Decidability of Tiling with One or Two Prototiles

Determine whether the plane tiling decision problem is decidable when the input consists of either one or two simple-polygon prototiles whose tiles may be placed as isometric copies (with rotations and with reflections either allowed or forbidden).

Background

The paper proves that determining whether three given simple-polygon prototiles tile the plane is co-RE-complete (and hence undecidable), improving the prior best upper bound of five prototiles. This sharpens the frontier at which undecidability is known to occur for polygonal prototiles.

The authors note that periodic tilings can be found algorithmically by enumerating fundamental domains, implying that undecidability requires prototiles admitting only aperiodic tilings. Recent work by Smith, Myers, Kaplan, and Goodman-Strauss produced a single aperiodic monotile, removing an obvious barrier to undecidability with fewer prototiles. Nonetheless, whether the tiling problem is decidable for the cases of one or two prototiles remains unresolved.