using the ammann-beenker tiling to model...

USING THE AMMANN-BEENKER TILING TO MODEL

QUASICRYSTALS

BRITTANY LIVSEY, JASON MIFSUD, AND FRANCESCA ROMANO

Abstract. The Ammann-Beenker tiling is a mathematical model for qua-

sicrystals with eightfold symmetry. This tiling can be constructed through a

matching technique, substitution, or the cut and project method. We use thesubstitution method to generate subsets of this octagonal tiling from which we

calculate the spectrum of the associated Laplacian. We use these calculations

to approximate the Laplacian of the complete Ammann-Beenker tiling. Wealso approximate the Hausdorff dimension of the spectrum of this tiling.

Contents

1. Introduction 21.1. Outline and Statement of Results 21.2. Acknowledgements 32. The Ammann-Beenker tiling 32.1. Definitions 32.2. Generating a Tiling 32.3. Code for Generating the Tiling 63. The Laplacian Operator and Spectral Results 83.1. Definitions 83.2. Code for Making the Laplacian 93.3. Results 94. Hausdorff Dimension 114.1. Definitions 114.2. Code for Approximating the Hausdorff Dimension 124.3. Results 125. Future Work 12Appendix A 13Appendix B 15Appendix B.1 genABTiling.m 15Appendix B.2 makeLaplacian.m 26Appendix B.3 TrianglePairs.m 28Appendix B.4 makeSquare.m 29Appendix B.5 Hausdorff.m 32References 35

Date: August 1, 2013.This work was supported by NSF grant DMS-0739338.

1

2 B. LIVSEY, J. MIFSUD, AND F. ROMANO

1. Introduction

The discovery of quasicrystals in the 1980s raised many questions in the physicalsciences, as well as in mathematics. It was shown in [10] that a two-dimensionalquasicrystal with an eightfold symmetry could be produced in a laboratory. Theelectron diffraction pattern of this quasicrystal does not correlate to a periodiclattice structure, but does display eightfold symmetry, and its “corresponding high-resolution electron microscopic image agrees well with the 2D eightfold quasilatticeconsisting of squares and 45◦ rhombi” [10]. This particular quasicrystal is modeledmathematically as an aperiodic octagonal tiling, known as the Ammann-Beenkertiling.

The Ammann-Beenker tiling, a subset of which is shown in Figure 1, is namedafter the two mathematicians who independently discovered an aperiodic octagonaltiling, Robert Ammann and F.P.M. Beenker. Ammann discovered the aperiodicoctagonal tiling in the 1970s by building the tiling from prototiles using matchingand substitution methods [1], while Beenker generated his aperiodic octagonal tilingthrough the cut and project method [3]. These results were published in the 1980s.Their findings contributed not only to the understanding of aperiodic tilings, buthave also provided insight to the structure of quasicrystals, which can be modeledwith this tiling.

Figure 1. A subset of the Ammann-Beenker tiling modified from [2]

1.1. Outline and Statement of Results. In Section 2.1, we will provide back-ground information on the Ammann-Beenker tiling and discuss its properties. Thenin Section 2.2, we will discuss the three common methods of generating a tiling:matching, substitution, and cut and project. In Section 2.3, we present MatLabcode for generating iterations of the Ammann-Beenker tiling through the substitu-tion method. We use these iterations to approximate the spectrum of the Laplacian

USING THE AMMANN-BEENKER TILING TO MODEL QUASICRYSTALS 3

of the octagonal tiling in Section 3, as well as calculate the eigenvalues of approx-imations of the spectrum of the Ammann-Beenker tiling and find the integrateddensity of states. In Section 4, we use the Hausdorff dimension to show that thespectrum has fractal properties. We also determine the Hausdorff dimension of thespectrum of subsets of the Ammann-Beenker tiling to be between 0.75 and 0.95.This implies that the spectrum has fractal properties, which provides informationabout the quantum diffusion rates in a quasicrystal [7]

1.2. Acknowledgements. We would like to thank our mentor, May Mei, for herguidance throughout this research project and our project TA, Drew Zemke, forassistance throughout this project, especially with MatLab. We would also liketo thank the Cornell University Mathematics Department and the NSF for fundingthis project. We also thank Baake, Grimm, and Moody for their detailed exposition,“What is aperiodic order?” and for inspiring the images used in this paper. Finally,we thank Derek Young for his contribution to Section 3.

2. The Ammann-Beenker tiling

2.1. Definitions. We will use the following definitions throughout this paper:

Definition 1 (Taken from [8]). A simple tiling is a tiling in which

(1) There are only a finite number of tile types, up to translation. Put anotherway, there exists a finite collection of prototiles pi such that each tile is atranslated copy of one of the pi.

(2) Each tile in R2 is a polygon.(3) Tiles meet full-edge to full-edge. An edge of one tile cannot partially overlap

with an edge of a neighboring tile.

Definition 2 (Taken from [2]). A tiling is nonperiodic if it lacks translationalsymmetry.

Definition 3 (Taken from [2]). A set of tiles is aperiodic if it admits only non-periodic tilings.

The Ammann-Beenker tiling has several distinguishing properties. First, thetiling has eightfold symmetry, meaning that if the tiling is rotated by π

4 radians, weget the same tiling. A second property of this tiling is that it is nonperiodic. Thefinal important property to note is the repeating subsets of the tiling. Althoughthe tiling is not periodic, every subset of the tiling can be found infinitely manytimes in other regions of the tiling.

2.2. Generating a Tiling. We will discuss three ways to generate the Ammann-Beenker tiling: matching, substitution, and cut and project.

