Generative Density Approximation for Deterministic Point Patterns: Hopalong Attractor

[ratwolf]

The "Hopalong" attractor (Martin’s map) shown as a Density Heatmap, which enables a 3D representation by mapping normalized density values ​​along the Z-axis.

The "Hopalong" attractor, authored by Barry Martin of Aston University in Birmingham, England, was popularized by A.K. Dewdney
in the September 1986 issue of Scientific American

For details and code, see Github

               

Important note: The programs use Numba @njit, currently only compatible up to Pyton 3.12.x, not yet 3.13.x!

[sakshi009]

The Hopalong Attractor is a chaotic system that generates deterministic point patterns, often resembling intricate, fractal-like shapes. It operates through iterative mathematical equations that determine the position of each point based on previous ones. Despite its deterministic nature, the attractor produces seemingly random, dense patterns in certain areas while leaving others sparse. Density approximation for such patterns involves estimating how densely points populate different regions, revealing underlying structure and symmetry. This can be done using computational techniques like kernel density estimation. The Hopalong Attractor's striking visual complexity makes it a popular study subject in mathematical visualization and computational artistry.

[ratwolf]

The Hopalong Attractor is a chaotic system that generates deterministic point patterns, often resembling intricate, fractal-like shapes. It operates through iterative mathematical equations that determine the position of each point based on previous ones. Despite its deterministic nature, the attractor produces seemingly random, dense patterns in certain areas while leaving others sparse. Density approximation for such patterns involves estimating how densely points populate different regions, revealing underlying structure and symmetry. This can be done using computational techniques like kernel density estimation. The Hopalong Attractor's striking visual complexity makes it a popular study subject in mathematical visualization and computational artistry.

KDE is computationally expensive, while my simple generative pixel-based density approximation is lightning-fast compared to KDE. Of course, this depends on the hardware, but on my Mac Mini with an M2 processor, one can go out for lunch while computing KDE with a grid size of 1000 and 1 million iterations of the Hopalong attractor equations. In contrast, the pixel-based approach, using the same parameters, does it in 0.3 seconds. And the density approximation is not that bad. I reduced KDE's smoothing to avoid predicting too many points :-)

Comparison Pixel-based versus KDE based approach

   


Comparison Pixel-based versus Histogram based approach