METALLIC RATIOS AND THEIR SYMMETRY GROUPS: A DEEP ARITHMETIC-GEOMETRIC PERSPECTIVE

This paper establishes profound connections between metallic ratios, Diophantine equations, and their underlying symmetry groups. Building on recent work on Diophantine equations and their solutions, we develop a comprehensive theory revealing the algebraic and geometric structures governing families of Diophantine equations associated with metallic ratios. Through detailed investigations of automorphism groups, continued fractions, and arithmetic geometry, we provide complete proofs and explicit examples that illuminate the deep arithmetic properties of metallic ratios. Our work offers a unified framework bridging number theory, group theory, and algebraic geometry, with applications to class field theory and computational number theory.