An excerpt from the parameter plane of this formula iterated in a loop (p = 2):


z := z^p + c
z := (|x| - i y)^p + c
z := (x - i |y|)^p + c


Another good iteration loop formula is:


z := z^p + c
z := (|x| + i |y|)^p + c
z := (|x| + i |y|)^p + c
z := z^p + c
`

(p = 2)

@mathr What's happening here?
Is this like Mandelbrot fractals except you use a different formula for each step of the iteration?

@gaeel yes that's it: the second one has first iteration Mandelbrot, then two iterations of Burning Ship, then one more iteration of Mandelbrot, then repeat the 4 steps in a loop until the pixel escapes (or you give up).

@mathr Nice!
Mind if I try running it through my Nebulabrot renderer tonight?

https://merveilles.town/@gaeel/101244663094053042

@mathr I tried Julias but they didn't look very good

@gaeel you mean Julia-style rendering (c constant, z_0 set to pixel) of the formulas I posted? I haven't tried, but I suspect using the Mandelbrot-style rendering (c set to pixel, z_0 set to 0) can help find good c values to try out.

@mathr No, I meant when I do the Nebula-style rendering of Julia (c constant, z_0 set to pixel) fractals, it doesn't give particularly pretty results

The way the Nebulabrot is rendered is you record all the points you visited while iterating, and you brighten those pixels if you escaped, it's a heat-map of the escape paths, rather than a map of which values escape

@gaeel oh ok. I've done some Nebulabrot/Buddhabrot stuff in the past, it's fun to have more than 3 channels to map to RGB later.

it's probably not surprising that it's not so interesting with Julia technique, because the Julia set is invariant under iteration, and the filled-in Julia set mostly goes to a periodic attractor (well, there are some aperiodic ones for the Burning Ship I think, see math.stackexchange.com/questio )

@mathr That certainly explains what I saw, yeah, usually a few EXTREMELY hot spots and nothing anywhere else

Welcome to post.lurk.org, an instance for discussions around cultural freedom, experimental, new media art, net and computational culture, and things like that. This is part of a family of services that include mailing lists, group chat, and XMPP.