Pixel-Based Density Approximation of Deterministic Chaos: The Hopalong Attractor

Historical Context

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. In Germany, the Hopalong attractor gained further popularity through an algorithm called HÜPFER in the article Psychotapeten published in Spektrum der Wissenschaft. Nicknamed by A.K. Dewdney.*

Computational Approach

Pixel-based density approximation presents a promising alternative to histogram-based density estimation, each offering distinct advantages.

This generative method creates a density heatmap matrix by mapping trajectory points directly to discrete pixels. Each pixel corresponds to a specific region in continuous space, with emergent density patterns arising from the interaction between the attractor’s dynamics and the discretization process.

The resulting density matrix not only enables efficient 2D visualization but also serves as the foundation for 3D topographic maps, where density values define elevation for spatial pattern analysis.

The pixel-based approach is well-suited for visual exploration and supports fast algorithms that efficiently handle large numbers of iterations. This efficiency arises from its ability to directly update the density heatmap matrix sequentially.

A detailed description, Python code (JIT compiled) and a visual and quantitative comparison using a histogram-based approach can be found in the Github repository: **https://github.com/ratwolfzero/hopalong_python**

https://doi.org/10.5281/zenodo.18811881

The repository also includes detailed references for further reading and context.

From 2D Density Approximation to 3D Topographic Visualization (Image by the author)

Image by the author

Image by the author