The mathematics of tiling has undergone a transformation from its roots in "recreational mathematics" many years ago to its status today as a lively area of research with fundamental ties to combinatorics, group theory, and topology. I first encountered the charm of tiling questions in my own capstone undergraduate project, which centered on the groundbreaking paper by John Conway and JC Lagarias, [2]. Now I have mentored eight students on four student-inspired research projects in the mathematics of tiling. These research experiences, which have been funded partially by Linfield College Student-Faculty Collaborative Research Grants and partially by the National Science Foundation, have resulted in at least six regional and national presentations by students, and four publications.

What follows is a brief overview of the student-faculty collaborative research projects I have mentored, with links to our publications. Generally speaking, our research has focused on finding and using tile invariants to solve tiling questions in the integer lattice. A tile invariant is a linear combination among the number of copies of each tile that must persist in any tiling of a region. An overview of student-centered research on tiling questions will appear soon as a chapter in a book on undergraduate research [3].

Tiling modifed rectangles with ribbon tiles (2008) with Ben Coate at The College of Idaho. We worked on questions involving ribbon tile tetrominoes and rectangles whose lower-right squares have been removed. Ben presented our work at the Regional Meeting of the MAA at Carroll College, in June, 2009. Our main result can be summarized as follows:

Let \(\Gamma_m(a,b)\) be the “gamma”-shaped region obtained by removing from an \(a \times b\) rectangle an \(m \times m\) square from its lower right corner (for instance, \(\Gamma_3(7,15)\) is pictured below). For odd \(m \geq 1\) and \(a,b \gt m\) the tile set pictured below tiles the region \(\Gamma_m(a,b)\) if and only if \(a \equiv m\) (mod 4) and \(b \equiv m\) (mod 8); or \(a \equiv 3m\) (mod 4) and \(b \equiv 3m\) (mod 8).

Tiling with skew and T-tetrominoes (2010) with Cynthia Lester at Linfield College. We investigated which rectangles and modified rectangles (rectangles whose upper-left and lower-right squares have been removed) can be tiled by the set of eight skew and T-tetrmonioes pictured below. Cynthia presented her work at the Northwest Undergraduate Mathematics Symposium at Reed College in June, 2011, and published her results in [6].

Tiling annular regions (2013) with Amanda Bright, Greg Clark, Kyle Evitts, Brian Keating, Brian Whetter, and faculty colleague Chuck Dunn at Linfield College. We investigated which annular regions of the form \(A_n(a,b)\) can be tiled by the set of 8 skew and T-tetrominoes below, where \(A_n(a,b)\) denotes the rectangular annulus of width \(n\) enclosing an \(a \times b\) rectangle. For instance, \(A_2(3,4)\) is pictured below. We enumerated tilings for width 2-annuli (e.g., \(A_2(3,4)\) can be tiled in 64 distinct ways!), and determined the tile counting group for width-2 annuli. Amanda, Greg, and Kyle each gave a talk at the Joint Mathematics Meetings in Baltimore in January, 2014. We published our results in Involve, [1].

Competitive tiling (2013) with Levi Altringer and faculty colleague Chuck Dunn at Linfield College. Competitive tiling consists of two players, a tile set, a region, and a non-negative integer \(d\). Alica and Bob, our two players, alternate placing tiles on the untiled squares of the region. They play until no more tiles can be placed. Alice wins if at most \(d\) squares are untiled at the end of the game, and Bob wins if more than \(d\) squares are untiled. For given regions and tile sets we are interested in the smallest value of \(d\) such that Alice has a winning strategy. We call this the game tiling number. We found the game tiling number for dominoes and three families of regions: \(2 \times n\) rectangles, modified \(2 \times n\) rectangles, and width-2 annular regions. Levi gave a talk at the Joint Mathematics Meetings in Baltimore in January, 2014. More information can be found here.

These student-centered research projects led me to study the connection between tile invariants and the homotopy of 2-complexes, see [4].

Other good introductory articles on the mathematics of tiling, with an emphasis on tile invariants: [2, 5, 7, 8, 9].

References

[1]

Bright, Amanda; Clark, Gregory J.; Dunn, Charles; Evitts, Kyle; Hitchman, Michael P.; Keating, Brian; Whetter, Brian, Tiling annular regions with skew and T-tetrominoes. Involve 10 (2017), no. 3, 505–521. http://msp.org/involve/2017/10-3/p09.xhtml

In this paper, we study tilings of annular regions in the integer lattice by skew and T-tetrominoes. We demonstrate the tileability of most annular regions by the given tile set, enumerate the tilings of width-2 annuli, and determine the tile counting group associated to this tile set and the family of all width-2 annuli.

[2]

Conway, J.H.; Lagarias, J.C. Tiling with polyominoes and combinatorial group theory. J. Combin. Theory Ser. A, 53 (1990), 183-208.

[3]

Hitchman, Michael P. "A toolbox for tackling tiling questions." A Primer for Undergraduate Research: from Groups and Tiles to Frames and Vaccines Eds. Wootton, A.; Peterson, V.; Lee, C. Birkhauser, Cham, 2017.

This article provides an introduction to techniques developed to address three types of tiling questions in the integer lattice: questions of tileability - which regions in a collection \(\cal R\) can be tiled by a tile set \(T\text{;}\) connectivity - what can be said about how two tilings of a region in \(\cal R\) must be related; and enumeration - how many ways can a region in \(\cal R\) be tiled by \(T\text{.}\) We place an emphasis on tile invariants, linear combinations among the number of copies of each tile that must persist in any tiling of a region. A tiler's toolbox draws on content from combinatorics, number theory, group theory, and topology. Given their hands-on nature, their variety, and their relevance to the undergraduate curriculum, tiling questions continue to be a lively subject for undergraduate mathematics research.

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.

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.

[7]

Pak, I. Ribbon tile invariants, Trans. Amer. Math. Soc., 352 (2000), no. 12, 5525-5561.

[8]

Propp, J. A pedestrian approach to a method of Conway, or, A tale of two cities, Mathematics Magazine, 70 (1997), no. 5, 327-340.