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 Sources Further reading External links Numberphile video about Heesch Numbers. Tessellation