heesch’s problem (13. 083)

Ranging from zero (in the case of the circle) and infinity for squares—with seemingly few values in between—in the study of tessellations (see previously here, here and here) a Heesch number pertaining to a geometric shape is the maximum number of layers of identical copies of the same figure will bear with no gaps or overlaps. Named for the geometer and mathematician Heinrich Heesch, who also made significant contributions to the field of tiling patterns and then unproven for colour theorem (the first mathematical proof by a computer) for mapping boundaries, he noticed that a one sort of planar shape, a square fused with a triangle would only accommodate one extra layer, as illustrated with these spandrels (from an architectural space between the top of arch and the ceiling) term referring to the teardrop arranged by contemporary Walther Lietzmann, and posed it as a general puzzle. Beyond the core, one can only form a signal corona of identical shapes—and whilst blocky polyominoes and tetrominoes seem to hold limitless promise at first glance, there still seems to be a limiting factor with no more than six deep.