Mirrors, Escher, and Symmetries in Disguise: The Coxeter Group Isomorphism Problem-MAAD Talk-(Mathematical Adventures and Diversions)

Abstract: When the artist M.C. Escher created his famous Circle Limit prints—teeming with intricate repeating figures in non-Euclidean space—he was directly inspired by conversations with Canadian mathematician H.S.M. Coxeter. At the heart of Escher's patterns is a simple geometric idea: reflections. By placing mirrors across space, reflections act as building blocks to generate endless arrays of symmetry, forming algebraic objects known as Coxeter groups.
In this talk, we will trace the journey of Coxeter groups from visual art and geometric tilings (in Euclidean, hyperbolic,and line spaces) to linear algebra and abstract group theory. But a fascinating puzzle lies hidden in these structures: the isomorphism problem. How can two groups defined by completely different sets of mirrors, angles, and symbols end up describing the exact same underlying symmetry in disguise?
Using tools you already know—like vectors, inner products, and matrices—we will explore how mathematicians uncover these secret connections. No advanced abstract algebra is required—just an appreciation for geometry, history, and symmetry!