On the Spectrum of the Penrose Laplacianpersonal.denison.edu/~meim/SMI/...Presentation.pdf · History of Penrose tiling (Gardner) In 1973, Roger Penrose found a set of tiles that

Penrose TilingLaplacian Matrix and Spectrum

Hausdorff DimensionSummary of Results

On the Spectrum of the Penrose Laplacian

Michael Dairyko, Christine Hoffman,Julie Pattyson, Hailee Peck

Summer Math Institute

August 2, 2013

M. Dairyko, C. Hoffman, J. Pattyson, H. Peck On the Spectrum of the Penrose Laplacian



1 Penrose TilingDefinitionsSubstitution MethodComputational Methods

2 Laplacian Matrix and SpectrumDefinitionsExampleComputational Methods

3 Hausdorff DimensionDefinitionsExampleComputational Methods

4 Summary of Results




Background

History of quasicrystals (Shechtman)

Before 1982, crystalline structures were defined as havingperiodic structure.

In 1982, Dan Shechtman was able to create a crystal-likestructure that defied the property of periodicity in crystals,now known as quasicrystals.

Quasicrystalline structures have long-range order but do notexhibit the periodic patterns that characterized crystals.




Background

History of Penrose tiling (Gardner)

In 1973, Roger Penrose found a set of tiles that tile onlynonperiodically.

Penrose tiles generate a nonperiodic tiling which is the mostpopular two-dimensional model of a quasicrystal with fivefoldsymmetry.

The Laplacian on the Penrose tiling models an electronmoving through quasicrystalline matter.




Penrose Tiling

Figure: Subset of the Penrose tiling with rhombus and diamond




Goals

Generate a large finite iteration of the Penrose tiling

Construct the dual graph of this iteration to find the Laplacian

Approximate the spectrum and Hausdorff dimension for thePenrose tiling as they provide insight into the characteristicsof electron diffusion in quasicrystals




DefinitionsSubstitution MethodComputational Methods

Definition (Frank)

Prototiles - A set of finite inequivalent tiles (i.e. are notequivalent under rigid motions, expansions, or contractions).

Definition (Frank)

Tiling - An arrangement of tiles, such that their union covers andpacks R2 so that distinct tiles have non-intersecting interiors.

Definition (Grunbaum)

Nonperiodic- Tiling which does not have a period.

Definition (Grunbaum)

Aperiodic - Set of prototiles which admit infinitely many tilings ofthe plane, none of which are periodic.





Converting Penrose prototiles into Robinson triangles (Frank)

Raphael Robinson took the set of two Penrose prototiles, arhombus and diamond, and divided them into a set of fourtriangles.

These tiles, known as Robinson triangles, can also be used togenerate a Penrose tiling.

Once our tiling is generated, we must return to the original setof two prototiles to obtain an end result of a Penrose tiling.





Substitution Method (Frank)

We use the substitution method to generate the Penrose tilingwith Robinson triangles by starting with a finite subset of thetiling.

To generate new iterations of the Penrose tiling, we inflate

each triangle prototile by 1+√

52 and then subdivide it,

expanding the tiling to cover the plane.

The rules for subdividing each of the four inflated baseprototiles are shown on the next slide.





Inflate and Subdivide

Note: Triangles are not drawn to scale.





Substitution Rules

Rule Shared Edge External Adjacencies

1P3

P2W2B2

B3

P1 Y1 B1

adj

P31 − B3

2 , B11 − P1

2

2B1

B1Y1P1

P1

B2 W2 P2

adj

P21 − B2

2 , Y 31 − W 3

2

3Y 3

Y3P3

W 3

W4 B4

adj

Y 23 − W 2

4









Using MatLab to generate finite iterations

We created a function genPenTiling.m that generates finiteiterations of the Penrose tiling using this substitution method.





