Other Polynomial and Rational Maps

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Offline pauldelbrot

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« Reply #90 on: May 22, 2020, 05:57:03 PM »
Thanks!

Offline pauldelbrot

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« Reply #91 on: May 27, 2020, 10:39:30 AM »
Woodknot Brot

Offline FractalAlex

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« Reply #92 on: May 27, 2020, 04:37:55 PM »
Nice! Also, can you explain me the supernova parameter? I'm very curious what it is...
"I am lightning, the rain transformed."
- Raiden, Metal Gear Solid 4: Guns of the Patriots

Offline pauldelbrot

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« Reply #93 on: May 28, 2020, 02:31:59 AM »
\( z \leftarrow a\frac{z - c}{1 - zc} + b \)

When b = 0, 0 and infinity superattract. Move b, and while 0 remains a critical point the attractor moves off of 0 and weakens. It can eventually bifurcate or even disappear.

In images (both parameter space and Julia sets) made from this mapping, there tend to be "lakes" of 0-attracted points. These are simply-connected expanses if b = 0 but become more convoluted when b is perturbed. If the "zero attractor" bifurcates, the "lakes" become snakes, dragons, and such with multiply-connected borders and Fatou set shapes -- trifurcating the zero attractor produces rabbit Julia shapes, for example, meeting in threes. Destabilize it completely and the lakes fill in with the surrounding spirals and other structures (often governed by a; for example a on the unit circle near but not exactly on -1 gives seahorses everywhere) with the limit set they approach being a dendritic or disconnected Julia set.

Offline FractalAlex

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« Reply #94 on: May 28, 2020, 02:37:37 AM »
Thanks! You're the best, pauldelbrot, at explaining fractal behavior and functions! :thumbs: ;)

Online gerson

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« Reply #95 on: May 28, 2020, 03:29:38 AM »
@pauldelbrot all images on this post are great, colors and shapes, keep doing..
« Last Edit: May 28, 2020, 03:40:04 AM by gerson »

Offline pauldelbrot

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« Reply #96 on: May 28, 2020, 03:50:20 AM »
Thanks!

(I have been studying these things for three decades or so now. :))

Offline pauldelbrot

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« Reply #97 on: June 08, 2020, 01:50:21 PM »
Herman City Redux

Triskelion: the smooth blue "sea" is the basin of an attracting fixed point; the rest is the basin of a Herman ring. Some wacky harlequin trap coloring was used.

Offline pauldelbrot

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« Reply #98 on: June 20, 2020, 08:54:18 AM »
Silk Flowers

Offline pauldelbrot

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« Reply #99 on: July 03, 2020, 09:13:06 PM »
Bugs II

"Bugs" in black-and-white distance estimator coloring. Shallow z12 minibrot.


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