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 Author Topic: Toxic Spill  (Read 756 times) Description: Supernova c-plane M-Set image 0 Members and 1 Guest are viewing this topic.
Pauldelbrot
Fractal Senior

Posts: 2592

 « on: March 21, 2009, 01:31:02 AM »

One day I thought it would be nifty if the lake shapes in Antimatter fractals were more varied. In particular, if they could have petals. So I decided to make zero stop being always a superattracting fixed point.

Antimatter is the family of mappings z2$\frac{z - c}{1 - zc}$. Supernova just adds an innocent little "+ b" to that, not unlike how Nova was derived from the Newton's Method Julia fractal whose attractors were the roots of z3 - 1.

That innocent little "+ b" unleashes a veritable flood of exotic new forms. Of course the system remains capable of producing the full gamut of interesting dynamics, including Herman rings. The addition of a constant does not change the partial derivative with respect to z, so the critical points remain the same. However, only infinity is a permanent attractor now; zero superattracts only for b = 0 and for other b-values is an "interesting" critical point. The fixed point moves away from zero, though it still attracts for small b and small enough c.

In fact, the c-plane develops a Mandelbrot set looking like the original, but scaled and rotated depending on b, as viewed using the critical point zero as the lens. The most interesting images, however, come from studying one of the "usual" critical points for Antimatter, within the region near the border of the (now-"ghostly") Zero Mandelbrot. In these regions, the characteristic "lakes" in the centers of seahorses and similar structures implode as the attractor bifurcates and eventually disintegrates.

Numerous Supernova images will be forthcoming, interspersed with more Matchmaker and other images. To start things off, this image depicts a triple spiral inside which the "Zero Attractor" (in general not zero, but bifurcated from it and often catching zero) has become a period-3 one. It is shown in purple while the basin of infinity is shown in green.

Two of the three layers color the basin of infinity, one the lower iterations and the other the higher ones, for reasons of software limitations.

If you look closely at the lake in the lower left area (preferably in the 2048x1536 version of this image) you will see that the top petal has fractured into sub-petals; the attractor has bifurcated there into a 21-cycle. Part of that petal and most of the other two are whitish fuzz on dark; points here all go to infinity, but with high iterations. Zooming there results in structures like are found in quadratic Julia sets that are disconnected but close to the set.

This occurs because the lake here intersects the boundary of the Zero Mandelbrot. The period-3 bud's edge is between this lake and the main one of this image, and a period-21 bud attached to it overlaps the top petal of the lower-left lake. A period-105 bud attached to that overlaps the right petal, though it's not really evident in this image. (It is evident in one of my contest images, Ghost Bud, though.) Most of the rest of the lake petals lies outside the Zero Mandelbrot entirely, so only infinity attracts.

Freely redistributable and usable subject to the Creative Commons Attribution license, version 3.0.

Detailed statistics:
Name: Toxic Spill
Date: February 19, 2009
Fractal: Supernova Mandelbrot set
Location: c-plane; a = 0.61803398, b = 0.01 + 0.89635i
Depth: Shallow
Min Iterations: 23
Max Iterations: 288,423
Layers: 3
Anti-aliasing: 3x3, threshold 0.1, depth 1
Preparation time: 5 minutes
Calculation time: 4 minutes (2GHz dual-core Athlon XP)
 Logged

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