Pure Maths Colloquium: Action at a distance? The mysteries of aperiodic tilings.
24 May 2010 16:15 in CM221
How difficult could it be to tell whether a set of tiles can tile the plane? In fact it is impossible, in the 1960s Berger proved that the problem is undecidable. As a consequence there must be sets of shapes that can tile the plane but never periodically. Such sets of shapes are called aperiodic.
The original sets of aperiodic shapes were found by hand, with the star being the two Penrose tiles. Today there are a handful of general constructions. The earliest of these constructions involved hierarchical tilings generated by substitution rules. In fact (as proved by Goodman-Strauss in 1998) any substitution rule could be used to find a set of aperiodic tiles.