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Author Topic: The other dimension Mandelbrot  (Read 8862 times)
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bib
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« Reply #15 on: January 18, 2010, 01:27:08 AM »

Did I get that last part right? You're operating in complex numbers (or at least really directly in 2D) for those? cheesy

If that's the case, it shouldn't be too hard to do deep zooms of plane sections if the standard m-set formula and this one get generalized smiley

if the formula is r^n | phi*n | theta*n, the 2D-version should basically be where phi = 0 rather than theta... smiley

Does that colouring relate to the 3D-structure directly or is the 3D-effect archeived in a different way?

In UF I use a 3D Mandelbulb formula from Dave Makin and I clip the object. I am really curious if this 2D fractal, that naturally seems to have many properties similar to the Mandelbrot set, has ever been explored? I bet it exists somewhere in the UF database as a 2D formula, anyone knows?
For coloring I guess this is a side effect of the 3D coloring.

Here is another minibrot, this one is not distorted:

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« Reply #16 on: January 18, 2010, 03:20:21 AM »

What kind of coloring is that? It looks incredible!!!  w00t !! :surprise: w00t !!
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« Reply #17 on: January 18, 2010, 11:20:32 AM »

What kind of coloring is that? It looks incredible!!!  w00t !! :surprise: w00t !!

It's very special smileysmiley in the above image, the black, yellow and green areas are on the image plane and colored by iterations. The greyish areas are "holes" behind the image plane that show the 3D structure of the Mandelbulb, with a high solid threshold.
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« Reply #18 on: January 18, 2010, 03:39:02 PM »

If I try to program it in UF (that would be a big challenge!), is this logic correct :

I work in the xz plane, so theta=0 and y=0

I will use complex numbers to represent the xz plane.

First step is to convert into "spherical" coordinates at the intersection of the y=0 plane:

x+iz=r*e^i*phi

with r=(x²+z²)^1/2
and phi = atan2(x,z)

Then
newr= r²

and
if phi belongs to [0;pi/4] then newphi = 2 * phi
if phi belongs to [pi/4;pi/2] then newphi = pi - 2 * phi

Then translate back into complex, add c and iterate.

It's still a bit fuzzy to me. Does that sound correct?

Edit : I forgot to take into account that theta is not always 0. It is 0 if x>0 and pi if x<0.

can anyone give a hand ? smileysmiley
« Last Edit: January 18, 2010, 03:44:28 PM by bib » Logged

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« Reply #19 on: January 18, 2010, 06:50:31 PM »

Hmm, as far as I can tell, theta would act the same, but *phi* would be pi/2. If you think, theta is the latitude and phi is the longitude (correct me if it's the other way around), then in the regular MSet phi=0 and theta=atan2(z). Here, you still include the x-axis, so yes, it's in the xz plane. But theta still is the the latitude, and acts like the direction from the origin. Phi is set for the entire plane at 90 degrees, or pi/2. I don't think that could be canceled out in any way... After one iteration, it doubles to pi. The you add c, which could already bring it to anything like 3*pi/4, or any other number - after just one iteration it has a unique value  sad Then y would also have a non-zero value... I guess you'd have to compute the entire thing normally, like the mandelbulb; it wouldn't be nearly as slow, though, because you only have one plane.
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« Reply #20 on: January 18, 2010, 10:39:34 PM »

Thanks, but tonight I preferred looking for nice spots than doing maths smileysmiley Maybe one day I will give it a more serious try!

In the meantime, here is a spiral, again taken from the "perpendicular Mandelbrot"

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« Reply #21 on: January 18, 2010, 10:49:30 PM »

amazing cheesy

I must admit, there rarely are images of the standard Mset which look equally interesting cheesy
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Jesse
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« Reply #22 on: January 18, 2010, 11:36:09 PM »

If I try to program it in UF (that would be a big challenge!), is this logic correct :
...
can anyone give a hand ? smileysmiley

Hmm, istnt it just like:

  x' = x*x - y*y + cx
  y' = -2 * y * Abs(x) + cy

(If you want it as 2D formula)

But dont ask me about UF programming  cheesy
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« Reply #23 on: January 18, 2010, 11:57:18 PM »

If I try to program it in UF (that would be a big challenge!), is this logic correct :
...
can anyone give a hand ? smileysmiley



Hmm, istnt it just like:

  x' = x*x - y*y + cx
  y' = -2 * y * Abs(x) + cy

(If you want it as 2D formula)

But dont ask me about UF programming  cheesy

Yes it works smiley


* PerpendicularM.jpg (282.78 KB, 1024x768 - viewed 413 times.)
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« Reply #24 on: January 19, 2010, 12:11:48 AM »

Btw, very beautiful images and coloring.

Is the image size limiting to 512kB new, or is this just a personal degredation  huh?
However, i had to test this slice also, without that great coloring of UF, scaled down to 1/4 size:
(Seems to be magna cave underneath the mountains, may give a vulcan in a 4D version  undecided)


* minZslice_ds.jpg (310.29 KB, 1567x812 - viewed 427 times.)
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« Reply #25 on: January 19, 2010, 02:02:28 AM »

From the pictures so far, it seems like you might have discovered the Holy Grail of the 2D Mandelbrot  grin. Can't wait to see some more pics!
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« Reply #26 on: January 19, 2010, 11:09:01 AM »

I think this fractal and the function Jessy provided are very interesting for those looking for the holy grail.

Think it this way :

In the Twinbee/bugman Mandelbulb, the iteration formula in the xy plane is :

  x' = x*x - y*y + cx
  y' = 2 * x * y + cy

And in the xz plane it is :

  x' = x*x - z*z + cx
  z' = -2 * z * Abs(x) + cz

It seems to me that part of the 3D fractal effect comes from this difference and transformation of x into -Abs(x), because if we don’t use the -Abs() transformation we get the lathed Mandelbrot. So how could could find a way to keep the same classic formula in the xz plane, and avoid the lathed Mandelbrot?  

here is a picture of the same fractal, without the minus sign. It's funny to note that this shape is very often found as a minibrot in the Perpendicular Mandelbrot.


* PM-.jpg (251.41 KB, 1024x768 - viewed 373 times.)
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« Reply #27 on: January 19, 2010, 08:23:41 PM »

Using directly Jessy's 2D formula allows to leverage the power of UF coloring, and rendering speed, as in this attached example smiley


* PM6.jpg (250.57 KB, 1024x768 - viewed 645 times.)
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kram1032
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« Reply #28 on: January 19, 2010, 08:25:36 PM »

beautiful cheesy
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« Reply #29 on: February 02, 2010, 07:03:15 PM »

Hi
Another vid using this time the +sine variation of the Mandelbulb. The idea stays the same : cut it in half across the perpendicular plane to the classic 2D M set, then zoom/explore...

<a href="http://www.youtube.com/v/xB3vITRz-f8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/xB3vITRz-f8&rel=1&fs=1&hd=1</a>
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