2.2.1. Matching. Of these three methods, the simplest is matching. In this method,we start with the prototiles for the Ammann-Beenker tiling shown in Figure 2, whichare a rhombus with 45◦ and 135◦ angles and a square divided into two isoscelestriangles. We divide the square because this is consistent with the substitutionmethod, see Section 2.2.2 for further explanation. The prototiles will have shapemarkings on the edges and decorations on the vertices. These markings and deco-rations correspond to the matching rules for this tiling. The double arrow markingalong side 1 of the triangle prototiles must match together with a double arrowmarking along side 1 of another triangle prototile. The single arrow marking along


the remaining edges of the triangle and the rhombus must be matched with a singlearrows matching of another prototile. Vertex decorations must align in a way thatthey form houses. See [2] for further exposition about this matching rule.

Figure 2. The matching and substitution tiles for the Ammann-Beenker tiling

2.2.2. Substitution. The second method of construction for the Ammann-Beenkertiling is substitution [2]. We use the same prototiles as before with matching, i.e. arhombus with 45◦ and 135◦ angles and a square divided into two isosceles triangles.In this method, we inflate each prototile by α = 1 +

√2, which is the “silver ratio,”

and then subdivide the inflated tiles into prototiles of the original size, as shownin Figure 3. Note that the subdivision agrees with the matching rules presentedin Subsection 2.2.1. We then iterate this process, the first of which is shown inFigure 4.

Figure 3. The substitution process for the octagonal tiling


Figure 4. The first two iterations of the substitution method forthe Ammann-Beenker tiling

2.2.3. Cut and Project. A third common method of construction for the Ammann-Beenker tiling is the cut and project method, which was discovered by F. P. M.Beenker in the 1980s. The idea behind the cut and project method is to “projectthe points of a certain lattice in four dimensions, swept out by an octagon” [2]. Thedescription of the cut and project method is taken from [2].

We begin the cut and project method of construction with an intial set of integerlinear combinations, M = {u1a1 + u2a2 + u3a3 + u4a4|u1, u2, u3, u4 ∈ Z}, wherea1, a2, a3, a4 are the four unit vectors shown to the left in Figure 5. Now, the ∗-map


sends ai to a∗i for i ∈ {1, 2, 3, 4}, where a∗1, a∗2, a∗3, a∗4 are the four unit vectors shown

to the right in Figure 5. The ∗-map triples the angle between each ai and the x-axisof every linear combination in M . We let M∗ be the new set of linear combinationswhere M∗ = {u1a∗1 + u2a

∗2 + u3a

∗3 + u4a

∗4|u1, u2, u3, u4 ∈ Z}. Now, if we consider

the set P = {(x, x∗)|x ∈ M,x∗ ∈ M∗}, we observe that P forms a lattice in fourdimensions. We want to construct a new set A, which consists of the set of points,x, whose image x∗ under the ∗-map lies inside a regular octagon. Each point inthis set will correspond to the prototiles used to create the Ammann-Beenker tilingthrough the matching and substitution construction methods.

Figure 5. The set of unit vectors used to contruct the octagonaltiling (shown to the left) and their corresponding unit vectors underthe ∗-map (shown to the right).

2.3. Code for Generating the Tiling. In this section, we discuss a MatLabprogram, genABTiling, that generates iterations of the Ammann-Beenker tilingusing the substitution method from Section 2.2.2. We begin by labeling the sidesof the prototiles as shown in Figure 2. We then label the triangles and rhombi(referred to as children tiles) after one substitution for each of the parent tiles asshown in Figure 6.

Figure 6. The labeling of the internal adjacencies

From this one substitution, we determine the rules for the internal adjacenciesof each prototile (see Appendix A). We understand internal adjacencies to be thetile adjacencies that occur between tiles generated from the same parent tile, andexternal adjacencies to be the tile adjacencies that occur between two tiles generatedfrom different parent tiles. Next we generate a finite list of external adjacenciesthat occur during various iterations of the substitution method. We begin with


two triangles matched along side 1 into a square as shown in Figure 7. After oneinteration of the substitution method, we develop rules for the adjacencies betweentiles along the original Triangles’ side 1 as follows:

(1) Triangle 1 side 2 is neighbors with Triangle 1 side 2(2) Triangle 2 side 1 is neighbors with Triangle 2 side 1(3) Rhombus 2 side 4 is neighbors with Rhombus 2 side 4,

where the first tile in the rule is from parent Triangle 1 and the second tile in therule is from parent Triangle 2.

Figure 7. The starting structure for substitution of the Ammann-Beenker tiling

Given the rules we found for the internal adjacencies and the rule for two trian-gles connected along side 1, we generate the complete list of rules for substitutioniterations. Each external adjacency is iterated and catalogued until an iteration isreached in which all neighbor interactions have been previously observed. For ex-ample, in the rule for Triangle’s aligning along side 1, we set an external adjacencyof Rhombus 2 side 4 with Rhombus 2 side 4. In the next iteration, the externaladjacency rule for Rhombus side 4 with Rhombus side 4 is needed. This rule setstwo external adjacencies:

(1) Triangle 3 side 1 is neighbors with Triangle 3 side 1(2) Rhombus 1 side 4 is neighbors with Rhombus 1 side 4.

Notice that we now have rules for both of these external adjacencies for the nextiteration.

This complete list of rules was programmed into MatLab as a function calledgenABTiling to generate subsequent iterations of the tiling (see Appendix B forall code). The function genABTiling takes a cell array, oldTiles, as input. Theelements of oldTiles are structures representing tiles. Each structure is assignedan id, shape, children, and n1, n2, n3,n4 if such neighbors exist. The id is apositive integer unique to the tile in the current iteration. The shape is an ‘r’

or ‘t’ corresponding to a rhombus or a triangle. The children field is an arraycontaining the ids of the tiles in the next iteration. The neighbors (n1, n2, n3,n4)are the ids of the tiles adjacent on a particular side, if they exist.

