Simetr\'ias Algebraicas en Geometr\'ia y Arquitectura: Un Enfoque desde Grupos de Transformaciones

Abstract

This article develops an algebraic-geometric theoretical framework for the study of central, axial, and rotational symmetries in R2 and R3, with applications in the classification of conic and quadric surfaces through transformation groups. An analytical methodology based on group theory and geometric invariants is employed, complemented by numerical modeling using the Finite Element Method. As a result, it is demonstrated that the D8 symmetry of the Pantheon structurally optimizes stress distribution, while the hexagonal symmetry rho_{\pi/3} of the Soumaya Museum offers both aesthetic and structural advantages. The approach establishes a rigorous link between algebraic abstraction, geometry, and architectural design.