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 Author Topic: The other dimension Mandelbrot  (Read 9015 times) Description: 0 Members and 1 Guest are viewing this topic.
bib
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Posts: 2070

At the borders...

 « 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?

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

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

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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Between order and disorder reigns a delicious moment. (Paul Valéry)
Timeroot
Fractal Fertilizer

Posts: 362

The pwnge.

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

What kind of coloring is that? It looks incredible!!!  :surprise:
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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
bib
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Posts: 2070

At the borders...

 « Reply #17 on: January 18, 2010, 11:20:32 AM »

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

It's very special 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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Between order and disorder reigns a delicious moment. (Paul Valéry)
bib
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Posts: 2070

At the borders...

 « 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 ?
 « Last Edit: January 18, 2010, 03:44:28 PM by bib » Logged

Between order and disorder reigns a delicious moment. (Paul Valéry)
Timeroot
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Posts: 362

The pwnge.

 « 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  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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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
bib
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Posts: 2070

At the borders...

 « Reply #20 on: January 18, 2010, 10:39:34 PM »

Thanks, but tonight I preferred looking for nice spots than doing maths 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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Between order and disorder reigns a delicious moment. (Paul Valéry)
kram1032
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Posts: 1863

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

amazing

I must admit, there rarely are images of the standard Mset which look equally interesting
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Jesse
Fractal Schemer

Posts: 1013

 « 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 ?

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
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bib
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Posts: 2070

At the borders...

 « 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 ?

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

Yes it works
 PerpendicularM.jpg (282.78 KB, 1024x768 - viewed 414 times.) Logged

Between order and disorder reigns a delicious moment. (Paul Valéry)
Jesse
Fractal Schemer

Posts: 1013

 « 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
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  )
 minZslice_ds.jpg (310.29 KB, 1567x812 - viewed 429 times.) Logged
Timeroot
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Posts: 362

The pwnge.

 « 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  . Can't wait to see some more pics!
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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
bib
Global Moderator
Fractal Senior

Posts: 2070

At the borders...

 « 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 374 times.) Logged

Between order and disorder reigns a delicious moment. (Paul Valéry)
bib
Global Moderator
Fractal Senior

Posts: 2070

At the borders...

 « 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
 PM6.jpg (250.57 KB, 1024x768 - viewed 648 times.) Logged

Between order and disorder reigns a delicious moment. (Paul Valéry)
kram1032
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Posts: 1863

 « Reply #28 on: January 19, 2010, 08:25:36 PM »

beautiful
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bib
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Posts: 2070

At the borders...

 « 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...