Darkfall I

[Pauldelbrot]

This Matchmaker Mandelbrot image shows an area of the parameter space where there is one attractor of period 3 overlapping an area where there is an attracting fixed point. The main blue area at top is the intersection, with the downward-hanging buds being where the fixed point bifurcates into various attracting cycles of fairly long period. Below these is an area with only the period-3 attractor; the complex dendrites here reflect the periods of the nearby buds, the way elephants in the classic M-set do, but the extra "elephant tails" relate to the stable cycle here being a period-3 cycle instead of a period-1 one as in the classic M-set's elephant valley. At top left can just be seen some buds sticking up, black on dark blue; this is the edge of the period-3 attractor's zone of stability, with the black buds representing bifurcations of that attractor, and only a stable fixed point being present outside them.

There are three layers. One provides the blue shades in the top area and buds (and a few mini Mandelbrots visible among the dendrites below) and is applied to all parameter points in the image with two attractors. The gradient reflects the smoothed iterations calculated for both critical points and averaged. The other two layers color points with only one attractor, one coloring averaged smoothed iterations for values from 1 to 158 with a repeating linear gradient, the other iterations higher than 158 with a nonrepeating logarithmically-mapped gradient. (Limitations of the software I use prevent easily using a single gradient and a single layer to do this. I am planning to write new software that will be far more efficient; instead of calculating every point six times, three for each critical point, it would do it only twice, once for each critical point, and do this same image in under an hour. Besides enabling more complexly-defined iteration-to-color mappings, it will enable multiple gradients per layer, one per region -- attractor, Mandelbrot component, or similarly. Potentially every bud could get its own gradient!)

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

Detailed statistics:
Name: Darkfall I
Date: February 27, 2009
Fractal: Matchmaker Mandelbrot set
Location: b-plane, a = -1.316107i
Depth: Shallow
Min Iterations: 1
Max Iterations: 1,000,000
Layers: 3
Anti-aliasing: 3x3, threshold 0.10, depth 1
Preparation time: 1 hour
Calculation time: 5 hours (2GHz dual-core Athlon XP)