This function takes oldTiles as input and applies substitution rules to generatethe new tiles for the next iteration and establish the internal adjacencies. We setthe internal adjacencies before the external ones because we need the entire list oftiles for the new iteration to be generated before any external adjacecnies can beset. Then genABTiling uses the external adjacency rules to define any remaining


neighbors in the tiling. Finally, the function stores the tiles in a new cell array ofstructures representing the tiles of the next iteration.

To create the data structures representing the square in Figure 7, we enter thefollowing in the terminal window:

t r i a n g l e 1 = s t r u c t ( ‘ id ’ , 1 , ‘ shape ’ , ‘ t ’ ) ;t r i a n g l e 2 = s t r u c t ( ‘ id ’ , 2 , ‘ shape ’ , ‘ t ’ ) ;t r i a n g l e 1 . n1 = t r i a n g l e 2 . id ;t r i a n g l e 2 . n1 = t r i a n g l e 1 . id ;

3. The Laplacian Operator and Spectral Results

3.1. Definitions.

Definition 4. A graph is a pair G = (V,E), where V is a set of vertices and Eis a set of unordered pairs of vertices.

Definition 5. Let G be a graph and let f : V → R, where V is the set of thevertices of G. The Laplacian ∆ (of G) acts on f by

∆f(v) :=∑

d(v,w)=1

f(v)− f(w),

where d(v, w) is the shortest edge distance from u to v. In particular, d(v, w) = 1if (v, w) ∈ E.

Definition 6. Equivalently, the Laplacian ∆ = D − A, where D is the degreematrix of G and A is the adjacency matrix of G. A degree matrix, D, is a zeromatrix with the number of neighbors of tile i in the (i, i) entry. An adjacencymatrix, A, is a zero matrix with a 1 in the (i, j) entry if tile i and tile j areneighbors.

Remark 1. Note that Definition 5 works for all graphs of finite degree while Defi-nition 6 depends on a graph with a finite number of nodes.

Example 1. Given a graph with 3 nodes, let us compute the Laplacian. Considerthe graph P3,

and the function f(1) = a, f(2) = b, f(3) = c where a, b, c ∈ R. By definition:

∆f(1) =∑

d(1,w)=1

f(1)− f(w) = f(1)− (2) = a− b.

With similar computations for ∆f(2) and ∆f(3), we get the following column vec-tors:

f =

abc

and ∆f =

a− b(b− a) + (b− c)

c− b

.


Thus the Laplacian matrix is

∆ =

1 −1 0−1 2 −10 −1 1

.

Notice that this matrix is equivalent to subtracting the adjacency matrix A from thedegree matrix D:

D =

1 0 00 2 00 0 1

, A =

0 1 01 0 10 1 0

, (D −A) =

1 −1 0−1 2 −10 −1 1

.

Definition 7. The spectrum of a matrix is the set its eigenvalues (with multiplic-ity).

3.2. Code for Making the Laplacian. From the tilings generated by genABTiling,we form an adjacency matrix in makeLaplacian. This function takes newTiles asan input, which is the output of genABTiling. From this we make an n × n zeromatrix, A, where n is the number of elements in newTiles. We loop through allof the elements in the cell array, and if an element has a neighbor on side 1, thenwe set the matrix entry (i, j) equal to 1, where i is the id of the current tile and jis the id of the neighboring tile on side 1. Similarly, we set matrix entries to 1 forneighbors on sides 2, 3, and 4 if the tile has a neighbor on sides 2,3, and 4.

Since this tiling has conventionally been studied with rhombi and squares [3],we call the function trianglePairs, which outputs a cell array of pairs of idscorresponding to 2 triangles meeting along side 1. The input for trianglesPairs

is a cell array of structures representing the tiles in a particular iteration.Now, we pass the adjacency matrix A and triangleList to the makeSquare

function, where triangleList is the cell array output from trianglePairs. In themakeSquare function, we loop through triangleList and add the column (resp.row) corresponding to the id of the second element in the pair to the column (resp.row) corresponding to the id of the first element in the pair and then set thecolumn (resp. row) corresponding to the id of the second element in the pair toall zeros. We also set the diagonal entries back to zero since the graph contains noloops. After this loop, we create a new matrix containing only the nonzero rowsand columns. Thus, makeSquare outputs finalAdjacency, which is the adjacencymatrix of the current substitution iteration.

Next, in makeLaplacian, we create a zero matrix, D, of the same size asfinalAdjacency, and set the diagonal entriy (i, i) equal to the sum of all entries ofrow i in finalAdjacency, where i ∈ {1, ...,m} and m is the size of finalAdjacency.Recall from Section 3 that we create the Laplacian, L = D - finalAdjacency.

3.3. Results. Now, we discuss the numerical approximation of the spectrum forthe Laplacian of the octagonal tiling. Using the code from Section 3.2, we findthe eigenvalues of the Laplacians associated with the first five iterations of thesubstitution method. Plotting these eigenvalues yields Figure 8. We observe thatthe spectrum is bounded by 8, which is consistent with [9], which states that thespectrum of any graph’s Laplacian is bounded by twice the degree of the largestdegree node in the graph. Notice the gap that occurs around eigenvalues equal to5. This will be discussed further in this section.


Figure 8. Plot of the eigenvalues of the Laplacian for iterationsof the Ammann-Beenker tiling

Definition 8. A cumulative distribution function (CDF) describes the prob-ability that a real-valued random variable X with a given probability distributionwill be found at a value ≤ x ∈ R.

Definition 9. The support of a CDF is the smallest closed set whose complementhas probability zero.

To help us further analyze the spectrum, we use a CDF of the set of eigenvalueswith uniform probability to plot the probability that each eigenvalue will be belowa given number in the interval [0, 7] as shown in Figure 9. This plot is of interestbecause by Theorem 3.2 of [5], these functions have a limiting function. The limitingfunction is the integrated density of states of the spectrum of the Laplacian of theoctagonal tiling. The derivative of this function is the density of states, which is ofinterest to physicists because it provides information on the location of electrons ina substance.

