Exotica

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Offline pauldelbrot

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« on: April 26, 2018, 06:17:25 AM »
Exotica: Not mappings of single variables in the complex plane. Some have multiple real variables, some multiple complex variables, and some might just be uncategorizable.

Image 1: Taffy

The Hénon attractor is produced by a mapping of two real variables that has two real parameters. This is a Julia set, but with different parameter values than for the "classic" Hénon attractor. The parameter change has broken the strange attractor in two (dark squiggles on yellow), and caused the appearance of a second, non-strange attractor, a cycle of seven points (darkest regions in biggest brown patches). Their attracting basins (yellow and brown areas, respectively) resemble two differently colored toffees swirled together. Green points escape to infinity. The Julia set, unlike in the one-complex-variable case, is not identical to the border of each attracting basin. There are no green areas dotted along the borders between brown and yellow areas. Much of the Julia set does not border A(∞), though, interestingly, all of the border of A(∞) seems to be approached in the limit by both of the other attracting basins, and by the border between them.

A(∞) is colored by smoothed iterations. Each of the other basins is colored by the squared distance to the nearest point of that basin's attractor. All gradients shade to black for zero iterations/distance.

Offline pauldelbrot

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« Reply #1 on: May 06, 2018, 07:15:43 AM »
Mandelbrot Foam

This is a parameter-space image of a system of two complex variables:

w -> cw/z
z -> z2 + w2 + d

The image is in the d-plane with fixed values for c, initial z, and initial w. It resembles a foam outlined by Julia set shapes and containing little filled-in-Julia-like nuggets. Many Julia and Mandelbrot shapes can appear in systems like this, jumbled up. There's not, to my knowledge, a way to get a tidy Mandelbrot-like map, and all parameter space slices resemble this, or Mandelbrot images with z starting out away from a critical point, or stranger stuff indeed, such as (I kid you not) banded agate.

Offline pauldelbrot

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« Reply #2 on: May 17, 2018, 05:02:27 PM »
Y

Offline pauldelbrot

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« Reply #3 on: June 11, 2018, 06:01:23 AM »
Banded Agate

Parameter-space image from the same formula as Mandelbrot Foam. This is a rational mapping of two complex variables. In this instance it seems to have produced multiple parallel Herman mirrors, among other fascinating excursions from the realm of the usual.

Offline pauldelbrot

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« Reply #4 on: June 19, 2018, 01:02:25 AM »
Lava Bomb

Some red glows are seen inside this irregular black rock that was erupted by the Mandelbrot Foam formula.

Offline pauldelbrot

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« Reply #5 on: July 02, 2018, 12:14:17 AM »
Burnt Mandelbrot Printout

Kinda looks like one, but it's actually a Mandelbrot Foam image with no postprocessing, by fire or otherwise.

Offline pauldelbrot

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« Reply #6 on: July 12, 2018, 06:29:04 AM »
Interwoven

A complex-Hénon Julia set. The dark regions contain a 2-cycle; pink regions go to a 10-cycle; and the narrow green regions satellite to those go to a third, even longer cycle. The main exterior green shading to cyan region escapes to infinity. There's a sizable piece of the parameter space where the nonescaping region breaks into these two "counterrotating vortex" patterns with a similar herringbone pattern of escaping vs. trapped points in between.