THE CONVERGENCE OF BEAUTY: AN ANALYTICAL AND FRACTAL APPROACH TO THE SANGAKU CHAIN OF CIRCLES

Abstract

Considering the aesthetic and technical problem of the "Chain of Circles in an Acute Angle," recurring in the Japanese Sangaku tradition, and the need for didactic resources that connect synthetic geometry to the rigor of infinite series to overcome curricular fragmentation, it aims to analytically demonstrate the convergence of the total area occupied by this infinite succession of tangent circles, reducing the visual complexity of the problem to an elegant trigonometric solution. To this end, we proceed to an exploratory theoretical research, grounded in the deductive method and geometric modeling. In this way, it is observed that, through the identification of similarity properties and the application of geometric series summation, the total area results in a finite function strictly dependent on the initial radius and the opening angle. Which allows us to conclude that the beauty of the solution lies in the economy of the mathematical argument, reaffirming the effectiveness of synthetic geometry in simplifying iterative phenomena and providing a solid foundation for pedagogical practices aimed at integrating limits and plane geometry.