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Mathematics Colloquium

University of Tennessee

Title: Conformal Tilings of the Plane: Foundations, Theory, and Practice
Date: Friday, October 5, 2018
Place and Time: Room 101, Love Building, 3:35-4:25 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. (Joint work with Phil Bowers) The most famous tiling of the plane is probably the Penrose tiling. It is an aperiodic tiling that uses just a finite number of euclidean tile shapes and is visually fascinating. It is also a "subdivision tiling", one having a rule for subdividing tiles into subtiles of the same shapes. What happens if one disregards the geometry of such a tiling entirely and considers only the "combinatorics" --- the abstract pattern of tiles, who is next to whom? We start with such combinatorics, impose canonical conformal structures, and find a wonderfully rich 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 (plus, Phil will be happy to show you how to run CirclePack).