Additionally, the support of this limiting function is the spectrum of the Lapla-cian. Notice the steep slope of the fifth iteration in this plot around test value equalto 4. This is a result of a high number of eigenvalues in the fifth iteration equal to4. Also, notice the flat region of iteration 5 around test value 5. This correspondswith the gap around eigenvalues close to 5 in the eigenvalues plot.


Figure 9. Plot of the CDF of the spectrum of the Laplacian

4. Hausdorff Dimension

4.1. Definitions.

Definition 10 (Taken from [6]). Let X ⊆ Rn. Let |U | := sup{|x− y| : x, y ∈ U}.We say {Ui} is a δ-cover of X if X ⊂

⋃∞i=1 Ui and 0 ≤ |Ui| ≤ δ. For all δ > 0,

Hsδ(X) := inf

{ ∞∑i=1

|Ui|s : {Ui} is a δ - cover of F

}.

The s-dimensional Hausdorff measure of X is

Hs(X) := limδ→0Hsδ(X).

The Hausdorff dimension of X is

dimH X := inf{s ≥ 0 : Hs(X) = 0} = sup{s : Hs(X) =∞}.

Example 2. As an illustrative example, let us consider the unit interval [0, 1]. Letδ = 1

n , n ∈ N∗, and let Ui = [ in ,i+1n ] for all 1 ∈ {0, 1, ..., n − 1}. Note that {Ui}

forms a δ-cover of [0, 1]. Observe that

∞∑i=1

|Ui|s = n1

ns=

1

ns−1.

Now, let δ → 0, i.e. let n→∞. Notice that

limn→∞

1

ns−1=

∞ s < 10 s > 11 s = 1

This indicates that the Hausdorff dimension of the unit interval is 1, which agreeswith the standard topological dimension of the unit interval.


4.2. Code for Approximating the Hausdorff Dimension. We input the listof eigenvalues corresponding to our largest substitution iteration of the Ammann-Beenker tiling into the MatLab function Hausdorff. Since we know from Sec-tion 3.3 that the eigenvalues fall within [0, 8], we examine δ-covers of the interval[0, 8]. We will begin by looking at intervals in [0, 8] of length 1

2i , where i ∈ {1, ..., 7}.For each of these intervals, we determine whether it contains an eigenvalue and thenkeep track of the least and greatest eigenvalues found in the interval with the vari-ables least and greatest. We then determine the smallest length of this intervalcontaining all eigenvalues in the interval by subtracting least from greatest. Westore this length in a vector delta. After each interval has been examined for agiven i, we approximate the Hausdorff dimension.

4.3. Results. We want to find the s value in the definition of Hausdorff dimensionwhere Hs(eigenvalues) jumps from ∞ to 0. To do this we use a methodologysimilar to the one detailed by Chorin in [4] and output the following table:

si 0.65 0.75 0.85 0.95 1.051 8.73 8.12 7.55 7.03 6.542 10.70 9.24 7.98 6.89 5.953 13.03 10.48 8.50 6.78 5.464 15.74 11.80 8.85 6.64 4.995 18.73 13.07 9.12 6.37 4.456 21.82 14.13 9.15 5.94 3.857 24.01 14.36 8.59 5.15 3.09

The leftmost column in the table represents the varying values of i, and the toprow represents the test values for the Hausdorff dimension. The remaining columnsrepresent the values of Hsδ(eigenvalues) as i increases, which is decreasing thediameters of the δ-covers. Notice that when s = 0.75, we see a strictly increasingpattern as i increases, and when s = 0.95, we see a strictly decreasing pattern asi increases. However, the column between 0.75 and 0.95 has neither a strictly in-creasing nor strictly decreasing pattern. From this, we can infer that the Hausdorffdimension of the spectrum of the Laplacian of the Ammann-Beenker tiling is be-tween 0.75 and 0.95. This implies that the spectrum has fractal properties, whichprovides information about the quantum diffusion rates in quasicrystals [7].

5. Future Work

Due to the limited computing power of MatLab, we were only able to generatethe first five substitution iterations of the Ammann-Beenker tiling. In the future,a better approximation of the Hausdorff dimension could be calculated by usinglarger subsets of this tiling. There is also room for future work on this project byusing the cut and project method to approximate the Hausdorff dimension. Addi-tionally, the same approach that we use for the Ammann-Beenker tiling can be usedto study other aperiodic tilings modeling quasicrystals with different symmetries.Finally, since the approximation of the spectrum that we have calculated providesinformation to physicists about the movement of the electrons in a quasicrystal,more research can be performed in that area as well.


Appendix A

The internal adjacencies of the triangle are as follows:

(1) Triangle 1 side 3 is neighbors with Rhombus 1 side 2(2) Triangle 2 side 3 is neighbors with Rhombus 2 side 1(3) Triangle 2 side 2 is neighbors with Rhombus 1 side 1(4) Triangle 3 side 3 is neighbors with Rhombus 2 side 2(5) Triangle 3 side 2 is neighbors with Rhombus 1 side 4

The internal adjacencies of the rhombus are as follows:

(1) Triangle 1 side 2 is neighbors with Rhombus 1 side 2(2) Triangle 1 side 2 is neighbors with Rhombus 2 side 4(3) Triangle 2 side 2 is neighbors with Rhombus 2 side 3(4) Triangle 2 side 3 is neighbors with Rhombus 3 side 2(5) Triangle 3 side 2 is neighbors with Rhombus 2 side 2(6) Triangle 3 side 3 is neighbors with Rhombus 1 side 1(7) Triangle 4 side 2 is neighbors with Rhombus 2 side 2(8) Triangle 4 side 3 is neighbors with Rhombus 3 side 1

The external adjacencies are as follows, where the first (resp. second) prototile inthe lettered rules are children of the first (resp. second) prototile in the numberedrule:

