Reenigne blog

Fractals like z=z²+c and z=z³+c are easy to draw because they just involed 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
z^w = (e^{log z / log e})^w = e^{w log z / log e}
e^x+iy = e^x(cos y + i sin y)
log z / log e = log |z| / log e + i(arg z + 2nπ)

The trouble is, we have to pick a value for n. Given a parameter θ we can pick n such that θ – π <= arg(x+iy) + 2nπ < θ + π (i.e. choose a branch cut of constant argument and a sheet of the Riemann surface): n = ⌊((θ - arg(x+iy))/π + 1)/2⌋.

Then as we gradually vary θ the fractal we get will change. For w = u+iv, v ≠ 0, increasing θ will cause z^w to “spiral in” from infinity towards the origin (v<0) or out (v>0). When v = 0, increasing θ will have a periodic effect, which will cause the fractal to appear to “rotate” in fractally sort of ways with a period of 2π/u.

It would be interesting to plot these as 3D images (like these with θ on the z axis. These would look sort of helical (or conical, for v ≠ 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.