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Author Topic:  Quadratic Julia sets  (Read 2407 times)

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Offline pauldelbrot

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Quadratic Julia sets
« on: April 14, 2018, 07:42:33 PM »
Patchwork Rabbit

Linkback: https://fractalforums.org/image-threads/25/quadratic-julia-sets/1215/

Offline gerrit

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Re: Quadratic Julia sets
« Reply #1 on: April 15, 2018, 05:56:07 AM »
Nice. What are those bright spots?

Offline pauldelbrot

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Re: Quadratic Julia sets
« Reply #2 on: April 15, 2018, 06:57:19 AM »
Thanks. One of the central two bright spots in the central red area is the point in that area that's a member of the attractor. The other bright spots in the red areas are preimages of that point. The other of the two central ones lands on it after one iteration, since (-z)2 + c = z2 + c. The rest take larger numbers of iterations -- there will be two more that land on -z in one iteration, another two that land on each of those two for four more in the next tier, then eight, then sixteen, and so forth. The limit points of the union of all these sets will be the Julia set proper -- which is how IIM works to plot images of the things.


Offline pauldelbrot

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Re: Quadratic Julia sets
« Reply #4 on: April 15, 2018, 08:05:39 PM »
You'd have to ask the author of Ultrafractal, but I doubt it. Of course, one of my ucl files is responsible for the colorings used...

Offline pauldelbrot

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Re: Quadratic Julia sets
« Reply #5 on: April 25, 2018, 07:51:07 AM »
Foil Dragon

Offline pauldelbrot

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Re: Quadratic Julia sets
« Reply #6 on: May 05, 2018, 08:29:40 AM »
Triplex Flower

Offline pauldelbrot

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Re: Quadratic Julia sets
« Reply #7 on: May 17, 2018, 06:36:15 AM »
Dragon

Offline pauldelbrot

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Re: Quadratic Julia sets
« Reply #8 on: June 08, 2018, 01:46:09 AM »
Black Mesa

Offline pauldelbrot

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Re: Quadratic Julia sets
« Reply #9 on: June 18, 2018, 04:13:39 PM »
Lollipop

Coloring: Angle + smoothed iterations, modulo 2 pi, using a suitable circular color gradient. "Angle" here means the argument of zlast - attractor point, where zlast is the first orbit point to hit the tiny target disk around the attractor point. In fact, in math terms it couldn't be much simpler: sum of integer iterations, imaginary part of that difference, and the real part of that distance rescaled to where the log of the disk radius maps to 1 and the log of the radius of the first forward image of the disk that lands inside the disk maps to 0.

Offline gerrit

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Re: Quadratic Julia sets
« Reply #10 on: June 18, 2018, 05:10:45 PM »
Very nice!

Offline pauldelbrot

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Re: Quadratic Julia sets
« Reply #11 on: June 18, 2018, 06:15:08 PM »
Thanks!

Offline pauldelbrot

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Re: Quadratic Julia sets
« Reply #12 on: November 23, 2019, 08:30:12 PM »
Julia Mosaic

An array of tiny Julia sets, arranged according to their parameters. Squint and you can see the outline of the Mandelbrot set.

Offline pauldelbrot

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Re: Quadratic Julia sets
« Reply #13 on: August 04, 2020, 08:08:32 PM »
Dendrite



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