(1) If Triangle side 1 is neighbors with Triangle side 1(a) Triangle 1 side 2 is neighbors with Triangle 1 side 2(b) Triangle 2 side 1 is neighbors with Triangle 2 side 1(c) Rhombus 2 side 4 is neighbors with Rhombus 2 side 4

(2) If Triangle side 2 is neighbors with Triangle side 2(a) Triangle 3 side 1 is neighbors with Triangle 3 side 1(b) Rhombus 2 side 3 is neighbors with Rhombus 2 side 3

(3) If Triangle side 2 is neighbors with Rhombus side 1(a) Triangle 3 side 1 is neighbors with Triangle 4 side 1(b) Rhombus 2 side 3 is neighbors with Rhombus 3 side 4






(9) If Rhombus side 3 is neighbors with Rhombus side 3(a) Triangle 1 side 1 is neighbors with Triangle 1 side 1


(b) Rhombus 1 side 3 is neighbors with Rhombus 1 side 3(10) If Rhombus side 3 is neighbors with Rhombus side 4

(a) Triangle 1 side 1 is neighbors with Triangle 3 side 1(b) Rhombus 1 side 3 is neighbors with Rhombus 1 side 4

(11) If Rhombus side 4 is neighbors with Rhombus side 4(a) Triangle 3 side 1 is neighbors with Triangle 3 side 1(b) Rhombus 1 side 4 is neighbors with Rhombus 1 side 4


Appendix B

Appendix B.1 genABTiling.m.

% genABTiling .m

% Author : Br i t tan y Livsey , Jason Mifsud , and Francesca Romano

% Given the data f o r an i t e r a t i o n o f the Ammann−Beenker t i l i n g , we compute

% the next i t e r a t i o n .

% A t i l i n g i s s t o r e d as a l i s t o f s t r u c t s ; each has the f o l l o w i n g f i e l d s :

% i d A p o s i t i v e i n t e g e r t h a t i s unique to t h i s t i l e in t h i s

% i t e r a t i o n

% shape ’ r ’ or ’ t ’ , based on whether the t i l e i s a rhombus or a

% t r i a n g l e

% c h i l d r e n An array t h a t con ta i ns t h i s t i l e ’ s c h i l d r e n ’ s i d s

% n1 , n2 , A l i s t o f the t i l e ’ s n e i g h b o r s on each s i d e

% n3 , n4

function [ newTiles ] = genABTiling ( o l d T i l e s )

% The f i r s t s t e p i s to c r e a t e the new t i l e s w i thou t worrying about

% e x t e r n a l a d j a c e n c i e s . In order to make c r e a t i n g a d j a c e n c i e s e a s i e r ,

% we w i l l keep t r a c k o f the o l d t i l e s ’ c h i l d r e n and each new t i l e ’ s

% parent .

% R e c a l l the r u l e s : Each rhombus turns i n t o four t r i a n g l e s and t h r e e

% rhombi , and each t r i a n g l e becomes t h r e e t r i a n g l e s and two rhombi .

newTiles = {} ;

idCounter = 1 ;

% A l i s t o f c o p i e s o f the t i l e s in o l d T i l e s . The c h i l d r e n o f the t i l e s

% in t h i s l i s t w i l l be updated i n s t e a d o f updat ing the c h i l d r e n o f the

% t i l e s in o l d T i l e s

tempTiles = {} ;

for i = 1 : length ( o l d T i l e s )

t h i s T i l e = o l d T i l e s { i } ;

% t h i s T i l e i s a t r i a n g l e .

i f t h i s T i l e . shape == ’ t ’

% Make t h r e e new t r i a n g l e s and two new rhombi .

newTriangle1 = s t r u c t ( ’ id ’ , idCounter , ’ shape ’ , ’ t ’ ) ;

idCounter = idCounter + 1 ;






newRhombus1 = s t r u c t ( ’ id ’ , idCounter , ’ shape ’ , ’ r ’ ) ;




% Set c h i l d r e n o f t h i s T i l e .

t h i s T i l e . c h i l d r e n = [ newTriangle1 . id , newTriangle2 . id , . . .

newTriangle3 . id , newRhombus1 . id , newRhombus2 . id ] ;

% Set i n t e r n a l a d j a c e n c i e s .

newTriangle1 . n3 = newRhombus1 . id ;

newRhombus1 . n2 = newTriangle1 . id ;









% Store new t i l e s .

newTiles { length ( newTiles )+1} = newTriangle1 ;



newTiles { length ( newTiles )+1} = newRhombus1 ;


% t h i s T i l e i s a rhombus .

else

% Make new four new t r i a n g l e s and t h r e e new rhombi
















% Set c h i l d r e n o f t h i s T i l e .

t h i s T i l e . c h i l d r e n = [ newTriangle1 . id , newTriangle2 . id , . . .

newTriangle3 . id , newTriangle4 . id , newRhombus1 . id , . . .

newRhombus2 . id , newRhombus3 . id ] ;

% Set i n t e r n a l a d j a c e n c i e s .

















% Store new t i l e s .









end

% Store the t i l e t h a t we j u s t processed and a l t e r e d .

tempTiles { length ( tempTiles )+1 } = t h i s T i l e ;

end

% Now we w i l l i t e r a t e through temoTi les again to c r e a t e the e x t e r n a l

% a d j a c e n c i e s .

for i = 1 : length ( tempTiles )

t h i s T i l e = tempTiles { i } ;

% t h i s T i l e i s a t r i a n g l e .

i f t h i s T i l e . shape == ’ t ’

% Rule : t r i a n g l e has a ne ighbor 1 , which i s a lways connected to

% another t r i a n g l e on the 1 s i d e

i f i s f i e l d ( t h i s T i l e , ’ n1 ’ )

T = tempTiles { t h i s T i l e . n1 } ;

% t r i a n g l e 1 s i d e 2 connected to t r i a n g l e 1 s i d e 2

c h i l d = newTiles { t h i s T i l e . c h i l d r e n ( 1 ) } ;

c h i l d . n2 = newTiles {T. c h i l d r e n ( 1 ) } . id ;

newTiles { t h i s T i l e . c h i l d r e n (1)} = c h i l d ;





% rhombus 2 s i d e 4 connected to rhombus 2 s i d e 4




end

% Rule : t r i a n g l e has a ne ighbor 2 .




