Generation of fractals from incursive automata, digital diffusion and wave equation systems

Area of Science:

• * Formal Systems Design
• * Computational Mathematics
• * Fractal Geometry

Background:

• * Existing formal systems design tools lack methods for complex, time-dependent behaviors.
• * Fractal patterns and self-referential systems are areas of interest in both information and physical sciences.

Purpose of the Study:

• * To introduce and apply the concepts of incursion and hyperincursion as novel modeling tools.
• * To demonstrate the generation of fractal patterns using these new methods.
• * To explore the application of these concepts in formal systems design for information and physical systems.

Main Methods:

• * Application of incursion and hyperincursion to a fractal machine (hyperincursive cellular automata).
• * Utilizing sequential computations with exclusive OR and incorporating future time steps.
• * Generating fractals through scaling symmetries, holographic-like simulations, and digital diffusion/wave equations.

Main Results:

• * Successful simulation and generation of fractal patterns using the incursion/hyperincursion model.
• * Demonstration of fractal generation with varying scaling symmetries and self-referential properties.
• * Fractals generated from image interference and digital equations, showing complex pattern formation.

Conclusions:

• * Incursion and hyperincursion provide a powerful framework for modeling complex systems and generating fractals.
• * The methods offer a new perspective on self-referential systems and their relation to hypersets.
• * These tools have practical applications in the design of information and physical systems.