An Aperiodic Set of Six Square Prototiles

We present Q, an aperiodic tileset consisting of six edge-marked square prototiles with standard nearest-neighbor matching rules. All six prototiles are freely rotatable; two (T1 and T2) admit reflection, while the remaining four (T0, T3, T4, T5) do not. A central challenge in the theory of aperiodic tilings is reducing the number of prototiles re- quired to force aperiodicity. For square Wang tiles under the standard convention of translation- only placement, Jeandel and Rao established an eleven-tile minimal set; Robinson’s earlier six-tile construction, recast as a freely rotatable edge-marked alphabet, requires fourteen deco- rated prototiles. Jeandel and Rao’s minimality result, however, is proved specifically for tilesets restricted to translation, and we are not aware of a comparable minimality result once free ro- tation is permitted – a distinction the field treats as substantive, marking one of the principal relaxations of Wang’s original convention. The aperiodicity of Q is established via mutual local derivability (MLD) with the trilobite and crab tileset of Goodman-Strauss. To our knowledge, Q is the smallest known freely rotatable aperiodic set of edge-marked squares with standard nearest-neighbor matching rules, improving on the fourteen-tile freely rotatable form of Robinson’s tileset; the eleven-tile Jeandel-Rao set is smaller still, but is defined without free rotation, so the two counts are not directly comparable.