tim_hutton
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« on: September 05, 2012, 10:57:28 PM » |
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Hello, Following some hints by John Baez I've made this: http://code.google.com/p/mandelstir/The key idea is to visualize each iteration as a smooth distortion. In each video the value shows the (non-integer) iteration count. With Julia sets this shows how the shape maps exactly onto itself. With the Mandelbrot set the situation is more complex but visually very informative. And surprising, for me. In both cases the separate influences of the z 2 and +c components can be seen. The z 2 giving the 'stirring' of course. It's not rocket science but it's not something I've seen done before so I thought it might be of interest. All feedback welcome. Thanks! Tim
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cKleinhuis
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« Reply #2 on: September 06, 2012, 07:15:54 AM » |
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interesting animations, on first sight it reminds on increasing the power values of the formulas, on second sight i see that it isnt the power, but some weird affine reordering, or mapping, dont really understood what the method is and how it is done, ... i am wondering wasnt it clear that julia sets are self similar ?! the john baez animation is funny, showing exactly how the julia maps onto itself ....
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divide and conquer - iterate and rule - chaos is No random!
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tim_hutton
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Posts: 6
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« Reply #3 on: September 06, 2012, 11:08:54 AM » |
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Thanks Syntopia. cKleinhuis: John Baez's post suggests this formula: z=z 2t+ct where t goes from 0 to 1 in each iteration. In my implementation I'm using this: z=z t+1+ct to get a smoother feel but it is the same idea - make intermediate steps between each iteration. The formula only works for t between 0 and 1 though - for higher iterations we need to perform the integer steps first and then the leftover fractional part, else the points rotate too fast. The code explains this. I find it helpful to see how the distortion gets from one iteration to the next, instead of just a jump from here to there. Especially for the Julia set video shown, where the symmetric central blob with two spirals turns into an asymmetric one with only one spiral - it is hard to understand how that could happen, without seeing the distortion. This page talks about the distortion caused by the z 2 term. Here's a new video showing the same distortion but on a checkerboard pattern instead of on the Mandelbrot image:
http://www.youtube.com/v/8op_yA6vNsw&rel=1&fs=1&hd=1Regards, Tim
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« Last Edit: September 10, 2012, 06:08:38 PM by tim_hutton »
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tim_hutton
Forums Newbie
Posts: 6
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« Reply #5 on: September 10, 2012, 06:06:28 PM » |
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An alternative approach for animating fractional iterations is to simply use linear interpolation between each iteration. This makes the movement less confusing, if less mathematically correct.
http://www.youtube.com/v/mdjrKFI9GRk&rel=1&fs=1&hd=1
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« Last Edit: September 10, 2012, 11:13:55 PM by tim_hutton »
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cKleinhuis
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« Reply #6 on: September 10, 2012, 08:49:44 PM » |
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so, do get it right, you are "just" using each iteration step as seed for a completely new iteration, after that it is repeated for the next frame of animation using one iteration further ... ok ... at first i thought it would be some kind of hybrid ... *hijack starts* and it brought me to the idea how to visually edit hybrid fractals, for sure each fractal has its own rendering of its own, wich should be shown as thumbnail, but additionally a transformed checkerboard could visualise how it transforms a plane/cube which is the input the next formula, and this checkerboard view would render both formulas and so on ... ... lol sorry for letting my mind freely flow
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---
divide and conquer - iterate and rule - chaos is No random!
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tim_hutton
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Posts: 6
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« Reply #7 on: September 10, 2012, 11:11:29 PM » |
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cKleinhuis: yes, sort of. I'm not doing anything other than animate the orbits of the points in the usual Mandelbrot set. Each point moves with each iteration - I'm just showing that movement. The only difference is that previously we've only seen that movement as a jump.
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matsoljare
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« Reply #8 on: September 11, 2012, 05:15:08 PM » |
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Very interesting, can you do this with a Julibrot?
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tim_hutton
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Posts: 6
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« Reply #9 on: September 12, 2012, 11:16:30 AM » |
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It would work with any escape-time or IFS fractal. In 3D it will be harder to render of course.
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M Benesi
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« Reply #10 on: October 07, 2012, 10:48:06 PM » |
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Thanks Syntopia. cKleinhuis: John Baez's post suggests this formula: z=z 2t+ct where t goes from 0 to 1 in each iteration. In my implementation I'm using this: z=z t+1+ct to get a smoother feel but it is the same idea - make intermediate steps between each iteration. Makes me wonder about the summations of the function from t=0 to 1, with step size delta t. In fact, why not sum dt (z t+1+ct)? End up with something like: [c*log(z)+2*z^2-2*z] / [2*log(z)]
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« Last Edit: October 07, 2012, 11:01:13 PM by M Benesi »
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tim_hutton
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Posts: 6
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« Reply #11 on: October 07, 2012, 11:00:26 PM » |
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Makes me wonder about the summations of the function from t=0 to 1, with step size delta t. In fact, why not sum dt (z t+1+ct)? I'm not sure what you mean but maybe just give it a go and see what happens.
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M Benesi
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« Reply #12 on: October 07, 2012, 11:02:47 PM » |
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It might be interesting. I'll check it out with the formula above- my editing of the post could have had better timing, ehh?
Glad you're here. I've seen a little bit of your work.
Update: Nothing interesting, at least for 2 dimensions.
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« Last Edit: October 08, 2012, 01:49:22 AM by M Benesi »
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