E² and Gamma distributions in polygonal networks

Area of Science:

• Physics
• Materials Science
• Biology

Background:

• Polygon-based networks are found in diverse natural phenomena, from solar supergranulation to biological structures.
• Despite varied origins, these networks display consistent patterns and complexities.

Purpose of the Study:

• To demonstrate the ubiquity of an exponential distribution in the squared norm of the deformation tensor (E²) across different polygonal tessellations.
• To analytically explain the origin of Gamma distributions in polygon aspect ratios, linked to E².
• To introduce a pseudo-temperature concept for thermodynamic ensemble applications.

Main Methods:

• Analysis of 99 polygonal tessellations from various physical origins.
• Statistical analysis to identify distribution patterns of the deformation tensor.
• Development of an analytical approach to derive polygon aspect ratio distributions.

Main Results:

• Demonstrated the universal exponential distribution of E² in polygonal networks.
• Confirmed the resulting ubiquitous presence of Gamma distributions in polygon aspect ratios.
• Established a link between E² and energy forms, defining a pseudo-temperature.

Conclusions:

• The study reveals a fundamental, unifying principle governing the geometry of natural polygonal networks.
• The findings provide a thermodynamic framework for understanding these complex systems.
• The identified pseudo-temperature offers new avenues for research in statistical physics and materials science.