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Geometric and Algebraic Properties of Polyomino Tilings Michael Robert Korn

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Geometric and Algebraic Properties of Polyomino Tilings Michael Robert Korn View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DSpace@MIT Geometric and algebraic properties of polyomino tilings by Michael Robert Korn B.A., Princeton University, 2000 Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2004 c Michael Robert Korn, MMIV. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author.............................................................. Department of Mathematics May 4, 2004 Certified by. Igor Pak Assistant Professor of Mathematics Thesis Supervisor Accepted by . Rodolfo Ruben Rosales Chairman, Committee on Applied Mathematics Accepted by . Pavel Etingof Chairman, Department Committee on Graduate Students 2 Geometric and algebraic properties of polyomino tilings by Michael Robert Korn Submitted to the Department of Mathematics on May 4, 2004, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis we study tilings of regions on the square grid by polyominoes. A polyomino is any connected shape formed from a union of grid cells, and a tiling of a region is a collection of polyominoes lying in the region such that each square is covered exactly once. In particular, we focus on two main themes: local connectivity and tile invariants. Given a set of tiles T and a finite set L of local replacement moves, we say that a region Γ has local connectivity with respect to T and L if it is possible to convert any tiling of Γ into any other by means of these moves. If R is a set of regions (such as the set of all simply connected regions), then we say there is a local move property for T and R if there exists a finite set of moves L such that every Γ in R has local connectivity with respect to T and L. We use height function techniques to prove local move properties for several new tile sets. In addition, we provide explicit counterexamples to show the absence of a local move property for a number of tile sets where local move properties were conjectured to hold. We also provide several new results concerning tile invariants. If we let ai(τ) denote the number of occurrences of the tile ti in a tiling τ of a region Γ, then a tile invariant is a linear combination of the ai’s whose value depends only on Γ and not on τ. We modify the boundary-word technique of Conway and Lagarias to prove tile invariants for several new sets of tiles and provide specific examples to show that the invariants we obtain are the best possible. In addition, we prove some new enumerative results, relating certain tiling prob- lems to Baxter permutations, the Tutte polynomial, and alternating-sign matrices. Thesis Supervisor: Igor Pak Title: Assistant Professor of Mathematics 3 4 Acknowledgments I would first and foremost like to thank my advisor, Professor Igor Pak, for sug- gesting interesting problems, keeping me on track, and putting up with my crazy American attitudes about work. His knowledge of the subject and cheerful demeanor were greatly appreciated during the course of writing this thesis. Hearty thanks also go to Professor Daniel Kleitman and Professor Richard Stanley for being part of my thesis committee, providing helpful comments, and being very supportive of my studies. They have been excellent role models throughout my time at MIT. Thanks should also go to the Department of Defense and the American Society for Engineering Education, for providing me with a generous fellowship which made my graduate studies, and ultimately this thesis, possible. Furthermore, I would like to thank my officemates Andrew Brooke-Taylor, Rados Radoicic, Karen Bernhardt, and Frederic Rochon, the entire MIT math department, as well as Monica Linden, Carol Meyers, Ben Walter, and everyone else I’ve known here, for making this experience a pleasant one. Perhaps most importantly, I thank my parents for giving me reason to believe in myself. 5 6 Contents 1 Introduction 17 1.1Polyominoes................................ 17 1.2 Tilings ................................... 18 2Localmoves 23 2.1Introduction................................ 23 2.2Localconnectivityforsetsofregions.................. 25 2.2.1 Thesetofallregions....................... 25 2.2.2 Thesetofsimply-connectedregions............... 26 2.2.3 Thesetofrectangles....................... 27 2.3Provinglocalmoveproperties...................... 27 2.3.1 Heightfunctions......................... 27 2.3.2 Adhocmethods......................... 31 2.4Disprovinglocalmoveproperties.................... 31 3 Tile invariants 33 3.1Introduction................................ 33 3.2Provingtileinvariants.......................... 