Combinatorial and Computational aspects of Tilings

Fractal tilings ("f-tilings") are described based on single prototiles with a variety of shapes. These prototiles have one or two long edges and two or more short edges, and the angles and scaling factors between long and short edges are in many cases irrational. The f-tilings are constructed by iterative arrangement, according to a simple matching rule, of successively smaller generations of tiles about a central group of largest-generation tiles. For the most part, these f-tilings do not cover the infinite plane, but rather are bounded and contain singular points. Within their boundaries, which are in most cases fractal curves, they contain neither gaps nor overlaps, and the f-tilings presented here are all edge-to-edge. For some of these f-tilings, tiles of a single generation will tile the plane and allow inflation and deflation by substitution. More recently, unicursal fractal knots have been designed using iterative substitution of a patch of tiles. This is made possible by first arranging a starting knot as a patch of tiles that contains individuals tiles similar in shape to the overall patch.