Exotica

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Offline pauldelbrot

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« Reply #30 on: February 27, 2019, 07:32:14 PM »
Launch

An early, experimental rendering of the discrete Volterra-Lotka parameter space. This is, basically, its Mandelbrot set. Red-to-white points: no finite attractors. Orange: there's a cyclic attractor, brighter orange = the stablest such is more stable. Dark maroon in the border region = no cyclic attractor but there's a limit circle attractor. Purple = strange attractors dominate. The image was generated by sampling a distribution of points in the dynamic space for each parameter point. The fuzzy/dusty little structures occur where finite attractors exist but their basins have small enough area to not reliably capture at least one sampling point.

I have much more sophisticated images of this fractal, including some shallow zooms, which will be posted here over time, mixed with other "exotica" fractals.

Offline pauldelbrot

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« Reply #31 on: March 08, 2019, 11:17:45 PM »
The Swarms of Azudar

Mandelbrot Foam formula, Julia fractal.

Offline pauldelbrot

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« Reply #32 on: March 21, 2019, 03:17:37 AM »
Beyond Arcturus

This was an experimental blending of two formulae. A Matchmaker Julia was calculated that had no stable attractors, using elliptic harlequin coloring calculations, but instead of just rendering this, the color components were (after an affine transform) used to determine the parameters for a Mandelbrot Foam calculation, making these vary more-or-less smoothly from pixel to pixel. The result was what you see here: not really structured like many fractals, but with detail everywhere and resembling an artist's impression done for some NASA presser or science-magazine article.

Offline pauldelbrot

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« Reply #33 on: April 01, 2019, 11:42:35 PM »
Julia Foam

A Julia set of Mandelbrot Foam. It seems to fit inside the outline of a Julia set from the normal Mandelbrot's largest bulb, but it's full of holes with inversions of that same outline as boundaries, and these holes form swirling patterns of bubbles. There are some trapped points, the chains of small dark nuggets in the interior of the "foam" region.

Offline pauldelbrot

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« Reply #34 on: April 16, 2019, 10:20:24 PM »
Mandelbrot Foam

Offline pauldelbrot

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« Reply #35 on: April 28, 2019, 12:55:54 PM »
The Dark Lotus Opens

A zoom into a Mandelbrot-like feature of Mandelbrot Foam. The Dark Lotus can have aspects that seem disturbingly squid-like.

Offline pauldelbrot

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« Reply #36 on: May 11, 2019, 11:22:52 PM »
Julia Island

Mandelbrot foam, parameter space. There's a little island off the coast whose own coastline looks like a warped quadratic Julia set from inside the cardioid a bit left of center, but its interior is full of more Mandelbrot foam.

Offline pauldelbrot

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« Reply #37 on: May 26, 2019, 10:23:52 PM »
Cosmic Knot

Part of a strange attractor produced by discrete Volterra-Lotka. Red points escape; blue points go to the strange attractor, shown in white and extending beyond the image borders down and left.

Offline pauldelbrot

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« Reply #38 on: June 04, 2019, 10:49:42 PM »
At the Mountains of Madness

What seems from a distance to be a mountain in a range seems, on closer inspection, to dissolve into mist and cloud not quite concealing shadowy nameless beasts ...

Formula: Ibiza parameter space. Ibiza is:

\( x_{n+1} = 1 - ax_n^2 + y_n \)
\( y_{n+1} = bx_n y_n + cy_n^3 - 1 \)

which is a two-real-variable system with three parameters. The image is in the a/b plane with c fixed at 0.1. Coloring is 2D combining average atan x over the attractor with ln(average of 1/|P|2 over the orbit) where P is the orbit point and |P| is the usual Euclidean norm. It is possible to see the "swallowtails" and other related structures here and there.

Offline Dinkydau

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« Reply #39 on: June 10, 2019, 01:03:03 PM »
Nice

Offline pauldelbrot

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« Reply #40 on: June 10, 2019, 03:38:36 PM »
Thanks!