% t h i s T i l e ’ s ne ighbor 2 i s a t r i a n g l e

i f ( T. shape == ’ t ’ )





% rhombus 2 s i d e 3 connected to rhombus 2 s i d e 3




% t h i s T i l e ’ s ne ighbor 2 i s a rhombus

e l s e i f ( T. shape == ’ r ’ )

% t r i a n g l e s i d e 2 connected to rhombus s i d e 1

i f ( i s f i e l d ( T, ’ n1 ’ ) && T. n1 == t h i s T i l e . id )

% t r i a n g l e 3 s i d e 1 to t r i a n g l e 4 s i d e 1




% rhombus 2 s i d e 3 to rhombus 3 s i d e 4




% t r i a n g l e s i d e 2 i s connected to rhombus s i d e 2

e l s e i f ( i s f i e l d ( T, ’ n2 ’ ) && T. n2 == t h i s T i l e . id )





















else









end

end

end

% Rule : t r i a n g l e has a ne ighbor 3 . This ne ighbor i s a lways a

% rhombus t i l e
























end

end

% t h i s T i l e i s a rhombus

e l s e i f ( t h i s T i l e . shape == ’ r ’ )

% rhombus has a ne ighbor on s i d e 1



% rhombus s i d e 1 i s connected to t r i a n g l e s i d e 2










% rhombus s i d e 1 connected to t r i a n g l e s i d e 3











end

end

























end

end





i f ( i s f i e l d ( T, ’ n2 ’ ) && T. n2 == t h i s T i l e . id && T. shape == ’ t ’ )









% rhombus s i d e 3 i s connected to rhombus s i d e 3

e l s e i f ( i s f i e l d ( T, ’ n3 ’ ) && T. n3 == t h i s T i l e . id && T. shape == ’ r ’ )




















end

end





i f ( i s f i e l d ( T, ’ n2 ’ ) && T. n2 == t h i s T i l e . id && T. shape == ’ t ’ )






























end

end

end % ends d e c i d i n g the shape

end % ends the f o r loop

end % ends the f u n c t i o n


Appendix B.2 makeLaplacian.m.

% makeLaplacian .m


%

% Given newTiles as input , a c e l l array o f t i l e s t r u c t u r e s wi th ids , shapes ,

% c h i l d r e n , and neighbors , we output L , the Laplac ian matrix

% newTiles a c e l l array o f t i l e s t r u c t u r e s wi th ids , shapes , c h i l d r e n ,

% and n e i g h b o r s .

% L the Laplacian matrix corresponding to newTiles

function [ L ] = makeLaplacian ( newTiles )

% Adjacency matrix b e f o r e t r i a n g l e s are merged i n t o squares .

A = zeros ( length ( newTiles ) ) ;

% Places a 1 in the i , j s po t o f matrix A i f i and j are n e i g h b o r i n g t i l e s

for i = 1 : length ( newTiles )

tempTile = newTiles { i } ;

% tempTile has a ne ighbor 1

i f ( i s f i e l d ( tempTile , ’ n1 ’ ) )

A( i , tempTile . n1 ) = 1 ;

end



A( i , tempTile . n2 ) = 1 ;

end



A( i , tempTile . n3 ) = 1 ;

end




A( i , tempTile . n4 ) = 1 ;

end

end

sparseA = sparse (A) ;

% Creates a l i s t o f p a i r s o f i n d i c e s f o r t r i a n g l e s ad jac en t on s i d e 1 .

p a i r s = t r i a n g l e P a i r s ( newTiles ) ;

% Removes the e x t r a rows and columns from A when we merge t o g e t h e r the

% t r i a n g l e s t h a t are ad j acen t on s i d e 1 .

f i na lAd jacency = makeSquare ( sparseA , p a i r s ) ;

% The degree matrix .

D = zeros ( length ( f i na lAd jacency ) ) ;

sparseD = sparse (D) ;

% Sums the v a l u e s in row i and s e t s t h i s number as the v a l u e o f the i , i

% s po t in the degree matrix .

for i = 1 : length ( sparseD )

row = f ina lAd jacency ( i , : ) ;

count = sum( row ) ;

% f o r j = 1: l e n g t h ( row )

%

% sum = sum + row ( j ) ;

%

% end

sparseD ( i , i ) = count ;

end

% Creates the Laplacian matrix by s u b t r a c t i n g the f i n a l A d j a c e n c y matrix

% from the degree matrix

L = sparseD − f i na lAd jacency ;

end


Appendix B.3 TrianglePairs.m.

% t r i a n g l e P a i r s .m


% Creates a c e l l array , t r i a n g l e L i s t , o f p a i r s o f the i d s o f t r i a n g l e s t h a t

% are ad ja cen t on s i d e 1 wi th each o th er in newTiles

% newTiles a c e l l array o f t i l e s t r u c t u r e s wi th ids ,

% shapes , c h i l d r e n , and n e i g h b o r s .

