Tilings with T and Skew Tetrominoes

We consider tiling problems in the integer lattice. Specifically, we look at a set of four T-tetrominoes and four skew tetrominoes and determine when this set can tile rectangles and modified rectangles. Local considerations and coloring arguments are the main methods used to prove the untileability of regions

Tilings of Annular Region

We present our summer research on mathematical tiling. We classified which rectangular annular regions are tileable by the set of T and skew tretrominoes. We present a partial proof of this result, and discuss some of the context for this problem

The Topology of Tile Invariants

In this note we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures we show that the tile counting group associated to a set $T$ of tiles and a set of regions tileable by $T$ is isomorphic to a quotient of the second homology group of a 2-complex built from $T$. In this topological setting we derive some well-known tile invariants, one of which we apply to the solution of a tiling question involving modified rectangles.Comment: 25 pages, 24 figure

Geometric and algebraic properties of polyomino tilings

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.Includes bibliographical references (p. 165-167).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.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 [Delta] has local connectivity with respect to T and L if it is possible to convert any tiling of [Delta] 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 r 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(t) denote the number of occurrences of the tile ti in a tiling t of a region [Delta], then a tile invariant is a linear combination of the ai's whose value depends only on t and not on r.(cont.) 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 problems to Baxter permutations, the Tutte polynomial, and alternating-sign matrices.by Michael Robert Korn.Ph.D

Ribbon Tilings and Multidimensional Height Functions

We fix $n$ and say a square in the two-dimensional grid indexed by $(x,y)$ has color $c$ if $x+y \equiv c \pmod{n}$. A {\it ribbon tile} of order $n$ is a connected polyomino containing exactly one square of each color. We show that the set of order-$n$ ribbon tilings of a simply connected region $R$ is in one-to-one correspondence with a set of {\it height functions} from the vertices of $R$ to $\mathbb Z^{n}$ satisfying certain difference restrictions. It is also in one-to-one correspondence with the set of acyclic orientations of a certain partially oriented graph. Using these facts, we describe a linear (in the area of $R$) algorithm for determining whether $R$ can be tiled with ribbon tiles of order $n$ and producing such a tiling when one exists. We also resolve a conjecture of Pak by showing that any pair of order-$n$ ribbon tilings of $R$ can be connected by a sequence of local replacement moves. Some of our results are generalizations of known results for order-2 ribbon tilings (a.k.a. domino tilings). We also discuss applications of multidimensional height functions to a broader class of polyomino tiling problems.Comment: 25 pages, 7 figures. This version has been slightly revised (new references, a new illustration, and a few cosmetic changes). To appear in Transactions of the American Mathematical Societ

A new mathematical model for tiling finite regions of the plane with polyominoes

We present a new mathematical model for tiling finite subsets of $\mathbb{Z}^2$ using an arbitrary, but finite, collection of polyominoes. Unlike previous approaches that employ backtracking and other refinements of `brute-force' techniques, our method is based on a systematic algebraic approach, leading in most cases to an underdetermined system of linear equations to solve. The resulting linear system is a binary linear programming problem, which can be solved via direct solution techniques, or using well-known optimization routines. We illustrate our model with some numerical examples computed in MATLAB. Users can download, edit, and run the codes from http://people.sc.fsu.edu/~jburkardt/m_src/polyominoes/polyominoes.html. For larger problems we solve the resulting binary linear programming problem with an optimization package such as CPLEX, GUROBI, or SCIP, before plotting solutions in MATLAB

Locked Polyomino Tilings

A locked $t$-omino tiling is a grid tiling by $t$-ominoes such that, if you remove any pair of tiles, the only way to fill in the remaining $2t$ grid cells with $t$-ominoes is to use the same two tiles in the exact same configuration as before. We exclude degenerate cases where there is only one tiling overall due to small dimensions. It is a classic (and straightforward) result that finite grids do not admit locked 2-omino tilings. In this paper, we construct explicit locked $t$-omino tilings for $t \geq 3$ on grids of various dimensions. Most notably, we show that locked 3- and 4-omino tilings exist on finite square grids of arbitrarily large size, and locked $t$-omino tilings of the infinite grid exist for arbitrarily large $t$. The result for 4-omino tilings in particular is remarkable because they are so rare and difficult to construct: Only a single tiling is known to exist on any grid up to size $40 \times 40$. Locked $t$-omino tilings arise as obstructions to widely used political redistricting algorithms in a model of redistricting where the underlying census geography is a grid graph. Most prominent is the ReCom Markov chain, which takes a random walk on the space of redistricting plans by iteratively merging and splitting pairs of districts (tiles) at a time. Locked $t$-omino tilings are isolated states in the state space of ReCom. The constructions in this paper are counterexamples to the meta-conjecture that ReCom is irreducible on graphs of practical interest