Animating fractional iterations: "stirring"

[tim_hutton]

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 z² and +c components can be seen. The z² 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

[Syntopia]

Great animations, Tim, and welcome to the Forums.

(If you don't know Tim, he has done some great work with Ready: http://code.google.com/p/reaction-diffusion/)

[cKleinhuis]

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 ....

[tim_hutton]

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 z² term.

Here's a new video showing the same distortion but on a checkerboard pattern instead of on the Mandelbrot image:

<a href="http://www.youtube.com/v/8op_yA6vNsw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/8op_yA6vNsw&rel=1&fs=1&hd=1</a>

Regards,

Tim

[kram1032]

Oh, that's a really nice technique.
Also, by visiting a link, linked via a link that was linked via the link you posted, I found this:
http://www.quora.com/Fractals/Why-is-the-Mandelbrot-set-a-fractal
Which explains mathematically, the notion of self-similarity, including pretty images. Really nice.

[tim_hutton]

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.

<a href="http://www.youtube.com/v/mdjrKFI9GRk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/mdjrKFI9GRk&rel=1&fs=1&hd=1</a>

[cKleinhuis]

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

[tim_hutton]

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.

[matsoljare]

Very interesting, can you do this with a Julibrot?

[tim_hutton]

It would work with any escape-time or IFS fractal. In 3D it will be harder to render of course.

[M Benesi]

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)]

[tim_hutton]

 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.

[M Benesi]

  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.