On Tilings of Quadrants and Rectangles and Rectangular Pattern

Viorel  Nitica

doi:10.4236/ojdm.2016.64028

Viorel Nitica

Abstract: 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.

References

[1] Golomb, S.W. (1954) Checker Boards and Polyominoes. American Mathematical Monthly, 61, 675-682.
http://dx.doi.org/10.2307/2307321

[2] Golomb, S.W. (1994) Polyominoes, Puzzles, Patterns, Problems, and Packings. Princeton University Press, Princeton.

[3] Golomb, S.W. (1989) Polyominoes which Tile Rectangles. Journal of Combinatorial Theory, Series A, 51, 117-124.
http://dx.doi.org/10.1016/0097-3165(89)90082-4

[4] Klarner, D.A. (1969) Packing a Rectangle with Congruent N-Ominoes. Journal of Combinatorial Theory, 7, 107-115.
http://dx.doi.org/10.1016/S0021-9800(69)80044-X

[5] Reid, M. (2014) Many L-Shaped Polyominoes Have Odd Rectangular Packings. Annals of Combinatorics, 18, 341-357.
http://dx.doi.org/10.1007/s00026-014-0226-9

[6] Chao, M., Levenstein, D., Nitica, V. and Sharp, R. (2013) A Coloring Invariant for Ribbon L-Tetrominoes. Discrete Mathematics, 313, 611-621.
http://dx.doi.org/10.1016/j.disc.2012.12.007

[7] Nitica, V. (2015) Every Tiling of the First Quadrant by Ribbon L n-Ominoes Follows the Rectangular Pattern. Open Journal of Discrete Mathematics, 5, 11-25.
http://dx.doi.org/10.4236/ojdm.2015.52002

[8] Nitica, V. (2016) Signed Tilings by Ribbon L n-Ominoes, n Odd, via Gröbner Bases. Open Journal of Discrete Mathematics, 6, 297-313.
https://arxiv.org/abs/1601.00558

[9] Gill, K. and Nitica, V. (2016) Signed Tilings by Ribbon L n-Ominoes, n Even, via Gröbner Bases. Open Journal of Discrete Mathematics, 6, 185-206.
http://dx.doi.org/10.4236/ojdm.2016.63017

[10] Calderon, A., Fairchild, S., Nitica, V. and Simon, S. (2016) Tilings of Quadrants by L-Ominoes and Notched Rectangles. Topics in Recreational Mathematics, 7, 39-75.

[11] Golomb, S.W. (1964) Replicating Figures in the Plane. Mathematical Gazette, 48, 403-412.
http://dx.doi.org/10.2307/3611700

[12] Nitica, V. (2003) Rep-Tiles Revisited. In: Katok, S., Sossinsky, A. and Tabachnikov, S., Eds., MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics, American Mathematical Society, Providence, 205-217.

[13] Yang, J. (2014) Rectangular Tileability and Complementary Tileability Are Undecidable. European Journal of Combinatorics, 41, 20-34.
http://dx.doi.org/10.1016/j.ejc.2014.03.008