Conformal Tilings of the Plane: Foundations, Theory, and Practice
Analysis and Geometry Seminar
8th May 2018, 3:00 pm – 4:00 pm
Howard House, 2nd Floor Seminar Room
(Joint with Phil Bowers and Rick Kenyon) Certainly one of the most famous tilings of the plane is the Penrose tiling. The common "kite/dart" form is an aperiodic tiling having just two euclidean tile shapes - it is visually quite fascinating. This is also a "subdivision tiling", one associated with a rule for subdividing tiles into subtiles of the same shapes. What happens if one disregards the geometry entirely and considers the "combinatorics" alone - the abstract pattern of tiles, who is next to whom? We start with such combinatorics, impose canonical conformal structures, and find a new family of geometric tilings, the "conformal" tilings. In this talk I will introduce these, mention some basic theory, and show how to visualize them in practice via circle packing. Using the resulting experimental capabilities, we will discover that the geometry of traditional tilings, like the Penrose, emerge spontaneously from their combinatorics. We will, indeed, find a whole new playground for those captivated by these intricate objects. This will be a visually based talk and no particular background is required.