A Transformative Geometric Framework for Dihedral Groups: The Symmetry Density Index in Three-Dimensional Space
Akshay Kumar Vyas
Department of Mathematics, Government Motilal Vigyan Mahavidyalay, Bhopal, India.
Ram Milan Singh *
Department of Mathematics, Institute for Excellence in Higher Education, Bhopal, India.
Amardeep Singh
Department of Mathematics, Government Motilal Vigyan Mahavidyalay, Bhopal, India.
*Author to whom correspondence should be addressed.
Abstract
This paper introduces the Symmetry Density Index (SDI), a pioneering metric that redefines the analysis of dihedral group Dn actions in three-dimensional (3D) spaces. Departing from the traditional planar focus, we develop a comprehensive framework integrating advanced geometric invariants, rigorous theorems, and computational validations, supported by vibrant, multi-colored visualizations. Our findings reveal intricate symmetry distributions across 3D volumes, offering profound implications for fields such as computational topology, quantum chemistry, robotic kinematics, and materials science. By synthesizing algebraic rigor with spatial intuition, this work establishes a transformative paradigm for symmetry studies, poised to inspire groundbreaking interdisciplinary research.
Keywords: Symmetry Distribution Index (SDI), computational topology, quantum chemistry, robotic kinematics, materials science, computer vision, graph theory, bioinformatics, astrophysics, symmetry analysis