Heesch's problem

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Amman's example of a polygon with Heesch number 4

Heesch's problem, named for geometer Heinrich Heesch, concerns the number of layers of congruent copies of a geometric figure (usually a polygon) that can surround that figure. For instance, a square may be surrounded by infinitely many layers of congruent squares in the square tiling, while a circle cannot be surrounded by even a single layer of congruent circles without leaving some gaps. More complicated examples such as the one shown here can be surrounded by a nonzero number of layers, but not infinitely many. Geometers have constructed examples of polygons that may surround themselves for k layers but no more, for 0≤k≤5, but it is not known whether similar results are possibly for any k≥6.

[edit] Formal Definitions

A tessellation of the plane is a partition of the plane into smaller regions called tiles. The zeroth corona of a tile is defined as the tile itself, and for k>0 the kth corona is the set of tiles sharing any boundary point with the (k-1)st corona. The Heesch number of a figure S is the maximum value k such that there exists a tiling of the plane, and tile t within that tiling, for which that all tiles in the zeroth through kth coronas of t are congruent to S. In some work on this problem this definition is modified to additionally require that the union of the zeroth through kth coronas of t is simply connected.

If there is no limit on the number of layers by which a region may be surrounded, we say that its Heesch number is infinite. In this case, an argument based on König's lemma can be used to show that there exists a tessellation of the whole plane by copies of the polygon.

[edit] Example

For example, consider the polygon shown in the figure, an example discovered by Robert Amman and 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 the polygon, one large infinite region, and four small diamond-shaped polygons. The first through fourth coronas of the central polygon consist entirely of congruent copies of that polygon, so it has Heesch number 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 the polygon have too many indentations relative to the number of projections that could fill them; by formalizing this argument one can prove that the polygon in the figure has Heesch number exactly four. According to the modified definition with simply connected coronas, its Heesch number is instead three.

[edit] References

Eppstein, David. The Geometry Junkyard: Heesch's Problem. Retrieved on 2006-09-05.

Fontaine, Anne (1991). "An infinite number of plane figures with Heesch number two". Journal of Combinatorial Theory Series A 57 (1): 151–156. doi:10.1016/0097-3165(91)90013-7.