MatLab Code

genPenTiling.m

Inputs oldTiles

Creates children tiles and adjacenciesUses the substitution method to generate the next iterationOutputs newTiles

Demonstration

Tile1 = struct(‘color’ , ‘p’ , ‘id’ , 1 , ‘n3’ , 2);

Tile2 = struct(‘color’ , ‘b’ , ‘id’ , 2 , ‘n3’ , 1);

oldTiles = ( {Tile1 , Tile2} );

newTiles = genPenTiling( oldTiles );

id: 1 id: 2 id: 3 id: 4 id: 5 id: 6

color: ‘b’ color: ‘w’ color: ‘p’ color: ‘p’ color: ‘y’ color: ‘b’

n1: 2 n2: 3 n2: 2 n1: 5 n2: 6 n2: 5

n3: 4 n1: 1 n1: 6 n3: 1 n1: 4 n1: 3




DefinitionsExampleComputational Methods

Definition (Sheng)

The Laplacian matrix ∆ of a finite graph G is

∆(G ) = D(G )− A(G ).

Definition (Chung)

In a graph G , let u and v be vertices and dv the degree of vertex v

∆(u, v) =

dv if u = v ,−1 if u and v are adjacent,

0 otherwise.

Definition

The spectrum of ∆ is the set of eigenvalues of ∆.





Definition

Let f : V (G )→ R be a function that assigns a real value to eachvertex of the graph G . The Laplacian operator is a function ∆acting on f , defined by

∆f (v) =∑

w :d(v ,w)=1

f (v)− f (w)

which sums the difference of the real values of adjacent nodes.

Remark

The above definition does not require that G has finitely manyvertices.





4

3

6 5

2 1

Example

degree of n Laplacian entry of (a, b)

deg(1) = 2 = ∆(1, 1) ∆(1, 2) = −1 = ∆(2, 1)deg(2) = 3 = ∆(2, 2) ∆(1, 5) = −1 = ∆(5, 1)deg(6) = 1 = ∆(6, 6) ∆(1, 3) = 0 = ∆(3, 1)





4

3

6 5

2 1

Example

D(G ) =

2 0 0 0 0 00 3 0 0 0 00 0 2 0 0 00 0 0 3 0 00 0 0 0 3 00 0 0 0 0 1

A(G ) =

0 1 0 0 1 01 0 1 0 1 00 1 0 1 0 00 0 1 0 1 11 1 0 1 0 00 0 0 1 0 0





Example

∆(G ) =

2 −1 0 0 −1 0−1 3 −1 0 −1 0

0 −1 2 −1 0 00 0 −1 3 −1 −1−1 −1 0 −1 3 0

0 0 0 −1 0 1

σ(∆) = {0, 0.7216, 1.6826, 3, 3.7046, 4.8912}





MatLab Code

tilingLaplacian.m

Inputs newTiles

Returns Robinson triangles to Penrose prototilesConstructs adjacency and degree matricesForms Laplacian matrixOutputs eigenvalues

Remark

The Laplacian matrix is constructed using the dual graph of thePenrose tiling. Each tile’s unique id number corresponds to avertex.





Demonstration


A =

0 1 0 1 0 01 0 1 0 0 00 1 0 0 0 11 0 0 0 1 00 0 0 1 0 10 0 1 0 1 0





Demonstration


A =

0 1 0 1 0 01 0 1 0 0 00 1 0 0 0 11 0 0 0 1 00 0 0 1 0 10 0 1 0 1 0

, A =

0 1 0 1 0 01 0 1 0 0 00 1 0 0 0 11 0 0 0 1 00 0 0 1 0 10 0 1 0 1 0





Demonstration


AJ =

0 1 0 1 01 0 1 0 00 1 0 0 11 0 0 0 10 0 1 1 0





Demonstration

∆(G ) =

2 −1 0 −1 0−1 2 −1 0 0

0 −1 2 0 −1−1 0 0 2 −1

0 0 −1 −1 2

σ(∆) = {0, 1.3820, 1.3820, 3.6180, 3.6180}





MatLab Code

plotMat.m

Inputs eigenvaluesPlots eigenvalues by iterationOutputs plot





0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Eigenvalues

Itera

tion

of P

enro

se T

iling

Spectrum of Finite Iterations of Penrose Tiling





Definition

The cumulative distribution function gives the probability that arandom eigenvalue is less than or equal to a given real-valuednumber.

