MandelbrotSet

This repository implements the algorithm of the Mandelbrot set to create both static and dynamic representations of the set on a complex plane.

Significance of the Mandelbrot Set

The Mandelbrot set is a famous fractal named after the mathematician Benoît B. Mandelbrot. It is a set of complex numbers that produces a distinctive, infinitely complex boundary when plotted. The set is defined by iterating the function:

[ f_c(z) = z² + c ]

where ( z ) starts at zero and ( c ) is a complex parameter. The Mandelbrot set includes all values of ( c ) for which the orbit of ( z ) does not escape to infinity.

Key Properties

• Self-Similarity: The boundary of the Mandelbrot set exhibits self-similarity at different scales, meaning smaller portions of the boundary resemble the whole set.
• Complexity from Simplicity: Despite being generated from a simple iterative process, the Mandelbrot set produces highly intricate and beautiful patterns.
• Connection to Julia Sets: Each point in the Mandelbrot set corresponds to a specific Julia set, another type of fractal.

Mathematical Calculation

To determine if a complex number ( c ) is in the Mandelbrot set, we iterate the function ( f_c(z) = z² + c ) starting with ( z = 0 ). The number ( c ) is part of the Mandelbrot set if the absolute value of ( z ) remains bounded (does not go to infinity) after many iterations. Typically, the following steps are used:

1. Initialize ( z = 0 ).
2. Iterate the function ( z = z² + c ).
3. Check if ( |z| > 2 ) (if so, ( c ) is not in the Mandelbrot set).
4. Repeat for a maximum number of iterations or until ( |z| ) exceeds 2.

The number of iterations before ( z ) exceeds 2 can be used to color the point ( c ), creating a visual representation of the set.

Features

• Static Representation: Generates a static image of the Mandelbrot set.
• Dynamic Representation: Creates an animated visualization of the Mandelbrot set's iterative process.

Getting Started

Prerequisites

• Python 3.x
• Required Python libraries: numpy, matplotlib, pyopengl

Installation

1. Clone the repository:

   git clone https://github.com/sparsh-karna/MandelbrotSet.git
   cd MandelbrotSet

2. Install the required libraries:

   pip install numpy matplotlib pyopengl

Usage

Static Representation

To generate a static image of the Mandelbrot set:

python static/mandelbrot_static.py

Dynamic Representation

To visualize the Mandelbrot set dynamically:

python dynamic/mandelbrot_dynamic.py

Project Structure

• static/: Contains the script for generating static images of the Mandelbrot set.
• dynamic/: Contains the script for creating dynamic visualizations of the Mandelbrot set.

Contributing

Contributions are welcome! Please fork the repository and submit a pull request.

License

This project is licensed under the MIT License. See the LICENSE file for details.

Contact

For any questions or suggestions, please contact Sparsh Karna.