An Aperiodic Set of Two Convex Polygons

Croft, Falconer and Guy asked whether an aperiodic set of two, or even one, convex poly- gons with no matching rules exists; the smallest previously known such sets have three tiles (Ammann). We give a set of two, with a computer-assisted proof of aperiodicity. Smith, Myers, Kaplan and Goodman-Strauss found a two-parameter family Tile(a,b) of non-convex polygons, containing the “hat”, each of which tiles the plane but never with a translational symmetry. We cut one member, Tile(299,240), into five convex pieces: one heptagon T0, three pentagons T1 and one mirror-image pentagon T1. Using exact arithmetic we enumerate, for each prototile, all ways it can be surrounded by tiles of the set (a “corona”) and prune those that cannot be extended. The surviving coronas show that in every tiling by T0, T1 and their reflections, with no matching rules and no edge-to-edge requirement, the tiles group uniquely into copies of Tile(299,240). Hence every such tiling is the dissection of a tiling by the parent tile and has no translational symmetry, and the set admits such tilings, so it is aperiodic. The argument is computer assisted and has not been peer reviewed; we describe the checks that were made (an independent second implementation, mutation tests, and a test against genuine tilings obtained from the hat substitution system) and the assumptions on which the result depends. Because no convex polygon is an aperiodic monotile (Rao, unrefereed), two prototiles is, conditionally, the smallest possible number for convex polygons.