From Plato to Plateau: Natural Tilings and Soft Cells

Abstract: Mosaic patterns and tilings are ubiquitous in nature, design, and art, appearing in systems ranging from cellular tissues, geological formations, biological shells as well as architecture. Traditionally, these structures have been modelled using polyhedral tilings composed of flat faces, straight edges, and sharp corners. There is an immense mathematical literature on the geometry of these structures going back thousands of year. However, careful observation reveals that many natural tilings deviate significantly from this paradigm: their boundaries are curved with smooth interfaces. In the spirit of applied mathematics, this realisation has motivated the introduction of a new class of shapes known as soft cells, which arise as smooth deformations of standard tilings. Such cells are also found in the geometry of metal and liquid foam as well as in many micro-structures modelled by triply periodic minimal surfaces. In this talk, I will explain the mathematics and physics of tilings, hard and soft, pure and applied, describe their construction and classification, and illustrate how they provide a more accurate geometric description of patterns found in biology, architecture, engineering, in the deepest sea and even in space.