Lollipop

[Pauldelbrot]

A standard-issue Julia set with a decidedly non-standard-issue coloring. The outside is ordinary smoothed iterations in greyscale. The inside is colored by a combination of phi angle and smoothed iterations. For this twist coloring, the phi angle is computed (and such that the origin has a phi of zero), after which the smoothed iteration value is added, multiplied by a twist factor. In this case, the twist factor is one full revolution per 10 iterations. Different Julia sets demand different twist factors to look good.

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

As usual, today's pictures are available at 2048x1536 at the links, and at that resolution in lossless PNG format upon request.

Detailed statistics:
Name: Lollipop
Date: Match 1, 2009
Fractal: Julia set
Location: c = 0.225365 + 0.259489i
Depth: Very Shallow
Min Iterations: 1
Max Iterations: 20
Layers: 2
Anti-aliasing: 3x3, threshold 0.1, depth 1
Preparation time: 10 minutes, mostly coding
Calculation time: 30 seconds (2GHz dual-core Athlon XP)

[Duncan C]

Very pretty image.

What does the iteration value mean for pixels inside the filled Julia set, however? Points inside the set never diverge, so the iteration value for all points should be infinity.

Duncan

[David Makin]

Hi Duncan - many of the usual "insides" of fractals consist of areas with orbits either converging to a point (like normal Newtons do) or converging to a periodic attractor. Paul is using a similar algorithm to that for bailout on Newtons to get a bailout for such areas - straightforward for point attractors but not so simple for periodic attractors (of period >1) - hence the description on some of his images of having to do very high iteration counts (well because of that and some of the other even more sophisticated colourings he's employing).