ON THE SPECTRUM OF THE PENROSE LAPLACIAN MICHAEL DAIRYKO, CHRISTINE HOFFMAN, JULIE PATTYSON, AND HAILEE PECK Abstract. Since the early 1960’s, aperiodic tilings have been a topic of par- ticular interest to mathematicians. The Penrose tiling is one example of such atiling.Inthispaper,wepresentbasicsubdivisionrulesforgeneratingthe Penrose tiling using a set of four Robinson tiles. We introduce code written in MatLab that creates and stores the Penrose tiling as a set of data struc- tures and produces the Laplacian matrix of the tiling. In doing so, we provide insight into the spectrum of the Penrose tiling. Contents 1. Introduction 1 2. Generating a Penrose Tiling 2 2.1. Substitution Method 3 2.2. Rules for Substitution with Robinson Triangles 4 2.3. Computational Methods for Penrose Tiling 7 3. The Laplacian and Spectrum 7 3.1. Computational Methods of the Laplacian and Spectrum 9 4. The Hausdor↵Dimension 10 4.1. Computational Methods for The Hausdor↵Dimension 11 5. Results 11 6. Conclusion and Future Work 13 7. Acknowledgments 14 Appendix A: MatLab Code 15 A.1. genPenTiling.m 15 A.2. tilingLaplacian.m 22 A.3. plotMat.m 25 A.4. cdf.m 25 A.5. ndsTable.m 26 References 28 1. Introduction Before 1982, crystalline structures were defined as having periodic lattice struc- ture. However, on April 8, 1982, Dan Shechtman discovered an aluminum-manganese quasicrystalline structure that had long-range order but did not exhibit the peri- odic patterns that characterized crystals [12]. Instead, the structure had icosahedral Date:August1,2013. This work was supported by NSF grant DMS-0739338. 1 2 M.DAIRYKO,C.HOFFMAN,J.PATTYSON,ANDH.PECK symmetry that was not possible, by definition, in crystal form. In 2011, Shecht- man was awarded the Nobel Prize in Chemistry for his discovery of and work with quasicrystals. Since his groundbreaking discovery, many examples of quasicrystals have been examined. However, mathematical interest in periodicity began before these discoveries of the physical applications occurred. The following discussion of this history comes from [8]. Questions relating to tiling and periodicity began with Hao Wang’s work in 1960 on tiling the plane with Wang dominoes, which are sets of unit squares with colored edges. This work began because of its relation to questions in symbolic logic. Wang conjectured that there was a way to tile the plane periodically given any set of tiles so that bordering edges share the same color. In 1964, Robert Berger provided a counterexample to the Wang conjecture. Berger showed that there is a set (of more than 20,000 Wang dominoes) that tiles only nonperiodically, and later much smaller sets (of 104 and 92 Wang dominoes) with the same property were found. Figure 1. Penrose rhombus and diamond prototiles divided into Robinson triangles Berger’s findings inspired further exploration with polygonal tiles. Raphael Robinson and Robert Ammann, in 1971 and 1977, respectively, constructed two di↵erent sets of 6 “square” tiles that tile only nonperiodically. These tiles are like jigsaw puzzle pieces which can be rotated and reflected. The projections and indentations on these tiles force specific tiles to match together. In 1973, Roger Penrose found a set of six “non-square” tiles that tile only nonpe- riodically. He later reduced it to a set of four tiles, and then two tiles. Each of these sets of Penrose tiles generate nonperiodic tilings, which are two-dimensional models of quasicrystals. Robinson took a set of two prototiles, a rhombus and diamond, and divided them into a set of four triangles as shown in Figures 1 and 2. These tiles are known as Robinson triangles. Our work is based on these Penrose tilings, which can be generated from any of the sets of prototiles that Penrose found. In Section 2, we discuss the method we use to construct the Penrose tiling and give an explanation of MatLab code that constructs finite tilings. In Section 3, we define the Laplacian matrix and spectrum and explain the corresponding MatLab code. In Section 4, we examine the Hausdor↵dimension and describe the MatLab code developed to estimate the Hausdor↵dimension of the finite Penrose tilings. 