% t r i a n g l e L i s t a c e l l array c o n t a i n i n g p a i r s o f t i l e i d s

% corresponding to t r i a n g l e t i l e s t h a t ne ighbor

% each o the r to make a square .

function [ t r i a n g l e L i s t ] = t r i a n g l e P a i r s ( newTiles )

t r i a n g l e L i s t = {} ;

% Loops through the l i s t o f t i l e s and adds a t r a i n g l e t i l e wi th an n1

% neighbor to the c e l l array

for i = 1 : length ( newTiles )

tempTile = newTiles { i } ;

% Checks i f the curren t t i l e i s a t r i a n g l e has a ne ighbor 1

i f ( i s f i e l d ( tempTile , ’ n1 ’ ) && tempTile . shape == ’ t ’ )

tempList = { tempTile . id , tempTile . n1 } ;

% Checks to not add d u p l i c a t e s to the c e l l array o f p a i r s

i f ( tempTile . id < tempTile . n1 )

t r i a n g l e L i s t { length ( t r i a n g l e L i s t ) + 1} = tempList ;

end

end

end

end


Appendix B.4 makeSquare.m.

% makeSquare .m


% Takes an adjacency matrix A r e p r e s e n t i n g a t i l i n g o f rhombi and t r i a n g l e s

% and merges the r e s p e c t i v e rows and columns t o g e t h e r o f t r i a n g l e s wi th n1

% n e i g h b o r s l i s t e d in the c e l l array t r i a n g l e L i s t . We then output the new

% matrix newA .

% A adjacency matrix o f our current i t e r a t i o n us ing

% t r i a n g l e s i n s t e a d o f squares

% t r i a n g l e L i s t an array c o n t a i n i n g p a i r s o f numbers corresponding to

% the i d number o f two t r i a n g l e t i l e s j o i n i n g to make

% a square

% sparseFinalA adjacency matrix o f our current i t e r a t i o n us ing squares

% i n s t e a d o f t r i a n g l e s

function [ f i na lA ] = makeSquare ( A, t r i a n g l e L i s t )

% New adjacency matrix

sparseNewA = A; % make A i n t o a sparse matrix

% For each t r i a n g l e pair , we make a square , so we w i l l d e l e t e one row

% from the adjacency matrix . tempFinalRow w i l l be the c o r r e c t

% s i z e to s t o r e t h i s new matrix .

tempFinalRowA = zeros ( length (A) − length ( t r i a n g l e L i s t ) , length (A) ) ;

finalRowA = sparse ( tempFinalRowA ) ; % make the above matrix sparse

% After removing an e q u a l number o f columns , we w i l l have a square matrix .

% tempFinalA i s the matrix we w i l l s t o r e our f i n a l matrix in .

tempFinalA = zeros ( length (A) − length ( t r i a n g l e L i s t ) ) ;

f i na lA = sparse ( tempFinalA ) ;

% Adds t o g e t h e r the e n t r i e s o f the columns ( resp . rows ) corresponding to

% the i n d i c e s in a p a i r o f the t r i a n g l e L i s t and s e t s the column ( row ) o f

% the second index in the p a i r to a l l z e r o s . Then we r e s e t the d i a g o n a l

% entry to 0 .

for i = 1 : length ( t r i a n g l e L i s t )

temp = t r i a n g l e L i s t { i } ; % g e t s the i d s o f the f i r s t p a i r o f t i l e s

index1 = temp {1} ; % id o f the f i r s t t r i a n g l e

index2 = temp {2} ; % id o f the second t r i a n g l e


% Add the row corresponding to the second t r i a n g l e to the f i r s t

sparseNewA ( index1 , : ) = sparseNewA ( index1 , : ) + sparseNewA ( index2 , : ) ;

% Add the column corresponding to the second t r i a n g l e to the f i r s t

sparseNewA ( : , index1 ) = sparseNewA ( : , index1 ) + sparseNewA ( : , index2 ) ;

% Set the d i a g o n a l entry back to 0

sparseNewA ( index1 , index1 ) = 0 ;

% Set the column corresponding to t r i a n g l e two to a l l z e r o s

sparseNewA ( index2 , : ) = 0 ;

% Set the row corresponding to t r i a n g l e two to a l l z e r o s

sparseNewA ( : , index2 ) = 0 ;

end

rowIndex = 1 ; % a counter f o r the number o f rows in finalRowA

for i = 1 : length ( sparseNewA )

% I f the i t h row i s a l l zeros , no a c t i o n

i f sparseNewA ( i , : ) == 0

else

% I f the i t h row has nonzero e n t r i e s , a s s i g n i t to the curren t

% row o f finalRowA

finalRowA ( rowIndex , : ) = sparseNewA ( i , : ) ;

rowIndex = rowIndex + 1 ; % Move to the next row in finalRowA

end

end

columnIndex = 1 ; % A counter f o r the number o f rows in f i n a l A

for i = 1 : length (A)

% I f the i t h column i s a l l zeros , no a c t i o n

i f finalRowA ( : , i ) == 0

else


% I f the i t h column has nonzero e n t r i e s , a s s i g n i t to the

% current column of f i n a l A

f i na lA ( : , columnIndex ) = finalRowA ( : , i ) ;

% Move to the next column in f i n a l A

columnIndex = columnIndex + 1 ;

end

end

end


Appendix B.5 Hausdorff.m.

% Hausdorf f .m


% The f u n c t i o n t a k e s e i g e n v a l u e s as input and c a l c u l a t e s an approximation

% o f the Hausdorf f dimension .

% e i g e n v a l u e s the e i g e n v a l u e s o f an i t e r a t i o n o f the t i l i n g

% dimension a matrix where the columns correspond to vary ing v a l u e s

% of D and the rows correspond to vary ing v a l u e s o f i .

function [ dimension ] = Hausdor f f ( e i g e n v a l u e s )

% S et s the output e q u a l to a 10 x 23 matrix ( rows correspond to the

% number o f i−v a l u e s + 1 and columns correspond to the number o f D

% v a l u e s we w i l l t e s t ) .

dimension = zeros (10 , 2 3 ) ;

% We w i l l t e s t wi th 9 d i f f e r e n t i v a l u e s .

for i = 1 :9

% Create an empty v e c t o r to ho ld the d i f f e r e n t i n t e r v a l l e n g t h s .

d e l t a = [ ] ;

% I n i t i a l lower bound .

lowerBound = 0 ;

% How the upper bound w i l l increment .

upperBound = 1 / (2 ˆ i ) ;

% While upperBound <= 8 , we cont inue to increment our i n t e r v a l s and

% check i f t h e r e are e i g e n v a l u e s in them .

while upperBound <= 8

% Represents the l a r g e s t e i g e n v a l u e in the i n t e r v a l .

