Negative multibrots

[xenodreambuie]

... or should I take it from xenodreambuie's comment that your [youhn's] images are julia sets which need the bailout to be higher? So does it remain the case that z^-2+c looks always like element90's picture, and the pretty patterns only turn up in the julia sets?

Yes, that is exactly right.

On the Devaney stuff:
I've just been looking at z+1/z+c instead (because I already have the capability without writing any new code). Very slow to render and hard to get good images. Here is most of the whole Mandelbrot set and a zoom into the valley of the smaller bud near the main bud.
It has good Julia sets too.

[element90]

Very nice images youhn... I don't understand how you found these when element90's image suggests the z-2+c is filled with a never ending dense cluster of z4+c-like minibrots.

Judging by the pictures it looks like the divergent bailout is used (the same bailout method as the standard Mandelbrot set), the picture with M4 (z^4) Mandelbrots was produced using convergent bailout.

Here are some more pictures using the divergent bailout and several colouring methods.





Changing the method to "absolute log of standard deviation of angle, scale = 1000".



Changing the method to "absolute log of exponential sum of ratio, smaller(real(z), imag(z)) divided by larger(real(z), imag(z)), scale = 100".



The patterns in these fractals are dependent on the bailout value used, the next three pictures show the effect of increasing the bailout value. Values used are 16, 32 and 64.







The final version uses a different method of selecting between inner and outer colouring methods. Titan's summary tab has a bug where "Colour selection" is shown as "unknown" so here is the colour tab from Saturn:





Adding transforms to this fractal produces some familiar features:

[Roquen]

Which of the two possible square roots do you pick?

There's only one "principle" square root.

[youhn]

Judging by the pictures it looks like the divergent bailout is used (the same bailout method as the standard Mandelbrot set),...

True. XaoS uses the standard Mandelbrot settings as default. I only added a minus sign as Tglad suggested and played around with the number of iterations and bailouts. Download XaoS for free and try for yourself. The program is pretty limited but has some cool features I did not find in other fractal software (real time zooming and color cycling).

This evening I'll try to create a video to show increasing bailout condition on the negative mandelbrot.

[youhn]

Not so exciting as i thought, but here the short HD video of the increasing bailout condition:

<a href="http://www.youtube.com/v/2IvEctTlPU8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/2IvEctTlPU8&rel=1&fs=1&hd=1</a>

Made using XaoS and ffmpeg.

[claude]

Using Lyapunov exponents for colouring negative multibrots seems to work well, even able to get "atom domains" similar to the regular Mandelbrot set by seeing when the derivative (as opposed to the iterate itself) gets really small (Fragmentarium source attached, each atom domain accumulated with cumulative additive blending, final Lyapunov exponent is used to colour the minibrots black):

[lkmitch]

I find that using a high bailout and few iterations is good for clearing up some of the visual noise with negative exponents.  At least, it'll allow you to see where the structure is.  This one is with an exponent of -2, 250 iterations, and a bailout of 1E12.

[knighty]

That's cool!
Thank you claude for the fragmentarium script.