Heesch's problem

In this case, an argument based on Kőnig's lemma can be used to show that there exists a tessellation of the whole plane by congruent copies of the tile.

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.

The first examples of polygons with Heesch number 2 were provided by Fontaine (1991), who showed that infinitely many polyominoes have this property.

Mann's tiles have Heesch number 5 even with the restricted definition in which each corona must be simply connected.