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 \leq \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.

This entry was posted on Thursday, May 22nd, 2008 at 4:02 pm and is filed under fractals. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Responses to “Rotating fractals”

  1. Rotating fractal « Reenigne blog says:
    October 29, 2009 at 4:05 pm

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

    Reply
  2. drew says:
    September 30, 2024 at 1:56 am

    Nice work, still interesting in 2024!

    Reply

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