Consider an rectangle and rectangular tiles of size 1×2 (dominoes). A domino tiling of the rectangle is a placement of dominoes that covers the rectangle completely without overlaps. A tiling exists if and only if and are not both odd, implying is even. One tiling can readily be found: suppose is even, place dominoes vertically in the first column and repeat for the next columns. This Demonstration generates random tilings of rectangles of chosen sizes and computes the total number of tilings possible.

If is the number of tilings of an rectangle, it is easy to see that . As and , is a Fibonacci number. The original motivation for tackling the tiling problem by dominoes (or dimers) was to provide a simple model to describe the thermodynamic behavior of fluids. The amazing formula for was first obtained by Kasteleyn in 1961: , which is the formula used in this Demonstration.

References

[1] P. Kasteleyn, "The Statistics of Dimers on a Lattice I. The Number of Dimer Arrangements in a Quadratic Lattice," Physica, 27, 1961 pp. 1209–1225.

[2] R. Kenyon and A. Okounkov, "What Is...a Dimer?"Notices of the American Mathematical Society, 52(3), 2005 pp. 342–343.