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Author Topic: 3D Mandelbrot (a newbies attempt)  (Read 2562 times)
Description: expanding the rules for complex multiplication and addition to the 3rd dimension
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gengis
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« on: September 08, 2010, 11:26:53 PM »

I have seen a few things in this forum looking something like what I have in mind, but only quite. What I have in mind is 3D-Mandelbrot which, when sliced at z=0 actually looks like the 2D-Mandelbrot.
here my attempt:

expand the rules for multiplication and summation of complex numbers to the 3rd dimension:
Z(n) = Z(n-1)^2 + C
would then mean convert Z(n-1) from Cartesian to spherical coordinates multiply the lengths and add the angles, convert it back to Cartesian and add C

I have done 256 iterations but only on a 154x154x154 array. here the results.

an isosurface on points which are finite:

count how many iterations it takes for vector length to exceed 4. plot a isosurface for each iteration 256 to 1 in steps of 4. each isosurface with a alpha of 0.1:
   
tell me what you think...


* isfinite_1.jpg (78.53 KB, 1582x983 - viewed 341 times.)

* iterations_1.jpg (85.42 KB, 1582x983 - viewed 329 times.)
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Tglad
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« Reply #1 on: September 09, 2010, 12:03:03 AM »

yes that's a mandelbulb. There are 3 mandelbulbs despite most images being of just one of them (an arbitrary choice because it looks a bit more bulby).
In fact there are as many mandelbulbs as there are orientations, the 3 are 90 degrees apart from each other in that sense... there is a video from about last dec showing the mandelbulbs in this space... the full set may have been called the hopfbulb or hopfbrot I forget.
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M Benesi
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« Reply #2 on: September 09, 2010, 03:15:43 AM »

  Yeah, an oldie but a goodie.

  It's the equivalent of taking:

the complex number z1=  sqrt(start_x^2 +start_y^2) + i start_z
  real_z1=  real part of z1
  imag_z1= imaginary part of z1

the complex number z2= start_x + i start_y
  real_z2=  real part of z2
  imag_z2= imaginary part of z2

magnitude of z2,  r =  sqrt (start_x^2 +start_y^2)

z1= z1 ^ n
z2= z2 ^ n
r=    r ^ -n

new_x = real_z1 * real_z2 *r
new_y = real_z1 * imag_z2 *r
new_z = imag_z1

  add in your pixel values appropriately, so on and so forth....

  However, the really detailed one is the "opposite" complex triplex:

 you take:
z1 = start_x + i sqrt(start_y^2+ start_z^2)...
etc...

  You don't get a 2d mandelbrot shape, but then again you shouldn't expect one due to the mathematics involved (as the 2d brot is skewed over the y axis, you get a double skew due to the 2 imaginary components for the 3d version, rather than a simple single skew)....
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Nahee_Enterprises
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nahee_enterprises Nahee.Enterprises NaheeEnterprise
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« Reply #3 on: September 12, 2010, 11:16:54 PM »

    I have seen a few things in this forum looking something like what I have in mind,
    but only quite.  What I have in mind is 3D-Mandelbrot which, when sliced at z=0
    actually looks like the 2D-Mandelbrot.   ...........
   
    tell me what you think...

Greetings, and Welcome to this particular Forum !!!    smiley

I think that people are still trying to define and discover the TRUE 3D-Mandelbrot.
 
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EvaB
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« Reply #4 on: January 03, 2011, 11:10:27 AM »

I think that people are still trying to define and discover the TRUE 3D-Mandelbrot.

So do I - another newbie's attempt:

z=a+bi+cj
z2=(a2-2bc) + i (2ab-c2) + j (2ac-b2)
with a special multiplication:
ii=-j jj=-i ij=ji=-1
e=1 ei=ie=i ej=je=j
http://members.aon.at/evab/
(german, but less text, more images)
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