Vortex

[Pauldelbrot]

A Julia set of the discrete Volterra-Lotka map-family found in The Beauty of Fractals. The slow render time for a low-iteration shallow image is due to the multiple layers combined with the aggressive antialiasing required to properly resolve the many fine, swirling filaments.

One layer colors the basin of infinity a bluish grey. There is a single finite attractor of period 7 which is highlighted by a second layer, which colors points by the closest approach of their orbits to the point itself -- points on the attractor get a closest approach distance of zero, obviously, and the further from the attractor a point is, the greater the minimum distance. The third layer gives the blue-purple background to the finite attractor's basin, and is responsible for the swirling pattern also; the immediate basins of the separate attractor points are discernible. The convergent smoothed iteration algorithm wasn't used; instead, orbit minimum distance from a fairly arbitrary point within the attracting basin was used.

I've got other Volterra-Lotka fractals on the render queue but the going is even slower with these, as all require aggressive antialiasing and most of them have aperiodic attractors, meaning large chunks of the image require 10,000 iterations or more. That translates into 10,000 iterations for a third or more of over 250 million samples -- over 2.5 trillion iterations in all, and of a fairly complex mapping, with additional running calculations of the Jacobian matrix to evaluate the ergodicity of the orbit. The Mandelbrots are even worse still, since they require up to a thousand samples per pixel, though fewer points in them are aperiodic and nondivergent. (One hundred samples in the dynamic plane per subpixel, and nine subpixels with weakish antialiasing.)

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

Detailed statistics:

Name: Vortex
Date: January 7, 2010
Fractal: Discrete Volterra-Lotka Julia Set
Location: h = 1.0375, p = 2.29375
Depth: Very Shallow
Min Iterations: 3
Max Iterations: 1006
Layers: 3
Anti-aliasing: 3x3, threshold 0.0, depth 2
Preparation time: 5 minutes
Calculation time: 1 hour 20 minutes (2.5GHz dual-core E5200)

[kram1032]

those julias look very organic and smooth and seem to have slightly similar behaviour to magnetic pendulum-style fractals..... (just more organic)