Stochastic webs and continuum percolation in quasiperiodic media

Area of Science:

• Physics
• Materials Science
• Mathematics

Background:

• Continuum percolation models are crucial for understanding disordered systems.
• Quasiperiodic structures exhibit unique geometric and scaling properties.
• Analyzing contour lines and surface geometry reveals fundamental characteristics of these systems.

Purpose of the Study:

• To investigate the geometric properties of contour lines and surfaces in quasiperiodic percolation models.
• To analytically determine fractal dimensions and scaling coefficients for 2D percolation.
• To explore the scaling characteristics of 3D isosurfaces with icosahedral symmetry.

Main Methods:

• Analytical methods were employed to study the fractal dimension and scaling coefficient (nu) in 2D percolation.
• Numerical simulations and computer graphic representations were used for the 3D potential function analysis.
• Focus on contour line and isosurface geometry in quasiperiodic systems.

Main Results:

• The fractal dimension of long isolines was analytically determined for the 2D percolation problem.
• The scaling coefficient nu was also analytically derived for the 2D case.
• Scaling characteristics of 3D isosurfaces with icosahedral symmetry were successfully obtained.

Conclusions:

• The study provides key insights into the geometric and scaling behaviors of quasiperiodic percolation.
• Analytical and numerical findings contribute to the understanding of complex system geometries.
• Results highlight the applicability of percolation theory to quasiperiodic phenomena.