Using the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionUsing the Ammann-Beenker Tiling to ModelQuasicrystalsBrittany Livsey, Jason Mifsud, and Francesca RomanoAugust 2, 2013Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Research GoalBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionA two-dimensional quasicrystal with an eightfold symmetrybut a nonperiodic diffraction pattern was produced in alaboratory in the 1980s [5].Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Research GoalBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionA two-dimensional quasicrystal with an eightfold symmetrybut a nonperiodic diffraction pattern was produced in alaboratory in the 1980s [5].This quasicrystal can be modeled mathematically as anaperiodic octagonal tiling, known as the Ammann-Beenkertiling.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Research GoalBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionA two-dimensional quasicrystal with an eightfold symmetrybut a nonperiodic diffraction pattern was produced in alaboratory in the 1980s [5].This quasicrystal can be modeled mathematically as anaperiodic octagonal tiling, known as the Ammann-Beenkertiling.Goal: Approximate the spectrum of the Ammann-BeenkerLaplacian and analyze its Hausdorff dimensionBrittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

DefinitionsBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionDefinition (Taken from [4])A simple tiling is a tiling in which1 There are only a finite number of tile types, up to translation.Put another way, there exists a finite collection of prototilesp i such that each tile is a translated copy of one of the p i .2 Each tile in R 2 is a polygon.3 Tiles meet full-edge to full-edge. An edge of one tile cannotpartially overlap with an edge of a neighboring tile.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

DefinitionsBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionDefinition (Taken from [4])A simple tiling is a tiling in which1 There are only a finite number of tile types, up to translation.Put another way, there exists a finite collection of prototilesp i such that each tile is a translated copy of one of the p i .2 Each tile in R 2 is a polygon.3 Tiles meet full-edge to full-edge. An edge of one tile cannotpartially overlap with an edge of a neighboring tile.Definition (Taken from [1])A tiling is nonperiodic if it lacks translational symmetry, and a setof tiles is aperiodic if it admits only nonperiodic tilings.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

BackgroundBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionThe Ammann-Beenker tiling is named after the twomathematicians who independently discovered an aperiodicoctagonal tiling, Robert Ammann and F.P.M. Beenker.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

BackgroundBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionThe Ammann-Beenker tiling is named after the twomathematicians who independently discovered an aperiodicoctagonal tiling, Robert Ammann and F.P.M. Beenker.Properties: eightfold symmetry, nonperiodic, repeating subsetsFigure : A subset of the Ammann-Beenker tiling modified from [1]Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

SubstitutionBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionPrototiles: a rhombus with 45 ◦ and 135 ◦ angles and a squaredivided into two isosceles trianglesBrittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

SubstitutionBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionPrototiles: a rhombus with 45 ◦ and 135 ◦ angles and a squaredivided into two isosceles trianglesInflate each prototile by α = 1 + √ 2Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

SubstitutionBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionPrototiles: a rhombus with 45 ◦ and 135 ◦ angles and a squaredivided into two isosceles trianglesInflate each prototile by α = 1 + √ 2Subdivide the inflated tile with prototiles of the original sizeusing the matching rulesBrittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

SubstitutionBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionBrittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

DefinitionsBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionDefinitionLet 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)=1f (v) − f (w),where d(v, w) is the shortest edge distance from u to v.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

DefinitionsBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionDefinitionLet 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)=1f (v) − f (w),where d(v, w) is the shortest edge distance from u to v.DefinitionEquivalently, the Laplacian ∆ := D − A. The degree matrix, D,is a zero matrix with the number of neighbors of tile i in the (i, i)entry. The adjacency matrix, A, is a zero matrix with a 1 in the(i, j) entry if tile i is neighbors with tile j.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

ExampleBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionExampleGiven a graph with 3 nodes, let us compute the Laplacian.Consider the graph P 3 , and the function f (1) = a, f (2) = b,f (3) = c where a, b, c ∈ R. By definition:∆f (1) =∑d(1,w)=1f (1) − f (w) = f (1) − f (2) = a − b.With similar computations for ∆f (2) and ∆f (3), we get thefollowing column vectors:⎛ ⎞ ⎛⎞af = ⎝b⎠ and ∆f = ⎝ca − b(b − a) + (b − c)c − b⎠ .Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

ExampleBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionExampleThus the Laplacian matrix is⎛1 −1⎞0∆ = ⎝−1 2 −1⎠ .0 −1 1Notice that this matrix is equivalent to subtracting the adjacencymatrix A from the degree matrix D:⎛ ⎞ ⎛ ⎞⎛⎞1 0 0 0 1 01 −1 0D = ⎝0 2 0⎠ , A = ⎝1 0 1⎠ , (D−A) = ⎝−1 2 −1⎠ .0 0 1 0 1 00 −1 1Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

DefinitionsBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionDefinitionThe spectrum of a matrix is the set of its eigenvalues (withmultiplicity).ExampleThe spectrum of⎛1 −1⎞0∆ = ⎝−1 2 −1⎠0 −1 1from the last example is {0, 1, 3}.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

LabelingBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionBrittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Internal AdjacenciesBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionThe internal adjacencies for the triangle:Tile 1 Side Tile 2 SideTriangle 1 3 Rhombus 1 2Triangle 2 3 Rhombus 2 1Triangle 2 2 Rhombus 1 1Triangle 3 3 Rhombus 2 2Triangle 3 2 Rhombus 1 4Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Starting IterationBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionThe external adjacencies for a triangle side 1 neighboring a triangleon side 1:Child of Tile 1 Side Child of Tile 2 SideTriangle 1 2 Triangle 1 2Triangle 2 1 Triangle 2 1Rhombus 2 4 Rhombus 2 4Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionGenerating the TilingStore tiles as data structures. Each structure is assigned anid, shape, children, and n1, n2, n3, n4 if such neighborsexist.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionGenerating the TilingStore tiles as data structures. Each structure is assigned anid, shape, children, and n1, n2, n3, n4 if such neighborsexist.We write a function genABTiling that takes a cell array oftiles as input. This function applies the substitution rules bygenerating the new tiles for the next iteration and establishingthe internal and external adjacencies.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionGenerating the TilingStore tiles as data structures. Each structure is assigned anid, shape, children, and n1, n2, n3, n4 if such neighborsexist.We write a function genABTiling that takes a cell array oftiles as input. This function applies the substitution rules bygenerating the new tiles for the next iteration and establishingthe internal and external adjacencies.To create the data structures representing the starting square,we enter the following 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 . i d ;t r i a n g l e 2 . n1 = t r i a n g l e 1 . i d ;Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionGenerating the TilingFrom the tilings generated by genABTiling, we form anadjacency matrix in makeLaplacian.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionGenerating the TilingFrom the tilings generated by genABTiling, we form anadjacency matrix in makeLaplacian.Since this tiling has conventionally been studied with rhombiand squares, we call the function trianglePairs to output acell array of pairs of ids corresponding to 2 triangles meetingalong side 1. We use these pairs to create an adjacencymatrix representing the tiling with rhombi and squares.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionGenerating the TilingFrom the tilings generated by genABTiling, we form anadjacency matrix in makeLaplacian.Since this tiling has conventionally been studied with rhombiand squares, we call the function trianglePairs to output acell array of pairs of ids corresponding to 2 triangles meetingalong side 1. We use these pairs to create an adjacencymatrix representing the tiling with rhombi and squares.Finally, we create the degree matrix and subtract theadjacency matrix from it to get the Laplacian.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

DefinitionsBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionDefinitionA cumulative distribution function (CDF) describes theprobability that a real-valued random variable X with a givenprobability distribution will be found at a value ≤ x ∈ R.DefinitionThe support of a CDF is the smallest closed set whosecomplement has probability zero.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Eigenvalues PlotBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionBrittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionCumulative Distribution FunctionBrittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Hausdorff DimensionBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionDefinition (Taken from [3])Let X ⊆ R n . Let |U| := sup{|x − y| : x, y ∈ U}. We say {U i } is aδ-cover of X if X ⊂ ⋃ ∞i=1 U i and 0 ≤ |U i | ≤ δ. For all δ > 0,{ ∞}Hδ s (X ) := inf ∑|U i | s : {U i } is a δ - cover of F .i=1The s-dimensional Hausdorff measure of X isH s (X ) := lim Hδ s (X ).δ→0The Hausdorff dimension of X isdim H X := inf{s ≥ 0 : H s (X ) = 0} = sup{s : H s (X ) = ∞}.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionHausdorff Dimension ExamplesThe Hausdorff dimension of the unit interval [0, 1] is 1.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionHausdorff Dimension ExamplesThe Hausdorff dimension of the unit interval [0, 1] is 1.The Hausdorff dimension of an open ball in R 2 is 2.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionHausdorff Dimension ExamplesThe Hausdorff dimension of the unit interval [0, 1] is 1.The Hausdorff dimension of an open ball in R 2 is 2.The Hausdorff dimension of the Cantor set is between 0 and 1.Figure : The Middle Third Cantor SetBrittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionApproximating the Hausdorff DimensionInput the list of eigenvalues corresponding to our largestsubstitution iteration of the Ammann-Beenker tiling.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionApproximating the Hausdorff DimensionInput the list of eigenvalues corresponding to our largestsubstitution iteration of the Ammann-Beenker tiling.Examine δ-covers of the interval [0, 8].Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionApproximating the Hausdorff DimensionInput the list of eigenvalues corresponding to our largestsubstitution iteration of the Ammann-Beenker tiling.Examine δ-covers of the interval [0, 8].Consider intervals in [0, 8] of length 1 2 i , where i ∈ {1, ..., 7}.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionApproximating the Hausdorff DimensionInput the list of eigenvalues corresponding to our largestsubstitution iteration of the Ammann-Beenker tiling.Examine δ-covers of the interval [0, 8].Consider intervals in [0, 8] of length 1 2 i , where i ∈ {1, ..., 7}.Keep track of the least and greatest eigenvalues found in theinterval. Determine the smallest length of this intervalcontaining all eigenvalues in the interval.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionApproximating the Hausdorff DimensionInput the list of eigenvalues corresponding to our largestsubstitution iteration of the Ammann-Beenker tiling.Examine δ-covers of the interval [0, 8].Consider intervals in [0, 8] of length 1 2 i , where i ∈ {1, ..., 7}.Keep track of the least and greatest eigenvalues found in theinterval. Determine the smallest length of this intervalcontaining all eigenvalues in the interval.For values of s ranging from 0 to 1.1, calculate Hδ s (X ) bysumming over the length of each interval raised to s.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Background and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionApproximating the Hausdorff DimensionWe want to find the s value in the definition of Hausdorffdimension where H s (X ) jumps from ∞ to 0, where X is the set ofeigenvalues. To do this, we use a methodology similar to the onedetailed by Chorin in [2] 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 3.09Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Future WorkBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionDue to limited computing power, we were only able togenerate the first five substitution iterations of the tiling. Inthe future, a better approximation could be calculated byusing larger subsets of the infinite tiling.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Future WorkBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionDue to limited computing power, we were only able togenerate the first five substitution iterations of the tiling. Inthe future, a better approximation could be calculated byusing larger subsets of the infinite tiling.The same approach that we used for the Ammann-Beenkertiling can be used to study other aperiodic tilings that modelquasicrystals with different symmetries.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

Future WorkBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionDue to limited computing power, we were only able togenerate the first five substitution iterations of the tiling. Inthe future, a better approximation could be calculated byusing larger subsets of the infinite tiling.The same approach that we used for the Ammann-Beenkertiling can be used to study other aperiodic tilings that modelquasicrystals with different symmetries.The approximation of the spectrum that we have calculatedprovides information to physicists about the movement of theelectrons in a quasicrystal. With the information we haveprovided, more research can be performed in that area as well.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

AcknowledgementsBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionWe would like to thank our mentor, May Mei, for herguidance throughout this research project and our project TA,Drew Zemke, for assistance throughout this project, especiallywith MatLab.We would also like to thank the Cornell UniversityMathematics Department and the NSF for funding thisproject.We also thank Baake, Grimm, and Moody for their detailedexposition, “What is aperiodic order?” and for inspiring theimages used in this presentation.Finally, we thank Derek Young for his contributions to theintroduction of this research project.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals

BibliographyBackground and MotivationGenerating the TilingAnalyzing the SpectrumHausdorff DimensionM. Baake, U. Grimm, and R. Moody.What is aperiodic order?arXiv:0203252, 2002.Alexandre Joel Chorin.Numerical estimates of hausdorff dimension.Journal of Computational Physics, 46(3):390 – 396, 1982.K. Falconer.Hausdorff Measure and Dimension, pages 27–38.John Wiley & Sons, Ltd, 2005.L. Sadun.Topology of tiling spaces, volume 46 of University LectureSeries.American Mathematical Society, Providence, RI, 2008.Brittany Livsey, Jason Mifsud, and Francesca RomanoUsing the Ammann-Beenker Tiling to Model Quasicrystals