William Thurston (1990) describes a test for determining whether a simply-connected region, formed as the union of unit squares in the plane, has a domino tiling. He forms an undirected graph that has as its vertices the points (x, y, z) in the three-dimensional integer lattice, where each such point is connected to four neighbors: if x + y is even, then (x, y, z) is connected to (x + 1, y, z + 1), (x − 1, y, z + 1), (x, y + 1, z − 1), and (x, y − 1, z − 1), while if x + y is odd, then (x, y, z) is connected to (x + 1, y, z − 1), (x − 1, y, z − 1), (x, y + 1, z + 1), and (x, y − 1, z + 1). The boundary of the region, viewed as a sequence of integer points in the (x, y) plane, lifts uniquely (once a starting height is chosen) to a path in this three-dimensional graph. A necessary condition for this region to be tileable is that this path must close up to form a simple closed curve in three dimensions, however, this condition is not sufficient. Using more careful analysis of the boundary path, Thurston gave a criterion for tileability of a region that was sufficient as well as necessary.