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This image results from zooming into the Herman Ring Mandelbrot c-plane, with a = 0.51803398. With this parameter value, there are seahorses everywhere, including clinging to diagonal linear features near the northeast, northwest, southeast, and southwest directions from the origin.

On each of those features, one side (depending on choice of critical point) has Mandelbrot buds and attached seahorses and the other has mangled buds where it is the other critical point that is bifurcating, and the selected critical point goes sometimes to zero, sometimes to infinity, and sometimes to the other finite attractor. So, in these "ghost buds" there are three basins.

In one of these the depicted seahorse tail is found. Blue points diverge, green points go to zero, and red points go to the other finite attractor.

The attracting cycle for the red points has period 19. The "cut gem" effect on the "rubies" results from the 19 points of the attractor, plus the critical point, moving about as c varies, and different attractor points moving closer to and further from the critical point.

Four layers are used: the blue and green layers are colored by smoothed iterations, the red layer by distance of closest approach of attractor to point of origin, and there's a black distance estimator layer made by taking a distance estimator layer like this:

• black further than one pixel from the M-Set;
• Shading logarithmically to white 1/10,000 of a pixel from the M-set; and
• white within 1/10,000 of a pixel of the M-Set

and subtracting it. This layer emphasizes the contours of the basin boundaries and M-set.

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

Detailed statistics:
Name: Ruby Necklace
Date: February 1, 2009
Fractal: Herman Ring Mandelbrot, c-plane
Location: angle parameter = 0.51803398, zoomed mildly about c = -2.9031072262295 + 2.805112405104i
Depth: Shallow
Min Iterations: 75
Max Iterations: 1,000,000
Layers: 4
Anti-aliasing: 3x3, threshold 0.10, depth 1
Preparation time: 30 minutes
Calculation time: 18 hours (2GHz dual-core Athlon XP)

That's a thing of beauty. Well done.

Based on the coloring you describe, I'd expect to see pure white surrounding the black edges of the set. I don't see much if any pure white however. Can you explain further?


Duncan