Visualizing Number Theory: Animated Proofs for Centered Polygonal Numbers

Abstract

Polygonal and centered polygonal numbers, rooted in ancient geometric traditions, bridge number theory and geometry through their elegant patterns. This article synthesizes their history and presents key identities, with a central focus on introducing animated proofs as a powerful tool. Moving beyond static “proofs without words,” these dynamic visualizations are designed to make the underlying geometric transformations intuitive and accessible. Each identity is demonstrated with both a formal algebraic proof and a corresponding animation, showcasing how this dual approach can significantly enhance conceptual understanding and engagement in number theory.