### Holomorphic Mandelbrot extensions

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#### quadralienne

• Fractal Friend
• Posts: 17
• infinite border, finite area

#### Re: Holomorphic Mandelbrot extensions

« Reply #30 on: June 25, 2019, 09:08:55 PM »
Here's some of the aforementioned messing around: p=1, q=-1, as in previous image, centred around ... uh, well, the projection was this:
xy:[-y,x] * 0.015 + xy:[-0.04,0.045]
and it's too hot to sort out what that means right now, but I think centred around (0.045+0.04i)

I made it escape only when *both* z and w were outside of standard Mandelbrot sets with the main circle transformed onto the normal circle of radius 2.

Tomorrow on an air conditioned computer I will make an animated version of this image where those two Mandelbrot sets spin around in opposite directions!

• 3f
• Posts: 1690

#### Re: Holomorphic Mandelbrot extensions

« Reply #31 on: June 26, 2019, 02:41:02 AM »
Here's from the p&q that I analysed. Exterior is black, interior min|z|.

#### quadralienne

• Fractal Friend
• Posts: 17
• infinite border, finite area

#### Re: Holomorphic Mandelbrot extensions

« Reply #32 on: June 27, 2019, 12:20:30 AM »
Ok so ... check this out https://youtu.be/eVyJZcf4Qz0

This one escapes if either z or w escapes from a plain-old-Mandelbrot main circle. Colouring ... well, it's complicated!

Oh, and be sure to right-click the video and turn on Loop!

• 3f
• Posts: 1690

#### Re: Holomorphic Mandelbrot extensions

« Reply #33 on: June 27, 2019, 02:36:44 AM »
The mini on the right is attacked from another dimension.
q=-1, p =0. This is the  best distance estimation coloring I got:
DE = 1/(1/dz+1/dw) with dz = |z|log|z|/|z'|, dw = |w|log|w|/|w'|.
Another image of same with the smooth iteration coloring discussed elsewhere.

• 3f
• Posts: 1690

#### Re: Holomorphic Mandelbrot extensions

« Reply #34 on: June 30, 2019, 09:35:46 AM »

• 3f
• Posts: 1690

#### Re: Holomorphic Mandelbrot extensions

« Reply #35 on: July 07, 2019, 04:43:01 AM »
$$z \leftarrow z^2 + aw +c\\ w \leftarrow 1/(w^2+c)\\ a=0.15$$

Cute but IMHO Mandelbrot foam is still the beauty queen of nontrivial 2 (complex) parameter iterated systems.

• 3f
• Posts: 1690

#### Re: Holomorphic Mandelbrot extensions

« Reply #36 on: July 07, 2019, 04:47:36 AM »
Mandelbrot foam variant:
$$w \leftarrow q (w/z)^5\\ z \leftarrow z^2 + w^3 + c\\ q = 0.1$$
Outside is black, inside colored by orbit trap min(|z|).

• 3f
• Posts: 1690

#### Re: Holomorphic Mandelbrot extensions

« Reply #37 on: July 12, 2019, 12:11:48 AM »
Coupled Mbrot:
$$z_0 = \sqrt{-pq/4}\\ w_0 = -z_0\\ z \leftarrow z^2 + qw +c\\ w \leftarrow w^2 + pz +c$$
with c = pixel.  p=0, q= 1/2. Plain (smooth) iteration coloring.

• 3f
• Posts: 1226

#### Re: Holomorphic Mandelbrot extensions

« Reply #38 on: July 12, 2019, 02:20:48 AM »
Beautiful.

#### gerson

• Fractal Feline
• Posts: 167

#### Re: Holomorphic Mandelbrot extensions

« Reply #39 on: July 12, 2019, 06:33:34 PM »
This topic is very interesting. Keep doing...

• 3f
• Posts: 1690

#### Re: Holomorphic Mandelbrot extensions

« Reply #40 on: July 13, 2019, 09:55:28 PM »
Complex general quadratic:
$$z \leftarrow a_1 z^2 + a_2 w^2 + a_3 zw + c\\ w \leftarrow a_4 z^2 + a_5 w^2 + a_6 zw + c$$
with $$a_i$$ random (too many to explore systematically).

• 3f
• Posts: 1690

#### Re: Holomorphic Mandelbrot extensions

« Reply #41 on: July 14, 2019, 09:14:06 AM »

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