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Welcome to Fractal Forums > Fractal Math, Chaos Theory & Research > Mandelbrot & Julia Set > Understanding the mandelbrot set
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Author Topic: Understanding the mandelbrot set  (Read 6901 times)
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cKleinhuis
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« Reply #30 on: March 22, 2013, 08:46:49 PM »
woot, on my homepc it is running like a charme!!

would you mind if i modify your script, one hand to remove the double iteration, and on the other hand to give it a formula selection wink
will send you an update...
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taurus
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« Reply #31 on: March 22, 2013, 09:37:53 PM »
Quote from: makc on March 22, 2013, 07:03:01 PM
may I suggest another tweak: color orbit vertices from, say, red to green, so we could see the direction easier at a glance, without having to trace the lines?
agree, a key feature of your (makc) version was, to see the development of the iteration. I spotted some places (difficult tom aim), where the iteration oscillated some times before bailing out to infinity. Syntopia's version is still static, showing no direction - no development.
And for sure the pragmatician wishes the current coordinates of the crosshair...
But to continue the eulogy. The ammount of creative crazyness here is unbelievable. For one who waited minutes for a 640x480 version of the basic mandelbrot set in his early days, this realtime exploration is great, great, great! No words left for this...
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Syntopia
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« Reply #32 on: March 22, 2013, 10:35:01 PM »
Quote from: cKleinhuis on March 22, 2013, 08:46:49 PM
woot, on my homepc it is running like a charme!!

would you mind if i modify your script, one hand to remove the double iteration, and on the other hand to give it a formula selection wink
will send you an update...

By all means play around with it. I don't think it is possible to remove the double iteration though, if you want to render the orbit path on the GPU side: even though it seems slightly excessive to calculate exactly the same path 250.000 times for each frame :-)

Quote from: makc on March 22, 2013, 07:03:01 PM
may I suggest another tweak: color orbit vertices from, say, red to green, so we could see the direction easier at a glance, without having to trace the lines?

It is slightly difficult, since I don't know the number of iterations before breakout (so it is hard to scale the gradient) -though I could do another pre-pass to calculate the number of iterations.

Besides that, it is still difficult to visually trace more complicated orbits:


I guess animating the trace would work better.
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taurus
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« Reply #33 on: March 22, 2013, 11:22:14 PM »
Quote from: Syntopia on March 22, 2013, 10:35:01 PM
It is slightly difficult, since I don't know the number of iterations before breakout (so it is hard to scale the gradient)

just a thought. Leave the scale equal to the max iteration number by default. an additional slider can adjust a possible gradient. maybe it is getting more complicated to use, but ignoring the gradient slider wouldn't make anything worse for anyone. the slider would only give an additional opportunity.
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makc
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« Reply #34 on: March 23, 2013, 01:39:04 AM »
Quote from: Syntopia on March 22, 2013, 10:35:01 PM
Besides that, it is still difficult to visually trace more complicated orbits
Indeed. But perhaps coloring edges could work better?
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taurus
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« Reply #35 on: March 23, 2013, 07:39:57 AM »
an example with colored nodes.
http://www.khanacademy.org/cs/mandelbrot-spirals-2/1030775610
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Furan
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« Reply #36 on: June 09, 2013, 07:49:01 PM »
Adding to the understanding of Mandelbrot Fractal, I was wondering, have you ever seen an animation of Minibrot evolution? If I understand correctly, any minibrot will transform into the main Mandelbrot after a finite number of transformations. What if we generated an animation, each time changing the angle orientation, position and zoom of the image to see the Minibrot evolve? I will do it in a few days if it's not done already. However I have only double precision in F77. Can someone else do it more professionally?
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Furan
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« Reply #37 on: June 11, 2013, 11:36:50 PM »
Ok, that didn't work.
All this time I was under the impression that points of the M-set boundary jump from place to place on the boundary. I guess that happens in Julia only, so there is no Minibrot evolution whatsoever. Now playing with the Mandelbrot Spirals. Should have paid more attention.

I made some continuous [z_n+1(t) = z_n^t + (1-t)*z0; t=1...2 ] transformations of the M-set. Hopefully tomorrow I will post an animated gif. These are just some teaser frames, initial, one iteration, 6.9th iteration, all calculated from a much bigger image of the initial M-set. (12 times larger with 3x3 AA)


Coloring is not correct, just black-red-yellow-white that gets mixed. What I want to do later is to use original image without AA and follow the colors. They should slowly filter out, starting with the red diverging into the infinite.
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Furan
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« Reply #38 on: June 12, 2013, 07:30:08 AM »
500 frames, 0 to 50 iterations
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claude
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« Reply #39 on: June 14, 2013, 11:16:36 PM »
Quote from: eiffie on March 22, 2013, 05:23:36 PM
I believe you can tell the "lineage" of a minibrot by studying its period. It will be multiplied by its parent bulb and its grandparents. That is why the periods get so long. I may be wrong about this, can someone verify?

Yes, the periods multiply by denominator of internal angle as you go through touching components away from the island's root.  And for islands in the hairs, they have their own periods but the orbit will get close to 0 when a shorter nearby orbit gets close to 0 too.  You can use the indices of successive minima of the magnitude of the iterates to get a feel for which periods are influencing that point - I call that list of numbers "partials" and gave some examples here: http://www.fractalforums.com/programming/cheap-way-to-determine-angle-in-bulb-t16259/msg62419/#msg62419

Each island minibrot is like a copy of the whole set, with all its periods multiplied by a factor, plus additional hairs all over with even more details that aren't necessarily multiples of the island's root period.  Keywords to search for on this part are "renormalization" and "tuning", here's one page that gave me several flashes of intuition:  http://www.ibiblio.org/e-notes/MSet/Contents.htm (section on Renomalization)

Embedded Julia sets occur in the hairs, I'm planning on writing a couple of blog posts going into more detail on that topic in the next week or two but until then here's some rough notes I scanned: http://mathr.co.uk/mandelbrot/2013-06-14_embedded_patterns/
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youhn
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« Reply #40 on: January 05, 2014, 01:55:39 AM »
Playing around with the mirroriness of the Mandelbrot around the X-axis. Starting with the formula x^(800/400) + c in Gnofract4D. Then slowly drifting away from the perfect 2 by adding 1 to 800:



On the right you see the 3th biggest bulb of the Mandelbrot.
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