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Author Topic: Symmetric mandelbrots as templates  (Read 3852 times)
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msltoe
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« on: April 14, 2010, 05:36:27 AM »

Hi everyone,

 If you remember the Juliabulb I found recently, the problem I thought I had with the Mandelbrot set versions was that the traditional 2nd order M-set is not 4-fold symmetric. The 2D M-set symmetric along the x-axis. One way to get symmetric M-sets is to go to multibrots (order n>2). This leads to the Mandelbulbs. But it also moves away from the beauty of the 2nd order set.
 Yesterday, toying around with ChaosPro, I noticed I could generate symmetric M-sets by playing with functions of c (not unlike the lambda M-sets). (z = z^2 + c^2) or (z = z^2 + c^2 - c^4). These are more like the Julia set in their 2-d shape. I reasoned I could use this then as the template for my symmetrized (z*z<y*y) variant. Here's two close ups of the results. I call them "glimpses" of the true 3d mandelbrot. There's still some flaws. Probably the biggest is the need to symmetrize. But sometimes we have to enjoy where we are at...

Close-up of largest bulb:

The 1st minibrot:


-mike
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reesej2
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« Reply #1 on: April 14, 2010, 07:02:28 AM »

Well, it's certainly more symmetrical, less choppy. But there's still a lot of "specks" where your julias were smooth. It OUGHT to be possible to get rid of that, but I'm not sure how...
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KRAFTWERK
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« Reply #2 on: April 14, 2010, 08:43:38 AM »

Looks very nice, you probably did do this, but I'll ask anyway...

Did you try to take it further?, like z = z^2 + c^2 - c^4 + c^6 - c^8 +...

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msltoe
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« Reply #3 on: April 14, 2010, 01:09:05 PM »

KRAFTWERK: The extended series produces a variety of shapes in 2-D which might be fun to explore. I'm using z^2+c^2 here which has the best symmetry.

reesej2: The problem is the discontinuities by artificially splicing two mirror image formulas together. Simple interpolation hasn't worked in my hands. Lots of experimentation still leads me with the original splicing idea - which works nearly perfectly for Julias of the form (a,0,0) but leads to discontinuites when b,c are non-zero (a,b,c).
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Paolo Bonzini
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« Reply #4 on: April 14, 2010, 03:43:48 PM »

Looks very nice, you probably did do this, but I'll ask anyway...

Did you try to take it further?, like z = z^2 + c^2 - c^4 + c^6 - c^8 +...

That's z = z^2 + c^2 / (c^2 + 1) so it's probably quite different.
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hobold
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« Reply #5 on: April 14, 2010, 04:48:22 PM »

It's incredible how close you keep coming even with different approaches. Does the overall shape look similar to the largest minibrot?


I have been wondering recently what the "true" three-dimensional equivalent to certain features of the classical Mandelbrot set ought to be. For example, spirals are a recurring theme in 2d. But what is THE equivalent of a flat spiral in three-space? A helix? But that wouldn't have a center, unless it were conical. And it wouldn't fill space the way that a spiral can fill a plane.

What are the fundamental building blocks of fractal chaos in three dimensions? Have we already found them, but don't recognize them as fundamental? In other words, is the shape we are looking for really the shape that we should be looking for?
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reesej2
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« Reply #6 on: April 15, 2010, 12:35:13 AM »

Well, I can imagine a 3D spiral, something winding out from a center and coiling in 3 dimensions. It'd be basically rotating through phi and theta simultaneously, at different rates, as r increases.

But I agree, I think we're all a little foggy on what exactly we're looking for. I guess I just have a sense of the "style" of the Mandelbrot and I'm looking for the same style in the 3D version, if not the same specific patterns or even the same overall shape.
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Timeroot
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« Reply #7 on: April 15, 2010, 01:21:41 AM »

A picture the 3D spirals as being a little cylinder that "wraps up on itself" in increasing radii, somewhat like a nautilus shell (but fatter).

If you think how a spiral must look two a flatland viewer, it would be one weird shaped edge leading into a cave. Nothing too exciting. I think that the shape I just described would have a similar quality, of occluding itself. Looking at Mandelbulb slices with spirals can sort of illustrate this. This is why I, for one, really want 4D vision.
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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.
reesej2
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« Reply #8 on: April 15, 2010, 02:09:31 AM »

cheesy But if we had 4D vision, we'd be searching for the 4D Mandelbrot! Anyhow, I agree that it would block off vision from itself. That problem could be resolved by making a transparent 3D Mandelbrot with refraction (which, incidentally, would be obscenely difficult to manage  :surprise:) which would allow a decent view of the innards of the structure.
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KRAFTWERK
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« Reply #9 on: April 16, 2010, 04:02:25 PM »

Yeah, I mean if you look at images like this one by Krzysztof http://fc02.deviantart.net/fs70/i/2010/045/6/d/Sliced_seahorse___Mandelbulb_by_KrzysztofMarczak.jpg, it seems that the ^2 mandelbulb is a very nice 3D-version of the 2D mandelbrot set.

Only the fact that the outside surface is "UGLY" makes us think it is not "the holy grail"...

BTW msltoe, a sliced image of your latest bulb might be interesting...  afro
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reesej2
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« Reply #10 on: April 16, 2010, 08:06:53 PM »

Also, I think it's worth pointing out that everything we do has to do only with drawing the points actually in the set. But if you color the Mandelbrot set black for the interior and white for the exterior, you get an uninteresting beetle shape with dust on the edges. I'm currently working on rendering a version of the Mandelbulb with a "fog" colored according to escape time, and that will hopefully be helpful. Before discarding anything like what msltoe's got we should check out these angles, too.
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msltoe
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« Reply #11 on: April 17, 2010, 12:57:53 AM »

It turns out that f(c) = c^2, etc. formulae I've been playing with doesn't affect the nature of the fractal. Think of it as applying mirrors or other sorts of distortions to the z^n + c fractal. It produces pretty macroscropic shapes but has less effect on the zoomed in parts. An example of my Juliabulb with c^2 is below.

I think the real Mandelbrot should have spheroidal bulbs and branching dendrites. It should be clean, e.g. no sharp discontinuities as iteration count is increased. Also, the y=0 and z=0 planes should reduce to the 2D Mset. It should have infinite detail with no smooth areas (albeit the bumps may get very tiny). Even if we don't get exactly the right solution, we'll know it when we see it...

reesej2: I think it would be great to somehow visualize the gradations of iteration count in 3-D and hope you succeed.




* mandelspot.jpg (212.03 KB, 600x600 - viewed 474 times.)
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msltoe
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« Reply #12 on: April 17, 2010, 02:52:50 AM »

Another example: bulbstar. you get the idea smiley


* bulb_star.jpg (231.13 KB, 600x600 - viewed 407 times.)
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reesej2
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« Reply #13 on: April 17, 2010, 03:14:17 AM »

Ha, that's a lot of symmetry! But the discontinuities still aren't going away  cry Definitely making headway, though.
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