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Heesch's problem
In geometry, the Heesch number of a shape is the maximum number of layers of copies of the same shape that can surround it. Heesch's problem is the problem of determining the set of numbers that can be Heesch numbers. Both are named for geometer Heinrich Heesch,〔. As cited by and .〕 who found a tile with Heesch number 1 (the union of a square, equilateral triangle, and 30-60-90 right triangle) and proposed the more general problem.〔.〕
For example, 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. The Heesch number of the square is infinite and the Heesch number of the circle is zero. In more complicated examples, such as the one shown in the illustration, a polygonal tile can be surrounded by several layers, but not by infinitely many; the maximum number of layers is the tile's Heesch number.
== 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 ''k''th corona is the set of tiles sharing a boundary point with the (''k'' − 1)th 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 ''k''th 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 ''k''th coronas of ''t'' is a simply connected region.〔
If there is no upper bound on the number of layers by which a tile may be surrounded, its Heesch number is said to be 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 congruent copies of the tile.〔, 3.8.1 The Extension Theorem, p. 151.〕
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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