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

Altringer, Levi A.

Bright, Amanda

Clark, Greg

Dunn, Charles

Evitts, Kyle

Hitchman, Michael P.

Keating, Brian

Whetter, Brian

English

DigitalCommons@Linfield

Tilings of Annular RegionsLevi Altringer, Linfield CollegeAmanda Bright, Westminster College MOGreg Clark, Westminster College PACharles Dunn, Linfield CollegeKyle Evitts, Linfield CollegeMike Hitchman, Linfield CollegeBrian Keating, UC San DiegoBrian Whetter, University of Puget SoundWhat is Tiling?In our context a tile is a collection of unit cells in the plane. Let T be a set of tiles, thenwe say a region Γ is tileable by T if we can cover every cell of Γ once and only once bytiles from T . We can use as many copies of a tile from T in a tiling as we wish. Also,the tiles cannot be rotated or flipped, but if we would like to allow for rotations and flipswe can include them in the tile set.Common tiling questions for a family of regions R and a tile set T are:1. Which regions in R can be tiled by T ?2. How many ways can we tile a region Γ ∈ R by T ?3. How are two different tilings of a given region related?• Is there a local move property? That is, is there a set of local moves that allow oneto transform any tiling of a region to any other tiling of that region?• Are there certain relations among the tiles in T that must persist in any tiling of aregion Γ ∈ R? In other words, what are the tile invariants associated to T and R?•What is the tile counting group associated to T and R (the tile counting grouprecords the various tile invariants associated to T and R)?Goals of Our ResearchIn this project we investigate tilings of rectangular annular regions which we denoteAn(a, b) using the set of T and skew tetrominoes (from this point on T will refer to thistile set). Figure 5 below depicts our tile set.t3 t4 t5 t6 t7 t8t1 t2Figure 1: Our tile set T of T and skew tetrominoes.The region An(a, b) is defined as a×b rectangle with n rows of cells around it. The a×brectangle is not included in the region. Thus these regions are not simply connected(they have a hole in the middle). Figure 2 gives an example of two of these regions.Figure 2: The annular regions A2(3, 2) and A3(1, 5)It can be quite difficult to show that a region is untileable by a given tile set. Thankfullymany techniques exist to help decide if a region is tileable. However, many of thesetechniques require that the region is simply connected (no holes). We hope to developnew techniques that work for non-simply connected regions. This is why we decidedto look at the An(a, b) regions, as they seemed like easy examples of non-simply con-nected regions.We also sought to answer some of the common tiling questions listed above for our tileset and family of annular regions.Sponsored by the Linfield College Faculty Student Collaborative Research Grant Pro-gram and the National Science FoundationMain ResultTheorem 1. An(a, b) is tileable by T if and only if• n is even,• n = 3 and a ≡ b (mod 2) and it is not the case that a, b ≡ 0 (mod 4),• n ≥ 5 is odd and a ≡ b (mod 2).The theorem above completely classifies which of our An(a, b) regions are tileable by T .Below we present a proof of the first case when n is even. To do this we first develop afew definitions and lemmas.Definition 1. An extended T, Xn, is any rotation of a region formed by removing the twocorner squares from first row of a 2× n rectangle for all n ≥ 3 (see Fig. 3).nn − 2Figure 3: An extended T of length nLemma 2. The region Xn is tileable by T for all odd n ≥ 3.Proof. We proceed by induction. For n = 3 we have a T-tile.Now assume Xn is tileable by T for some odd n ≥ 3. Now we must show that Xn+2 istileable by T . Simply add a skew tetrominoe to one end of the region. The remainingarea is tileable by the induction hypothesis.22 nn − 2Figure 4: Illustration of the induction step.Lemma 3. The region A2(a, b) is tileable by T for all a, b ∈ Z+.Proof. Consider the following 3 cases:a, b ≡ 0 (mod 2),a ≡ 0 (mod 2) and b ≡ 1 (mod 2),a, b ≡ 1 (mod 2)(note a ≡ 1 (mod 2), b ≡ 0 (mod 2) is covered in the second case by symmetry).In each case the region can be broken down into four extended T’s of odd length, whichare tileable by lemma 2. Below is a picture of such a decomposition for the first case.b + 3bb + 3a + 3a + 3 aFigure 5: An arrangement of odd length extended T’s showing that the region is tileable(a, b ≡ 0 mod 2).From these lemmas, it is not difficult to show that An(a, b) is tileable by T for all evenvalues of n. This can be done by induction on n. The previous lemma provides the basecase.Other Results•We proved that the tile counting group for the A2(a, b) regions with respect to T isisomorphic to Z3 × Z2. This means that we know what all the tile invariants look likefor these regions with our tile set.•We were able to show that the extended T’s have a local move property.•We also proved that there are 2a+b+1 ways to tile A2(a, b) by T .References[1] J.H. Conway and J.C. Lagarias, Tiling with polyominoes and combinatorial grouptheory, J. Combin. Theory A 53 (1990), pp. 183-208.[2] M. Hitchman, Topology and Tile Invariants, Rocky Mountain J.of Math. (2013), toappear[3] M. Korn, Geometric and algebraic properties of polyomino tilings, Ph.D. thesis, MIT(2004), http://hdl.handle.net/1721.1/16628[4] C. Lester, Tiling with T and skew Tetrominoes, Querqus: Linfield Journal of Under-graduate Research: Vol. 1: Iss. 1, Article 3[5] C. Moore and I. Pak, Ribbon tile invariants from signed area, J. Combin. Theory A98 (2002), pp. 1-16.[6] R. Muchnik and I. Pak, On tilings by ribbon tetrominoes, J. Combin. Theory A 88(1999), pp. 188-193.[7] I. Pak, Ribbon tile invariants, Transactions of the American Mathematical Society,352 (2000), no. 12, pp. 5525-5561.[8] W.P. Thurston, Conway’s tiling groups, Amer. Math. Monthly 95 (1990), specialgeometry issue, 757-773.

Tilings of Annular Region

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Tilings of Annular Region

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