Heesch number 4 example (First discovered by W.R. Marshall, and then independently by me) Heesch number 5 example (discovered by me):
This last example has the largest known finite Heesch number. Of course, a prototile that can be used to tile the entire plane has infinite Heesch number, but I am interested in those prototiles that cannot be used to tile the plane. Also, it has been customary in the past to require that the coronal configurations be simply connected (that is, there are no holes), but if we relax this condition, we see some can have even higher Heesch numbers. For example, several people have noticed that Ammann's example under the relaxed rules has Heesch number 4: The Heesch number 5 example from above cannot, however, attain 6 coronas, even if we relax the simple connectivity requirement. So the big question is this: Is there a maximum Heesch number? That is, is there some number N so that when any tile surrounds itself N times, then it must tile the plane? If so, what is N? At present, there is not an answer to this question. It has been conjectured by well-known and talented people in the field of tiling theory that there is a maximum (although they cannot say what that maximum is). I have also heard it conjectured that there is no bound on the Heesch number! I am reluctant to make a conjecture myself. But I can say that after Ammann's example was given, many people thought that there would not be another example with higher Heesch number. Heesch's problem has connections to a few well-known unsolved problems -- the domino problem and the "Einstein" problem. The domino problem asks if there exists an algorithm that, when given a prototile as input, decides if the prototile can be used tile the entire plane. If the Heesch number is in fact bounded, this gives a simple algorithm for deciding if a prototile can be used to tile the plane: Suppose the maximum Heesch number is N. One just places tiles in a systematic fashion until either he cannot proceed any further, in which case the prototile cannot be used to tile the plane, or until he has placed more than N coronas of tiles, in which case the tile must tile the plane. The domino problem in turn has a deep connection with the "Einstein" problem. The Einstein problem asks if there exists a single aperiodic prototile (Ein = one, stein = tile). The nonexistence of a single aperiodic prototile implies the the existence of a decision method for the domino problem. |