hopalong

Abstract

Historical Context

The "Hopalong"* attractor, authored by Barry Martin of Aston University in Birmingham, England [2], was popularized by A.K. Dewdney in the September 1986 issue of Scientific American. In Germany, it gained further recognition through a translation titled "HÜPFER" in Spektrum der Wissenschaft [3].
_{*Nicknamed by A.K. Dewdney.}

The Hopalong Attractor Functions

This Rust program computes and visualizes the “hopalong” attractor by iterating the following system of recursive functions:

$$ \large \begin{cases} x_{n+1} = y_n - \text{sgn}(x_n) \sqrt{\lvert b x_n - c \rvert} \\ y_{n+1} = a - x_n \end{cases} \large $$

Where:

• The sequence starts from the initial point (x₀ , y₀) = (0 , 0)
• x_n and y_n represent the coordinates at the n-th iteration
• a, b, and c are parameters influencing the attractor's dynamics
• sgn is the signum function

Features and Further Information

The color scheme is based on the pixel density, i.e. how often a pixel of the image is hit during the iteration.

For more information in general and about “pixel density”, i.e. displaying the attractor as a density heatmap, see my Python versions repository.

https://github.com/ratwolfzero/hopalong_python

For information on the implementation of the Signum function in Rust, see:

https://docs.rs/num-traits/latest/num_traits/sign/fn.signum.html

You can run this program from the command line in a terminal.

The number of iterations (num) can be entered as integer or in exponential form such as 1e6.

Example: ./hopalong -2 -0.33 0.01 2e8 (MacOS)

If you are using a Mac with Apple Silicon you should be able to use the executable in the 'Binary' folder.

The binary was compiled on a Mac Mini with M2 processor.
The calculated image should be displayed but there will be an error regarding saving the image.

// Save the image with the generated name
let save_path = format!("/Users/ralf//Projects/hopalong_pictures/{}", image_name); // Specify your desired save path
if let Err(e) = image_buffer.save_with_format(&save_path, ImageFormat::Png) {
    eprintln!("Error saving image: {}", e);
} else {
    println!("Image saved to: {}", save_path);
}

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References

[1]
J. Lansdown and R. A. Earnshaw (eds.), Computers in Art, Design and Animation.
New York: Springer-Verlag, 1989.
e-ISBN-13: 978-1-4612-4538-4.

[2]
Barry Martin, "Graphic Potential of Recursive Functions," in Computers in Art, Design and Animation [1],
pp. 109–129.

[3]
A.K. Dewdney, Program "HÜPFER," in Spektrum der Wissenschaft: Computer Kurzweil.
Spektrum der Wissenschaft Verlagsgesellschaft mbH & Co., Heidelberg, 1988.
(German version of Scientific American).
ISBN-10: 3922508502, ISBN-13: 978-3922508502.