# Finding parameter paths for beautiful fractal animations

So I just got renewed interest in fractals and especially animations with fractals. To make an image or a frame, we usually need to evaluate a fractal for a subset of it's parameters. However for many fractals a large portion of the possible subsets yield an unexciting image or boring behavior. In the noble pursuit of beauty and awesome, it would be interesting to find methods which manage to find sequences of subsets of parameters which manage to:

1. Capture interesting behavior.
2. Accomplish smooth transitions.

By continuity to get a smooth animation it would be reasonable to explore the parameter space along locally connected continous paths, probably closed paths if we desire loopable animations.

Except for these kind of elementary observations, are there any theoretical results in dymanical systems or iterated function systems which could help us here? Herestics maybe?

Own work

On the famous Julia set, the set of $z_0 \in \mathbb{C}$ for which $$J(c) : z_{n+1} = {z_{n}}^2 + c \hspace{1cm} z_k,c \in \mathbb{C}$$ we measure how fast (for which $n$) this grows to infinity and colour a different gray scale value for each $n$. Doing discretized spirals for our parameter $c$: $$c = S(w) = r^{(w/w_0)}\exp(i (w/w_0)*(2\pi))$$ Where $w$ is frame number, $w_0$ is frames per revolution, $r$ is how fast spiral "shrinks" in towards it's midpoint. By following this trajectory, we can get animations like this:

As you can see, following such a fixed smooth path can be interesting, but also for many parameters along the path we explore large relatively uninteresting areas, and when the magic happens it's usually over.. well a bit faster than would be desirable. It would be nice to try and steer or control both the speed and direction of the exploration.

• Why don't you just make a slider instead of an animation? – Zach466920 Oct 13 '15 at 14:06
• I could make a slider, but I'm interested in ways to produce automated exploration. – mathreadler Oct 13 '15 at 14:21
• I don't understand. You'd prefer not to explore the fractals yourself? – Zach466920 Oct 13 '15 at 14:46
• You'd need a mathematical definition of "interesting". I'd assume you mean you'd calculate the fractal dimension of the fractals as a function of a parameter and then explore areas where the derivative of this function is high. – Zach466920 Oct 13 '15 at 14:59
• if you think about slider then check fragmentarium : en.wikibooks.org/wiki/Fractals/fragmentarium – Adam Nov 4 '15 at 16:51