Nonrecursive tilings of the plane. II

Abstract

We show that there is a finite set of tiles which can tile the plane but not in any recursive way. This answers a natural sequel to Hao Wang's problem of the existence of a finite set of tiles which can tile the plane but not in any periodic way. In the proof, an elaboration of Robinson's method of transforming origin-constrained problems into unconstrained problems is applied to Hanf's origin-constrained tiling of Part I. We will assume familiarity with §§2, 3, and 7 of [3].Following Robinson, we will mark the edges of our tiles with symbols and configurations of arrow heads and tails as well as with colors. The matching condition for abutting tiles will be that the symbols on adjacent edges must be identical, every arrow head must match with a tail, every tail with a head, and the colors must be the same. This is, of course, mathematically equivalent to the original condition which involved only colors. A tiling of the plane by a set of tiles is a covering of the plane with translated copies of the tiles such that adjacent edges of abutting tiles satisfy the above matching condition. A set of tiles is consistent if the plane can be tiled by the set. A set of tiles with a designated origin tile is origin-consistent if there is a tiling of the plane with the origin tile at the center.A square block of tiles is a tiling if every pair of abutting tiles satisfies the above matching condition. If an origin tile has been designated, a block of tiles is an origin-constrained tiling if it is a tiling with the origin tile at the center. Two blocks have the “same” center row if the blocks are of the same size and have identical center rows or if the smaller block's center row is a centered segment of the larger block's center row.