MatLab Code

cdf.m

Inputs eigenvaluesThe probability increases by 1

n at every eigenvalueOutputs CDF





0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

Real Numbers

Pro

babi

lity

Cumulative Distribution Function





Hausdorff Dimension

Definition

Let X ⊂ Rn be nonempty, and define|U| := sup {|x − y | : x , y ∈ U}. We say {Ui} is a δ-cover of X if

1 X ⊂∞⋃i=1

Ui

2 0 ≤ |Ui | ≤ δ





Definition (Falconer, Equations 2.1, 2.2)

Let X ⊂ Rn be nonempty and s be any nonnegative real number.For δ > 0 define

H sδ (X ) := inf

{ ∞∑i=1

|Ui |s : Ui is a δ-cover of X

}.

The s-dimensional Hausdorff measure of X , denoted H s(X ) is

limδ→0

H sδ (X ).





Definition

Let X ⊂ Rn be nonempty and s be any nonnegative real number.For δ > 0 define

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

We call dimH(X ) the Hausdorff dimension of X.





Hausdorff dimension

s0 dimH(X )

H s(X )

∞





Example

Let us divide the interval A = [0, 1] into n closed subintervals ofequal length. Let Ui :=

[in ,

i+1n

], where i = {0, 1, . . . , n − 1}.

Therefore {Ui} is a δ-cover of A. Let

∑|Ui |s = n ·

(1

n

)s

=

(1

n

)s−1

be an approximation of H sδ (A).





Example

As δ → 0, that is n→∞ we have

H s(A) =

0 if s > 1,

1 if s = 1

∞ otherwise.

This indicates that the Hausdorff dimension of A is 1 which agreeswith our intuition about the topological dimension of A.





s0 1

H s(X )

∞

Hausdorff Dimension for [0, 1] with δ = 1n gleaned from Falconer





MatLab Code

ndsTable.m

Inputs eigenvalues and s ∈ [0, 1]Partitions [0, 8] into n intervalsApproximates δ → 0 by 1

2i for i ∈ {1, 2, . . . , 10}Outputs columns of nδs values





MatLab Code

ndsTable.m

Inputs eigenvalues and s ∈ [0, 1]Partitions [0, 8] into n intervalsApproximates δ → 0 by 1

2i for i ∈ {1, 2, . . . , 10}Outputs columns of nδs values





Table: Values of nδs as Functions of i and s

s

i .55 .65 .75 .85 .95 .99 .99999

1 9.56 8.92 8.32 7.76 7.24 7.04 7.002 12.59 10.96 9.54 8.31 7.23 6.84 6.753 17.20 13.97 11.35 9.22 7.48 6.89 6.754 23.28 17.64 13.37 10.13 7.68 6.87 6.685 31.36 22.17 15.68 11.08 7.84 6.82 6.596 41.93 27.66 18.25 12.04 7.94 6.72 6.457 52.08 32.05 19.73 12.14 7.47 6.15 5.86

Remark

As δ → 0 (down the column), H sδ = 0, or ∞. If the values increase (resp.

decrease), the estimated s is not the Hausdorff dimension, as the values aregoing to ∞ (resp. 0). The values of s in which the columns neither clearlydecrease nor increase provide an interval for the Hausdorff dimension.




Summary of Results

We know that the spectrum of the Penrose Laplacian isbounded by eight, which is twice the highest degree of anyvertex (see, for example, Spielman).

The results on the first seven iterations illustrate this.

We estimate the Hausdorff dimension of the PenroseLaplacian spectrum to be between .85 and .99.

This implies that the spectrum of the Penrose Laplacian hasfractal structure.




Future Work

We would like to explore ways to generate more iterations ofthe Penrose tiling to improve the estimate of the Hausdorffdimension.

We hope to use the methods established here to generalizethese results to other aperiodic tilings.




We would like to thank Dr. May Mei and Drew Zemke for theirhelp with this project. We would also like to thank the SummerMath Institute and the Mathematics Department at CornellUniversity for the use of their resources. This work was supportedby NSF grant DMS-0739338.





Documents

On the Spectrum of the Penrose Laplacianpersonal.denison.edu/~meim/SMI/...Presentation.pdf · History of Penrose tiling (Gardner) In 1973, Roger Penrose found a set of tiles that