Julia Mosaic I and II

[Pauldelbrot]

A mosaic of tiny Julia set images whose seed values form a grid, thus approximating a low-resolution image of the Mandelbrot set, while graphically demonstrating how the M-set is related to Julia sets.

Attracting basins of infinity are colored by smoothed iterations with a gradient; finite attractor basins are solid black.



The same as Julia Mosaic I, except rendered at 20480x15360 and then downsampled to 1024x768, with the individual Julia tiles staying 20x20.

As expected, it took almost exactly 100 times longer to calculate.

The result is a much closer approximation to a decent-resolution Mandelbrot set image.

The value of this technique is in attacking multi-variable systems like Phoenix/Henon or discrete Volterra-Lotka, which don't have the nifty property that maps of a single complex variable have that the fates of a small number of critical points suffice to characterize the dynamics. Generating parameter-plane images of these systems is much more difficult, though it can be done either by aggressively sampling many points or by generating a Julia mosaic and downsampling it.

The latter has an effect not dissimilar from grid-sampling the dynamic plane, but merely averages the pixel colors produced for each point instead of using more sophisticated statistics about the points' fates. It can be useful for a "quick and dirty" overview of such a system's parameter space that can be done cheaply given one already has the code to generate dynamic-space images (Julia sets).

The second image is not available at 2048x1536.

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

Detailed statistics:

Names: Julia Mosaic I and II
Date: January 28, 2009
Fractal: Mandelbrot/Julia
Location: Whole set
Depth: Very shallow
Min Iterations: 1
Max Iterations: 323
Layers: 1
Anti-aliasing: 3x3, threshold 0.10, depth 1
Preparation time: 5 seconds
Calculation times: 5 seconds for I and 45 minutes for II (2GHz dual-core Athlon XP)

[Dinkydau]

interesting stuff!

[Nahee_Enterprises]

I like what you have done here.  And especially the graphical relationship of the M-set to Julia sets.

[Pauldelbrot]

Thank you. Although some of my research is leading in other, very interesting directions at this time. Stay tuned for some very interesting Julia sets and Mandelbrot views, as well as maybe an animation.

I have probed several new rational maps lately, with interesting characteristics. One of them can produce Herman rings. I also took the original Herman-ring-producing family of maps, Antimatter, and doubled its chromosome count, images pending.