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.

Offline pauldelbrot

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« Reply #7 on: July 25, 2018, 02:10:26 AM »
Julia Chondrules

Offline pauldelbrot

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« Reply #8 on: August 05, 2018, 03:20:52 PM »
Roiled

Offline pauldelbrot

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« Reply #9 on: August 15, 2018, 07:32:12 PM »
Purple Pearls

Offline pauldelbrot

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« Reply #10 on: August 30, 2018, 09:47:59 PM »
Frozen Coal

Mandelbrot Foam, on a parallel to some unseen minibrot.

Offline gerrit

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« Reply #11 on: August 30, 2018, 10:19:02 PM »
M-foam is really fascinating. Could there be a way to slice the 4D parameter space to see a Mandelbrot mini?

Offline pauldelbrot

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« Reply #12 on: August 30, 2018, 10:43:18 PM »
Different init values, or a suitable vector space projection of the init-value 4-space onto the screen, perhaps.

Offline pauldelbrot

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« Reply #13 on: September 10, 2018, 07:25:00 AM »
The Dau and the Y of It

A Hénon map Julia rendering. The green region goes to an attracting fixed point at infinity. The violet regions, including the prominent one shaped like a lowercase letter "y", go to a 2-cycle whose member points are in the two largest purple regions ("Dau" is Welsh for "two"). Finally, points in the yellow region converge to a strange attractor that is in many small segments, the dark brownish arcs inside some of the larger yellow areas. The strange attractor has a hybrid behavior: the fragments can be mapped onto one another via diffeomorphisms, and every iteration sends a point in one of the fragments to a point in another fragment in such a way that the fragments are cycled through like the points of a plain finite cyclic attractor, and the point moves about on the common shape of all of the fragments in the ergodic manner of a point on the Hénon attractor.

Offline pauldelbrot

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« Reply #14 on: September 20, 2018, 04:13:17 AM »
Satin

A Julia of the discrete Volterra-Lotka system. Colored by how close the orbit past the 50th iteration ever comes to the starting point.