% S t a r t g r e a t e s t a t the l e a s t i t cou ld ever be .

g r e a t e s t = 0 ;

% Represents the s m a l l e s t e i g e n v a l u e in the i n t e r v a l .

% S t a r t l e a s t a t the most i t cou ld ever be .

l e a s t = upperBound ;

% This f o r loop c y c l e s through each e ig env a lu e , checks i f i t i s

% in the i n t e r v a l , s e t s i t e q u a l to g r e a t e s t / l e a s t i f

% appropr ia te , and increments count f o r each i n t e r v a l t h a t


% has an e i g e n v a l u e in i t .

for j = 1 : length ( e i g e n v a l u e s )

% Condit ion to check i f our e i g e n v a l u e i s in the i n t e r v a l .

i f ( e i g e n v a l u e s ( j ) < upperBound && e i g e n v a l u e s ( j ) >= lowerBound )

% I f the current e i g e n v a l u e i s in the i n t e r v a l and l e s s

% than the curren t v a l u e o f l e a s t , s e t l e a s t e q u a l to the

% current e i g e n v a l u e .

i f ( e i g e n v a l u e s ( j ) < l e a s t )

l e a s t = e i g e n v a l u e s ( j ) ;

end

% I f the current e i g e n v a l u e i s in the i n t e r v a l and g r e a t e r

% than the curren t v a l u e o f g r e a t e s t , s e t g r e a t e s t e q u a l

% to the current e i g e n v a l u e .

i f ( e i g e n v a l u e s ( j ) > g r e a t e s t )

g r e a t e s t = e i g e n v a l u e s ( j ) ;

end

end

end

% I f l e a s t i s e q u a l to upperBound , then the l e n g t h o f the

% i n t e r v a l i s zero

i f l e a s t == upperBound

d e l t a = [ d e l t a 0 ] ;

else

% I f the i n t e r v a l i s nonzero , then the l e n g t h o f the

% i n t e r v a l where e i g e n v a l u e s are found

% i s the d i f f e r e n c e o f the g r e a t e s t and l e a s t e i g e n v a l u e s

% found in i t .

d e l t a = [ d e l t a ( g r e a t e s t − l e a s t ) ] ;

end

% Now, we increment lowerBound and upperBound a p p r o p r i a t e l y to


% l o o k at the next i n t e r v a l .

lowerBound = upperBound ;

upperBound = upperBound + 1 / (2 ˆ i ) ;

end

% We d e f i n e entryCounter to keep t r a c k o f what column of the matrix

% we are in .

entryCounter = 1 ;

% The D v a l u e s are the t e s t v a l u e s f o r Hausdorf f dimension .

for D = 0 : . 0 5 : 1 . 1

% The f i r s t row w i l l correspond to the D v a l u e we are c u r r e n t l y

% c o n s i d e r i n g .

dimension (1 , entryCounter ) = D;

% Create a v a r i a b l e to ho ld the sum of the l e n g t h s o f the

% i n t e r v a l s r a i s e d to the D power .

sum = 0 ;

for j = 1 : length ( d e l t a )

% We s t o r e the sum o f the l e n g t h o f each i n t e r v a l r a i s e d to

% the D in the v a r i a b l e sum .

sum = sum + ( d e l t a ( j ) ˆ D) ;

end

% We s e t the ( i + 1) th entry in the column corresponding to

% entryCounter e q u a l to sum .

dimension ( i + 1 , entryCounter ) = sum ;

% Last ly , increment entryCounter .

entryCounter = entryCounter + 1 ;

end

end

end


References

[1] R. Ammann, B. Grnbaum, and G.C. Shephard. Aperiodic tiles. Discrete & ComputationalGeometry, 8(1):1–25, 1992.

[2] M. Baake, U. Grimm, and R. Moody. What is aperiodic order? arXiv:0203252, 2002.

[3] F.P.M. Beenker. Algebraic Theory of Non-periodic Tilings of the Plane by Two Simple Build-ing Blocks: A Square and a Rhombus. TH report. Eindhoven University of Technology, 1982.

[4] Alexandre Joel Chorin. Numerical estimates of hausdorff dimension. Journal of Computa-

tional Physics, 46(3):390 – 396, 1982.[5] D. Damanik, M. Embree, and A. Gorodetski. Spectral properties of schr\” odinger operators

arising in the study of quasicrystals. arXiv preprint arXiv:1210.5753, 2012.[6] K. Falconer. Hausdorff Measure and Dimension, pages 27–38. John Wiley & Sons, Ltd, 2005.

[7] I. Guarneri. Spectral properties of quantum diffusion on discrete lattices. EPL (Europhysics

Letters), 10(2):95, 1989.[8] L. Sadun. Topology of tiling spaces, volume 46 of University Lecture Series. American Math-

ematical Society, Providence, RI, 2008.

[9] D. Spielman. Spectral graph theory. Lecture Notes, 2009.[10] N. Wang, H. Chen, and K. H. Kuo. Two-dimensional quasicrystal with eightfold rotational

symmetry. Phys. Rev. Lett., 59:1010–1013, Aug 1987.

Department of Mathematics, Georgetown College, Georgetown, KY 40324

E-mail address: [email protected]

Department of Mathematics, Binghamton University, Binghamton, NY 13902


Department of Mathematics, Siena College, Loudonville, NY 12211


using the ammann-beenker tiling to model...

Documents