A domino tiling of a region in the Euclidean plane is a tessellation of the region by dominos, shapes formed by the union of two unit squares meeting edgetoedge. Equivalently, it is a matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares.
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For some classes of tilings on a regular grid in two dimensions, it is possible to define a height function associating an integer to the nodes of the grid. For instance, draw a chessboard, fix a node A_{0} with height 0, then for any node there is a path from A_{0} to it. On this path define the height of each node A_{n + 1} (i.e. corners of the squares) to be the height of the previous node A_{n} plus one if the square on the right of the path from A_{n} to A_{n + 1} is black, and minus one else.
More details can be found in Kenyon & Okounkov (2005).
William Thurston (1990) describes a test for determining whether a simplyconnected 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 threedimensional 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), (x1,y,z+1), (x,y+1,z1), and (x,y1,z1), while if x+y is odd, then (x,y,z) is connected to (x+1,y,z1), (x1,y,z1), (x,y+1,z+1), and (x,y1,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 threedimensional 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.
The number of ways to cover a mbyn rectangle with mn / 2 dominoes, calculated independently by Temperley & Fisher (1961) and Kasteleyn (1961), is given by
The sequence of values generated by this formula for squares with m = n = 0, 2, 4, 6, 8, 10, 12, ... is
These numbers can be found by writing them as the Pfaffian of an mn by mn antisymmetric matrix whose eigenvalues can be found explicitly. This technique may be applied in many mathematicsrelated subjects, for example, in the classical, 2dimensional computation of the dimerdimer correlator function in statistical mechanics.
The number of tilings of a region is very sensitive to boundary conditions, and can change dramatically with apparently insignificant changes in the shape of the region. This is illustrated by the number of tilings of an Aztec diamond of order n, where the number of tilings is 2^{(n+1)n/2}. If this is replaced by the "augmented Aztec diamond" of order n with 3 long rows in the middle rather than 2, the number of tilings drops to the much smaller number D(n,n), a Delannoy number, which has only exponential rather than superexponential growth in n. For the "reduced Aztec diamond" of order n with only one long middle row, there is only one tiling.
Redirecting to Domino tiling