36 3.2.1 Coloringarguments........................ 36 3.2.2 Boundary word arguments . ................. 37 3.2.3 Localconnectivity........................ 39 3.3Dominoes................................. 39 3.4T-tetrominoes............................... 41 7 4 Tiling with T-tetrominoes 45 4.1Introduction................................ 45 4.2 Tiling rectangles with T-tetrominoes . ................. 46 4.3Chaingraphs............................... 47 4.4Heightfunctions.............................. 50 4.5Localconnectivityfromheightfunctions................ 54 4.6Thelatticestructureonheightfunctions................ 55 4.7Tileinvariants............................... 57 4.8Non-rectangularregions......................... 58 4.9 Enumeration of tilings and the Tutte polynomial ............ 60 4.10 Sampling of tilings ............................ 64 4.11Icegraphs................................. 66 4.11.1Heightontheicegraph...................... 68 4.12Non-quadruplicatedregions....................... 72 4.12.1Cutsandcornerlesspoints.................... 72 4.12.2Chaingraphsandheightfunctions............... 73 4.13 The diamond-shaped completable region Dk .............. 76 5Rectangles 83 5.1 Tiling rectangular regions with non-rectangular tiles .......... 83 5.2 Tiling thin rectangular regions with polyomino tiles .......... 86 5.3 Tiling with rectangular tiles ....................... 90 5.4Openquestions.............................. 92 6 Tiling rectangles with irrational rectangles 95 6.1Asetofirrationaltiles.......................... 95 6.2 Necessary conditions for tileability . ................. 96 6.3 Tilings and permutations ......................... 96 6.4Baxterpermutations........................... 98 6.5GeneralizingtootherrectanglesΓ.................... 100 6.6 Conditions for tileability revisited . ................. 103 8 6.7Localmoves................................ 104 7 Tiling with generalized dominoes 107 7.1Introduction................................ 107 7.2Theheightfunction............................ 108 7.3Thesizefunction............................. 112 7.4Localconnectivity............................. 114 7.5Finalremarks............................... 116 8 Tiling with polyominoes of height 2 117 8.1Introduction................................ 117 8.2HorizontalT-tetrominoesandhorizontalskew-tetrominoes...... 117 8.2.1 Tileinvariants........................... 118 8.3HorizontalT-tetrominoesandhorizontaldominoes........... 123 8.3.1 Theheightfunction........................ 124 8.3.2 Thelatticeofheightfunctions.................. 130 8.3.3 Localconnectivity........................ 132 8.3.4 Conclusion............................. 137 9 Tiling with skew-tetrominoes 139 9.1Introduction................................ 139 9.2Localmoves................................ 139 9.3Tileinvariants............................... 140 9.4Skewtetrominoesandsquaretetrominoes................ 145 10 A new approach to ribbon tiles 147 10.1Introduction................................ 147 10.2 Boundary words . ............................ 148 10.3Projections................................ 150 11 Augmenting untileable regions 155 11.1Introduction................................ 155 9 11.2Row-convexregions............................ 156 11.3Regionswherethisfails.......................... 157 11.4Threeormoredimensions........................ 160 12 A counterexample in three dimensions 163 12.1Introduction................................ 163 12.2Thecounterexample........................... 164 10 List of Figures 1-1Somepolyominoes............................. 17 1-2Someshapeswhicharenotpolyominoes................. 17 1-3Thetwelvefreepentominoes....................... 19 1-4 A tiling of a 6 × 10 rectangle using the twelve free pentominoes. 19 1-5 A regular checkerboard tiled with dominoes, and our modified checker- board.................................... 20 2-1 A local move applied to a domino tiling. ................ 23 2-2 A set of tiles which can tile any region in at most one way. Rotations arenotallowed............................... 26 2-3 A non-simply-connected region for which local connectivity does not hold..................................... 32 3-1 Pasting together copies of Γ and Γ to form the simply connected region Γ∗. ..................................... 35 3-2Dominoes.................................. 39 3-3 A region with two tilings which demonstrates that b2 mayvaryby2. 41 3-4T-tetrominoes............................... 41 3-5 A region with two tilings which demonstrates that b4 can vary freely. 42 3-6 A region with two tilings which demonstrates that b2 can vary by ex- actly4.................................... 43 4-1T-tetrominoes..............................
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