Aperiodic tilings: from the Domino problem to an aperiodic monotile

Mike Whittaker (University of Glasgow )

Thursday 28th March 16:00-17:00 Maths 311B

Abstract

Almost 60 years ago, Hao Wang posed the Domino Problem: is there an algorithm that determines whether a given set of square prototiles, with specified matching rules, can tile the plane? Robert Berger proved the undecidability of the Domino Problem by producing a set of 20,426 prototiles that tile the plane, but any such tiling is nonperiodic (lacks any translational symmetry). This remarkable discovery began the search for other (not necessarily square) aperiodic prototile sets, a finite collection of prototiles that tile the plane but only nonperiodically. In the 1970s, Roger Penrose reduced this number to two. Penrose's discovery led to the planar einstein (one-stone) problem: is there a single aperiodic prototile? In a crowning achievement of tiling theory, the existence of an aperiodic monotile was resolved almost a decade ago by Joshua Socolar and Joan Taylor. My talk will be somewhat expository, and culminate in a new direction in aperiodic tiling theory.

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