Rotating fractals

Fractals like $z=z^2+c$ and $z=z^3+c$ are easy to draw because they just involve complex addition and multiplication. What happens if we try to generalize these sorts of fractals to non-integer powers?

We can do this using the identities
$\displaystyle z^w = (e^{\frac{\log z}{\log e}})^w = e^{w\frac{\log z}{\log e}}$
$\displaystyle e^{x+iy} = e^x(\cos y + i \sin y)$
$\displaystyle \frac{\log z}{\log e} = \frac{\log |z|}{\log e + i(\arg z + 2n\pi)}$

The trouble is, we have to pick a value for $n$. Given a parameter $\theta$ we can pick $n$ such that $\theta - \pi <= \arg(x+iy) + 2n\pi < \theta + \pi$ (i.e. choose a branch cut of constant argument and a sheet of the Riemann surface): $\displaystyle n = \lfloor\frac{1}{2}(\frac{\theta - \arg(x+iy)}{\pi} + 1)\rfloor$.

Then as we gradually vary $\theta$ the fractal we get will change. For $w = u+iv$, $v \ne 0$, increasing $\theta$ will cause $z^w$ to "spiral in" from infinity towards the origin ($v<0$) or out ($v>0$). When $v = 0$, increasing $\theta$ will have a periodic effect, which will cause the fractal to appear to "rotate" in fractally sort of ways with a period of $\displaystyle \frac{2\pi}{u}$.

It would be interesting to plot these as 3D images (like these with $\theta$ on the z axis. These would look sort of helical (or conical, for $v \ne 0$.)

Using $u<0$ poses some additional problems - the attractor isn't bound by any circle of finite radius about the origin so it's more difficult to tell when a point escapes and when it is convergent but just currently very far out. Clearly some further thought is necessary to plot such fractals accurately - perhaps there are other escape conditions which would work for these.

One Response to “Rotating fractals”

1. [...] year, I wrote about a way to make rotating fractals. I implemented this and here is the [...]