Planar tilings and the search for an aperiodic prototile

The most surprising planar tiling result in recent decades was the discovery in the 1960s of aperiodic sets of prototiles, that is sets of tiles which admit tilings of the Euclidean plane but where all such tilings are nonperiodic. In 1914, Roger Penrose discovered his famous aperiodic set consisting of just two prototiles. Since then, the outstanding open question has been whether there exists an aperiodic set containing just a single prototile. In an attempt to find such a prototile, I developed some tiling software which takes an arbitrary polytile (a class of promising types of tiles made by attaching together congruent copies of basic tiles) and tries either to find a periodic tiling or to prove that it does not tile the plane. Using this software, I eliminated millions of potential aperiodic prototiles; and determined the tiling status of every polyomino up through order fourteen, every polyhex up through order seven, and every aligned polyklein up through order twelve. Additionally, I found five tiles where the size of a minimal translational patch in any tiling admitted by the tile is larger than for any previously known tile (including one tile where the size is three times as large as the previously published record).