Jewel

[Pauldelbrot]

This is the Herman mapping's version of one of the M-Set's many non-seahorse-like dendrites. It differs from the norm in that its arms don't simply meet at the core, but instead enter a complex structure that contains inverted versions of the dendrite and, at its center, a pocket of the other attracting basin.

In this instance, the red "jewel" contains points that diverge, as to the blue "gems" dotting the lacework; the violet lacework consists mainly of points that converge to zero. Minibrots occur as black specks dotting the arms of the dendrite.

Three layers were used: one for the divergent points, one for the points convergent to zero, and a distance estimator layer that brightens the filigrees and the borders between the basins.

The Herman mapping's alpha parameter determines one species of dendrite that occurs repeatedly throughout the c-plane image, and whose central Misiurewicz point is replaced with these pockets of the inverse basin and their complex border regions. Further dendrites occur if you delve closer to various buds or minibrots, but these don't get hollowed out cores.

For alpha parameters close to rational numbers of small denominator, the c-plane image can be full of seahorses or similar structures with hollow cores.

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

Detailed stats:
Name: Jewel
Date: January 30, 2009
Fractal: Herman Ring Mandelbrot c-plane
Location: angle parameter = 0.2809017, zoomed near c = 7.7708334335 + 4.458333433i
Depth: Very Shallow
Min Iterations: 25
Max Iterations: 1,000,000
Layers: 3
Anti-aliasing: 3x3, threshold 0, depth 2
Preparation time: 10 minutes
Calculation time: 6 hours (2GHz dual-core Athlon XP)