The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by p...The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by polyominoes. Indeed, if all tilings of a (skewed) quadrant by a tile set can be reduced to a tiling by congruent rectangles (parallelograms), this provides information about tilings of rectangles (parallelograms). We consider a class of tile sets in a square lattice appearing from arbitrary dissections of rectangles in two L-shaped polyominoes and from symmetries of these tiles about the first bisector. Only translations of the tiles are allowed in a tiling. If the sides of the dissected rectangle are coprime, we show the existence of tilings of all (skewed) quadrants that do not follow the rectangular (parallelogram) pattern. If one of the sides of the dissected rectangle is 2 and the other is odd, we also show tilings of rectangles by the tile set that do not follow the rectangular pattern. If one of the sides of the dissected rectangle is 2 and the other side is even, we show a new infinite family of tile sets that follows the rectangular pattern when tiling one of the quadrants. For this type of dis-section, we also show a new infinite family that does not follow the rectangular pattern when tiling rectangles. Finally, we investigate more general dissections of rectangles, with. Here we show infinite families of tile sets that follow the rectangular pattern for a quadrant and infinite families that do not follow the rectangular pattern for any quadrant. We also show, for infinite families of tile sets of this type, tilings of rectangles that do not follow the rectangular pattern.展开更多
Let and let be the set of four ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by . Our main result shows a remarkable property of this set of tiles: any tiling of the first quadran...Let and let be the set of four ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by . Our main result shows a remarkable property of this set of tiles: any tiling of the first quadrant by , n even, reduces to a tiling by and rectangles, each rectangle being covered by two ribbon L-shaped n-ominoes. An application of our result is the characterization of all rectangles that can be tiled by , n even: a rectangle can be tiled by , n even, if and only if both of its sides are even and at least one side is divisible by n. Another application is the existence of the local move property for an infinite family of sets of tiles: , n even, has the local move property for the class of rectangular regions with respect to the local moves that interchange a tiling of an square by n/2 vertical rectangles, with a tiling by n/2 horizontal rectangles, each vertical/horizontal rectangle being covered by two ribbon L-shaped n-ominoes. We show that none of these results are valid for any odd n. The rectangular pattern of a tiling of the first quadrant persists if we add an extra tile to , n even. A rectangle can be tiled by the larger set of tiles if and only if it has both sides even. We also show that our main result implies that a skewed L-shaped n-omino, n even, is not a replicating tile of order k2 for any odd k.展开更多
文摘The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by polyominoes. Indeed, if all tilings of a (skewed) quadrant by a tile set can be reduced to a tiling by congruent rectangles (parallelograms), this provides information about tilings of rectangles (parallelograms). We consider a class of tile sets in a square lattice appearing from arbitrary dissections of rectangles in two L-shaped polyominoes and from symmetries of these tiles about the first bisector. Only translations of the tiles are allowed in a tiling. If the sides of the dissected rectangle are coprime, we show the existence of tilings of all (skewed) quadrants that do not follow the rectangular (parallelogram) pattern. If one of the sides of the dissected rectangle is 2 and the other is odd, we also show tilings of rectangles by the tile set that do not follow the rectangular pattern. If one of the sides of the dissected rectangle is 2 and the other side is even, we show a new infinite family of tile sets that follows the rectangular pattern when tiling one of the quadrants. For this type of dis-section, we also show a new infinite family that does not follow the rectangular pattern when tiling rectangles. Finally, we investigate more general dissections of rectangles, with. Here we show infinite families of tile sets that follow the rectangular pattern for a quadrant and infinite families that do not follow the rectangular pattern for any quadrant. We also show, for infinite families of tile sets of this type, tilings of rectangles that do not follow the rectangular pattern.
文摘Let and let be the set of four ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by . Our main result shows a remarkable property of this set of tiles: any tiling of the first quadrant by , n even, reduces to a tiling by and rectangles, each rectangle being covered by two ribbon L-shaped n-ominoes. An application of our result is the characterization of all rectangles that can be tiled by , n even: a rectangle can be tiled by , n even, if and only if both of its sides are even and at least one side is divisible by n. Another application is the existence of the local move property for an infinite family of sets of tiles: , n even, has the local move property for the class of rectangular regions with respect to the local moves that interchange a tiling of an square by n/2 vertical rectangles, with a tiling by n/2 horizontal rectangles, each vertical/horizontal rectangle being covered by two ribbon L-shaped n-ominoes. We show that none of these results are valid for any odd n. The rectangular pattern of a tiling of the first quadrant persists if we add an extra tile to , n even. A rectangle can be tiled by the larger set of tiles if and only if it has both sides even. We also show that our main result implies that a skewed L-shaped n-omino, n even, is not a replicating tile of order k2 for any odd k.
