Heesch's problem

Consider the non-convex polygon P shown in the figure to the right, which is formed from a regular hexagon by adding projections on two of its sides and matching indentations on three sides. The figure shows a tessellation consisting of 61 copies of P, one large infinite region, and four small diamond-shaped polygons within the fourth layer. The first through fourth coronas of the central polygon consist entirely of congruent copies of P, so its Heesch number is at least four. One cannot rearrange the copies of the polygon in this figure to avoid creating the small diamond-shaped polygons, because the 61 copies of P have too many indentations relative to the number of projections that could fill them. By formalizing this argument, one can prove that the Heesch number of P is exactly four. According to the modified definition that requires that coronas be simply connected, the Heesch number is three. This example was discovered by Robert Ammann. == Known results ==

It is unknown whether all positive integers can be Heesch numbers. The first examples of polygons with Heesch number 2 were provided by , who showed that infinitely many polyominoes have this property. Casey Mann has constructed a family of tiles, each with the Heesch number 5. Mann's tiles have Heesch number 5 even with the restricted definition in which each corona must be simply connected. In 2020, Bojan Bašić found a figure with Heesch number 6, the highest finite number until the present. For the corresponding problem in the hyperbolic plane, the Heesch number may be arbitrarily large. ==References==