2. Generating a Penrose Tiling Let us first present relevant definitions about tilings. ON THE SPECTRUM OF THE PENROSE LAPLACIAN 3 Definition 1. [7] Prototiles are a set of finite inequivalent tiles (i.e. are not equivalent under rigid motions, expansions, or contractions). Definition 2. [7] A tiling is an arrangement of tiles, such that their union covers and packs R2 so that distinct tiles have non-intersecting interiors. Definition 3. [9] A nonperiodic tiling does not have a period. Definition 4. [9] An aperiodic set of prototiles admits infinitely many tilings of the plane, none of which are periodic. Remark 1. An iteration is one application of the substitution method to a set of tiles. There are three main methods that can be used to construct the Penrose tiling. The matching method assigns matching rules to the edges of the prototiles, only allowing pairings of certain edges. These rules guarantee a nonperiodic tiling. The substitution method starts with a finite subset of the tiling and inflates and sub- divides existing tiles to expand the tiling to cover the plane. The cut and project method takes a higher dimensional lattice and then reduces the order, projecting it to a lower dimension. For a more detailed explanation, see [5] and [1]. 2.1. Substitution Method. In this paper, we construct the Penrose tiling using the substitution method. The substitution we use is based on the set of two Penrose prototiles, the rhombus and diamond (see Figure 1). The two tiles are each sliced temporarily in half into two triangles (called Robinson triangles), resulting in a substitution with four base prototiles: B, P , Y , and W (see Figure 2). As the substitution method for the rhombus and diamond allows for overlap, we implement the Robinson triangle substitution to avoid this problem. This method inflates 1+p5 each triangle prototile by 2 and replaces the inflated tile with a predetermined subdivision. Figure 2. How to subdivide Robinson triangle prototiles (not drawn to scale) The rhombus is composed of two triangles, which we label B and P . As shown in Figure 1, Triangle B is inflated and subdivided into three triangles: P 1,Y1, and B1. Triangle P is also inflated and subdivided into three triangles: B2,W2, and 4 M.DAIRYKO,C.HOFFMAN,J.PATTYSON,ANDH.PECK P 2. Similarly, the diamond is composed of two triangles, which we label Y and W . The triangle Y is inflated and subdivided into two triangles: P 3 and Y 3. The triangle W is also inflated and subdivided into two triangles: W 4 and B4. Figure 2 shows this inflation and subdivision of the four prototiles. Figure 3. First three iterations with Robinson triangles We begin with ten prototiles (five triangle B tiles and five triangle P tiles) and use this substitution to generate larger iterations of the Penrose tiling. See Figure 3 for the first two iterations of the tiling using the substitution. For a more detailed explanation see [7] and [11]. Figures 1, 2, and 3 were inspired by and modified from [7]. 2.2. Rules for Substitution with Robinson Triangles. The sides of each tri- angle are marked with either one arrow, two arrows, or no arrows. In the tiling, a one arrow side can only be adjacent to another one arrow side, a two arrow side can only be adjacent to a two arrow side, and a side with no arrows can only be adjacent to another side with no arrows (we will refer to this as the third side). In addition, the arrows on the shared sides of adjacent tiles must point in the same direction. The nine rules in Table 1 are the only possible ways that tiles can share edges. For example, the second side of a W tile can only be matched to the second side of a P tile or to the second side of a Y tile (following rules 7 and 8, respectively). Initially, one might think that the second side of a W tile could be matched to the second side of a B tile. However, if you were to match these two sides, you would see that arrows of the second side of each of these tiles do not point in the same direction. It is not possible to rotate these tiles in any way and match them on their second side so that the arrows on the second side of these tiles point in the same direction on their shared edge. Therefore these two tiles will not meet on their respective second sides. In Table 1, the “Shared Edge” column contains a tree diagram for each of the nine rules mentioned above. The two vertices connected by the solid black line represent which two parent prototiles share an edge. The side that they share is the number in superscript. The dashed arrows denote the children of each adjacent ON THE SPECTRUM OF THE PENROSE LAPLACIAN 5 parent. Each child is directly connected only to its parent tile, where the subscripts index the children. The “External Adjacencies” column contains the external adjacencies formed be- tween the new child prototiles after subdivision of the two parent prototiles which shared an edge. We say an adjacency is external if it connects children of dis- tinct prototiles. The adjacencies are separated by commas. Again, the superscript represents the side on which the tiles are adjacent and the subscript denotes the corresponding child. For example, Rule 5 describes P and Y prototiles that are adjacent on the first side (superscript 1).