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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.

KEYWORDS

Polyomino, L-Shaped Polyomino, Skewed L-Shaped Polyomino, Tiling Rectangles, Tiling Quadrants, Tiling Parallelograms, Rectangular Pattern for Tiling Quadrants-Rectangles

Cite this paper

Nitica, V. 2016 On Tilings of Quadrants and Rectangles and Rectangular Pattern. Open Journal of Discrete Mathematics, 6, 351-371. doi: 10.4236-ojdm.2016.64028.





Autor: Viorel Nitica

Fuente: http://www.scirp.org/



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