### Variations on a theme by gerrit

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#### gerson

• Fractal Frogurt
• Posts: 424

#### Re: Variations on a theme by gerrit

« Reply #15 on: May 21, 2019, 08:01:11 PM »
Fractal Zoomer has a direct color method that let user define how in and out coloring will be (see file). Is possible too program a code using "Edit user code" and after compile it.
superheal is the programer, maybe he knows how to do it.
zip has my parameter's image file.

• 3f
• Posts: 2240

#### Re: Variations on a theme by gerrit

« Reply #16 on: May 21, 2019, 09:45:38 PM »
$$\frac{z^2+a}{z^2+1}+ c$$
a = 3 + 3i
The mini has the full structure of the Mandebrot set. Here's a warped mini, and an embedded Julia set near its cusp.
Coloring is min(|z|) (value, not index).

• 3f
• Posts: 2240

#### Re: Variations on a theme by gerrit

« Reply #17 on: May 22, 2019, 05:18:06 AM »
Julia set of $$\frac{z+a}{z+1}+ c$$ with a= 1.09524+2.85714i and c=-0.992761778642-1.03937254809i.
« Last Edit: May 22, 2019, 07:50:41 AM by gerrit »

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #18 on: May 22, 2019, 02:42:53 PM »
You could animate that and use it for hypnosis.
Earl Hinrichs, offering free opinions on everything.

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #19 on: May 22, 2019, 02:53:21 PM »
Here is a slight change to the formula we have been playing with:$\frac{z^2}{\frac{z^2}{a}+1}+c$ When a = $$\infty$$ this is the Mandelbrot set. What happens when a is smaller?

a = 20, center = 0, width = 60

A tiny brot is facing off against the antibrot.

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #20 on: May 22, 2019, 02:57:12 PM »
I have several of these, so I am posting them at 1/4 size to save your bandwidth. If you want a larger image, just ask.

a = 15, center = -2, width = 20

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #21 on: May 22, 2019, 02:59:22 PM »
a = 14, center = -2, width = 20

Two of the mini's on the real axis are about to collide. My how he has grown!

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #22 on: May 22, 2019, 03:00:28 PM »
a = 13.9, center = -2, width = 20

Crash!

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #23 on: May 22, 2019, 03:02:35 PM »
a = 13.8, center = -2, width = 20

And boom! Two minis are expelled above and below the axis.

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #24 on: May 22, 2019, 03:07:35 PM »
a = 11.2, center = -2, width = 20

Each island along the axis collides, splits and gets pushed out in turn. Now the largest minis collide.

(Real life interruption. My grandkids just arrived. The battle between the brot and antibrot will resume later.)

• 3f
• Posts: 2240

#### Re: Variations on a theme by gerrit

« Reply #25 on: May 22, 2019, 03:46:15 PM »
Good one. Which critical orbit (starting z)  did you use?

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #26 on: May 22, 2019, 10:16:14 PM »
a = 10.7, center = -2, width = 20

And the biggest mini explodes and two clones are expelled.

Gerrit: Starting z is 0. It is critical point as usual. I believe it is the only one (other than $$\infty$$). Everything cancels out nicely in the derivative, leaving just 2z in the numerator.

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #27 on: May 22, 2019, 10:29:25 PM »
a = 7.5 center = -2 width = 8

This is zoomed in 2.5x from the previous set.

Eventually the period doubling bulbs reach the front line. Growing in size along the way.

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #28 on: May 22, 2019, 10:37:41 PM »
a = 7, center = -2, width = 8

The antibrot absorbs the period doubling bulbs and fills up like a balloon. The period two bulb, which itself has expanded to nearly the size of the main cardioid, is the last line of defense.

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #29 on: May 22, 2019, 10:40:06 PM »
a = 6, center = -2, width = 8

The period two bulb pops and flows into the antibort.

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