Abstract: Case Studies in Biomechanics: Mathematics Inspired by Biology

Nature is filled with complex shapes and patterns, many of which arise from fundamental mathematical principles. In this talk, we explore how biomechanics and mathematics intersect to explain intriguing biological structures. First, we examine hyperbolic surfaces as seen in the natural curvature of lettuce leaves and the undulating forms of sea-slugs, where underlying geometric constraints shape both their structure and dynamic behavior. Next, we delve into the physics of structural coloration, where hierarchical buckling mechanisms create brilliant iridescence in butterfly wings. Through these case studies, we highlight the role of mathematical modeling and analysis in helping us understand these remarkable biological phenomena.