Two talks on kaleidoscopic tilings, for a general mathematical audience of students and faculty.  The purpose of the talks is to present an area of intriguing mathematical research, rich with problems suitable for undergraduate research.

A soccer ball has an attractive pattern of pentagons and hexagons on its surface, with a great deal of symmetry.  Baseballs and basketballs also have certain patterns of symmetry which are different from the soccer ball  pattern. Though the sportsman might never ask, a mathematician would be intrigued by the possibility of  a higher genus soccer ball (a soccer ball with patterned handles). It turns out that they exist in great abundance though we need to give up on having only hexagons and pentagons.

The key to creating and understanding “soccer balls” are kaleidoscopic tilings of the 2-dimensional geometries: the sphere, the euclidean plane and the hyperbolic plane.  The sphere tilings, of course, yield the patterns of  sports balls.  The tilings of the euclidean and  hyperbolic planes form beautiful patterns and have their own artistic interest, as in some of the art of Escher. The higher genus soccer balls, though impractical, are a  convenient mental hook for generating questions about patterned surfaces, e.g., constructing simple examples.

 In the first talk the relation between (higher genus) soccer balls and tilings will be explored, including an introduction to hyperbolic geometry.  In the second talk I will present some work completed jointly by undergraduates and myself on divisible kaleidoscopic tilings, i.e., simultaneous tilings of the plane by two different kaleidoscopic polygons.  It has a nice interplay between combinatorics (Catalan numbers) and geometry.