Fibonacci Fields

[Pauldelbrot]

Distance estimator rendering of the c-parameter plane of the rational map z->e^iaz²(z - c)/(1 - cz) with a equal to 0.61803398. This produces "flowers" with Fibonacci spiral structure, among other things.

The yellow-brown region is the region of divergent points. The darker-brown region contains parameter values where the critical point goes to zero, a superattracting fixed point for all parameter choices. Navy blue color was applied to the parameters where the selected critical point did not go to either (i.e., to points in the Mandelbrot-like set).

Noteworthy features of the image include the paired buds near the axes, the ring of distorted minibrots where the numbers on a clock would go, and the "lakes" in the cores of seahorse-type structures.

Some really technical notes can be found below, for those who are interested.

Full-size image 2048x1536 (follow link, then note there are links for other sizes).

3.5-megabyte PNG available upon request.

The axial lines with buds to either side are characteristic of this system's c-plane images when a is real and irrational, and the dynamic plane contains Herman rings for a dense set of c values of magnitude greater than 3 and pure real or pure imaginary, i.e. a dense set in those axial lines. Wherever a bud is attached a parabolic basin occurs and at all the other points a Herman ring, much as on a normal component boundary you'd get a parabolic basin or a Siegel disk.

It took over nine hours of work to implement the distance estimator math for this function, particularly for calculating distance estimates in component interiors. Component-interior distance estimates require four partial derivatives, f(z, c) with respect to z, f(z, c) with respect to c, f(z, c) with respect to z twice, and f(z, c) with respect to c and then with respect to z. The latter two get very nasty with rational maps due to double application of the quotient rule! They end up with (1 - cz)³ in the denominator. It would be (1 - cz)⁴ except that the numerator ends up having a factor of 1 - cz.

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

Detailed stats:
Name: Fibonacci Fields
Date: January 26, 2009
Fractal: Herman Ring Mandelbrot, c-plane
Location: angle parameter ~= phi (0.61803398), centered, moderately wide field.
Depth: Very shallow.
Min Iterations: 1
Max Iterations: 1,000,000
Layers: 4
Anti-aliasing: 3x3, threshold 0.10, depth 1
Preparation time: 10 hours
Calculation time: 1 hour (2GHz dual-core Athlon XP)

[Chillheimer]

pauldelbrot, if you read this, could you please reupload the picture? Doing some research on Fibonacci/Mandelbrot http://www.fractalforums.com/mandelbrot-and-julia-set/fibonacci-and-the-mandelbrot-set