Dynamical Systems Seminar

We consider tilings of deficient rectangles by the set \(\mathcal{T}_4\) of ribbon \(L\)-tetrominoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is \((4m+1) \times (4m+1)\) and in an even position if the square is \((4m+3)\times (4m+3)\). The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a \(2\times 4\) rectangle. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of \(3\times 3\) squares that overlap over one of the corner cells. The crack divides the square in two parts of equal area. The number of tilings of a \((4m+1)\times (4m+1)\) deficient square by \(\mathcal{T}_4\) is equal to the number of tilings by dominoes of a \(2m\times 2m\) square. The number of tilings of a \((4m+3)\times (4m+3)\) deficient square by \(\mathcal{T}_4\) is twice the number of tilings by dominoes of a \((2m+1)\times (2m+1)\) deficient square, with the missing cell placed on the main diagonal. In both cases the counting is realized by an explicit function which is a bijection in the first case and a double cover in the second. The crack in a square naturally propagates to a crack in a larger square.