|w|)
    w = 2.0*z*w
    z = sqr(z)*(1.0 - t/s)
    m = (s=|z|)+(t=|w|)
    w = 2.0*z*w
    z = sqr(z)*(1.0 - t/s)
endif
    endif
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David Makin
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« Reply #23 on: July 21, 2014, 05:36:40 AM »

I checked the unit vector multiplication table and it cane out:

Code:
     r    i    j    k
r    r    i    j    k
i    i   -r    k   -j
j    j    k    0    0
k    k   -j    0    0

And then I realised how to fix the zero problem - use both the c^2-1 and 1-s^2 versions of the cosine as follows:


Code:
    float s
    float t
    float m = (s=|z|)+(t=|w|)
    if m!=0.0
      complex wt = w
      w = 2.0*z*w
      if s>=t
        z = sqr(z)*(1.0 - t/s)
      else
        z = (s/t-1.0)*sqr(wt)
      endif
if true
      wt = w
      m = (s=|z|)+(t=|w|)
      w = 2.0*z*w
      if s>=t
        z = sqr(z)*(1.0 - t/s)
      else
        z = (s/t-1.0)*sqr(wt)
      endif
      wt = w
      m = (s=|z|)+(t=|w|)
      w = 2.0*z*w
      if s>=t
        z = sqr(z)*(1.0 - t/s)
      else
        z = (s/t-1.0)*sqr(wt)
      endif
endif
    endif

Whipped cream == gone (at least on the z^8)
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« Reply #24 on: July 21, 2014, 08:51:41 AM »

Pictures!!!!  afro
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David Makin
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« Reply #25 on: July 21, 2014, 01:18:43 PM »

Pictures - OK, OK - just one small point - although the (j,k)*(j,k) unit vector multiplication is all zeroes in fact this isn't the whole story - because I'm using the real value for the hypoteneuse instead of the complex value then with the exception of when the values on axes (i.j) is (0,0) the magnitude is still maintained correctly.
Also I have one more version to try - at the moment I'm essentially using either cos^2-1 or 1-sin^2 depending which avoids the zero issue but it could equally use cos^2 - sin^2 which gives the normal bicomplex on its own if I remember correctly - so instead of splitting the first two depending on the magnitudes of z and w we can split the calculation 3 ways -> when magz>total mag*2/3, magw>total mag*2/3 and the case when the magnitudes of magz and magw are similar.
Now to post another pic....
« Last Edit: July 21, 2014, 01:38:05 PM by David Makin » Logged

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« Reply #26 on: July 21, 2014, 01:32:54 PM »

This is from the z^8+c using the 4D formula where the hypotenuse of a complex sided triangle (i.e. 4D) is taken as the real magnitude rather than the actual complex value, this allows three choices of calculation for the cosine (and the sine too I guess !). In this image the cosine used is essentially either cos^2-1 or 1-sin^2 depending on which is most likely to avoid asymptotes or whipped cream effects (basically this depends on the relative magnitudes of the initial z and w).

I found some "Super Villains" in the plain 4D Mandelbrot for z^8+c using this method (view x,y,z of x,y,z,w). This is at magnification *252098 with solid at 1e-9.



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16432
« Last Edit: July 21, 2014, 01:40:27 PM by David Makin » Logged

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David Makin
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« Reply #27 on: July 22, 2014, 05:32:20 AM »

Another pic - "Evolution", this is another z^8 Mandy but this time with cos(2a) as 1-sin(a)^2 instead of cos(a)^2-1, just to repeat the two are different because the hypotenuse is taken as real (the real magnitude of a 4 dimensional number made up of two complex numbers) with the opposite and adjacent both being complex.



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16433
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« Reply #28 on: July 22, 2014, 04:43:18 PM »

I did a few pictures of the first code and now I have a time explain what I meant.
I twas thinking everything what is not a fractal in lower number of dimensions would not be a fractal in higher number of dimensions. And fractal in higher number of dimensions will be a fractal in lower number of dimensions. At least for for quaternions, all the 2D slices of quaternion are fractals, exept YZ and YW axis are circles. I don't think something could appear in higher number of dimensions unless its very strached and that is how it goes in lower numbers of dimension.
However unused 4th dimension allows it to look throught the slices of 4th dimension or even to switch dimensions.

Alsou I think if it's a 3D mandelbrot then pow2 version should show becouse 8th power allways generates nice overall shape just by being perfectly symmetric. But maybe a pow 4 3D zooms would look the best becouse pow 2 looks like ribbons and stalks and pow 8 have too much small round things.

Anyway, no DE rendering, 20 iterations. Pow8 version is nice crispy. Maybe could do better zooms but that would take more time and I forgotten all the Chaos Pro setting which must be changed to get something out of it. And dont wanted render -> antialiasing what renders its two times as big so for times as slow.

All the parameter files included. This would be downloadable on the Chaos Pro only after the day, so the formula too.


Pow 2. Not so visible but all the smaller bulbs stalks and the seahorse area present.



First smaller bulb:


Somewhere on the x<0 bulb:


Pow 4. Maybe the bests zooms? never tried throught.



Pow 8



Julia -0.3, 0.5, 0.7


Somewhere below and to the right.


Seems perfect exept area of some ^#%&^.


This is strange. ZY x=0. However if you compare 3D cuyouts this seems to be the wort place even for pow8.


Anyway, I think if it's now what we were seeking then its close to it.
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« Reply #29 on: July 22, 2014, 04:47:21 PM »

All the parameters and Chaos pro formula files of the post above. I hope they 'll appear. EDIT: now they uploaded.

* Grail.zip (6.26 KB - downloaded 142 times.)
« Last Edit: July 22, 2014, 05:01:04 PM by Alef » Logged

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