On Tilable Orthogonal Polygons∗

We consider rectangular tilings of orthogonal polygons with vertices located at integer lattice points. Let G be a set of reals closed under the usual addition operation. A G-rectangle is a rectangle at least one of whose sides is in G. We show that if an orthogonal polygon without holes can be tiled with G-rectangles then one of the sides of the polygon must be in G. As a special case this solves the conjecture that domino tilable orthogonal polygons must have at least one side of even length. We also explore separately the case of othogonal polygons placed in a chessboard. We establish a condition which determines the number of black minus white squares of the chessboard occupied by the polygon. This number depends exclusively on the parity sequence of the lengths of the sides of the orthogonal polygon. This approach produces a different proof of the conjecture of the non domino-tilability of of orthogonal polygons without even length sides. We also give some generalizations for polygons with holes and polytopes in 3 dimensions. ∗An extended abstract of this paper has appeared in the proceedings of the 11th Canadian Conference on Computational Geometry, CCCG’99, held in Vancouver, Aug. 15-18, 1999, pp. 157-161, 1999. ¶Math. Inst. of the Hungarian Acad. of Sci., P.O. Box 127, 1364 Budapest, Hungary and SUNY at Brooklyn, Brooklyn, NY 11203. email:[email protected] §Département d’Informatique, Université du Québec à Hull, Hull, Québec J8X 3X7, Canada. email: [email protected] ‖Department of Computer Science, University of Liverpool, Peach Street, L69 7ZF, Liverpool, UK. email:[email protected] ∗Carleton University, School of Computer Science, Ottawa, ON, K1S 5B6, Canada. email: [email protected] ††Departamento di Matematicas, Universiad Autonoma Metropolitana-Iztapalapa, Mexico, email: [email protected] ‡University of Ottawa, School of Information Technology and Engineering, Ottawa, ON, K1N 9B4, Canada. email: [email protected] †Research supported in part by NSERC (National Science and Engineering Research Council of Canada) grant. ∗∗Research supported in part by NUF-NAL (The Nuffield Foundation Awards to Newly Appointed Lecturers) award.