Growth behavior of helical cellular automata.

Area of Science:

• Complex Systems
• Computational Physics
• Mathematical Modeling

Background:

• Cellular automata (CA) are widely used for modeling complex systems.
• Investigating CA on non-standard geometries, like helical structures, can reveal novel emergent behaviors.
• Understanding pattern formation in dynamic systems is crucial across various scientific disciplines.

Purpose of the Study:

• To present and analyze a helical cellular automata (HCA) model.
• To investigate the impact of helix circumference on HCA pattern formation.
• To quantitatively characterize the emergent patterns and evolutionary behaviors using fractal dimension analysis.

Main Methods:

• Development of a helical cellular automata (HCA) model on a 2D grid with a helical structure.
• Extensive computer simulations to observe pattern evolution under varying helix circumferences (p).
• Application of fractal dimension analysis to quantitatively assess pattern complexity and evolution.

Main Results:

• HCA pattern formation is highly sensitive to the helix circumference (p).
• Diverse patterns emerge, including Sierpinski triangle gaskets, complex textures, and quasiperiodic structures.
• A sharp transition from fractal to compact patterns is observed when p approaches powers of 2.
• Vertical growth patterns vary with p, showing periodic, regular, and random structures over time.

Conclusions:

• Helix circumference is a critical parameter dictating HCA emergent behavior and pattern complexity.
• The HCA model exhibits rich dynamics, transitioning between ordered, fractal, and random states.
• Fractal dimension analysis provides a robust method for characterizing the complex evolutionary pathways in the HCA model.