Undecidability and nonperiodicity for tilings of the plane

This paper is related to the work of Hao Wang and others growing out of a problem which he proposed in [8], w 4.1. Suppose that we are given a finite set of unit squares with colored edges, placed with their edges horizontal and vertical. We are interested in tiling the plane with copies of these tiles obtained by translation only. The tiles are to be placed with their vertices at lattice points, and abutting edges must have the same color. Wang raised the question whether there is a general method of deciding which finite sets of colored squares can be used to tile the plane in this way. He also discussed the relation of this problem to the decision problem for certain classes of formulas of the predicate calculus, but we shall consider only the geometrical problem here. Suppose that we have a tiling of the plane of this type which has a horizontal period. That is, we assume that the tiling remains invariant under a certain horizontal translation. There will then be a vertical strip which can be repeated to cover the plane. This strip has only a finite number of different horizontal cross sections, and hence has two which are alike. Thus the same tiles may be used to construct a tiling which has a vertical period as well as a horizontal period. A similar argument can be used even when the given period is not horizontal. That is, if a set of tiles permits a periodic tiling, then it also permits a doubly periodic tiling. In any such tiling, we can find equal horizontal and vertical periods, and hence can find a square of some size which repeats to cover the plane. Wang made the conjecture, since proved false, that any set of tiles which permits a tiling of the plane also permits a periodic tiling. He pointed out that if this conjecture were true, then we would have a decision method for an arbitrary set of tiles. Indeed, it would be sufficient to form all possible squares from the given set of tiles, starting with the smaller squares and working up, until we either reach a square which can be repeated periodically, or we find a square of a certain size which cannot be tiled at all. The latter will always happen if tiling of the whole plane