On the Spectrum of the Penrose Laplacian

Transcription

1 On the Spectrum of the Penrose Laplacian Michael Dairyko, Christine Hoffman, Julie Pattyson, Hailee Peck Summer Math Institute August 2, 2013

2 1 Penrose Tiling Substitution Method 2 3 4

3 Background Penrose Tiling History of quasicrystals (Shechtman) Before 1982, crystalline structures were defined as having periodic structure. In 1982, Dan Shechtman was able to create a crystal-like structure that defied the property of periodicity in crystals, now known as quasicrystals. Quasicrystalline structures have long-range order but do not exhibit the periodic patterns that characterized crystals.

4 Background Penrose Tiling History of Penrose tiling (Gardner) In 1973, Roger Penrose found a set of tiles that tile only nonperiodically. Penrose tiles generate a nonperiodic tiling which is the most popular two-dimensional model of a quasicrystal with fivefold symmetry. The Laplacian on the Penrose tiling models an electron moving through quasicrystalline matter.

5 Penrose Tiling Figure: Subset of the Penrose tiling with rhombus and diamond

6 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 the Penrose tiling as they provide insight into the characteristics of electron diffusion in quasicrystals

7 Substitution Method Definition (Frank) Prototiles - A set of finite inequivalent tiles (i.e. are not equivalent under rigid motions, expansions, or contractions). Definition (Frank) Tiling - An arrangement of tiles, such that their union covers and packs R 2 so that distinct tiles have non-intersecting interiors. Definition (Grünbaum) Nonperiodic- Tiling which does not have a period. Definition (Grünbaum) Aperiodic - Set of prototiles which admit infinitely many tilings of the plane, none of which are periodic.

8 Substitution Method Converting Penrose prototiles into Robinson triangles (Frank) Raphael Robinson took the set of two Penrose prototiles, a rhombus and diamond, and divided them into a set of four triangles. These tiles, known as Robinson triangles, can also be used to generate a Penrose tiling. Once our tiling is generated, we must return to the original set of two prototiles to obtain an end result of a Penrose tiling.

9 Substitution Method Substitution Method (Frank) We use the substitution method to generate the Penrose tiling with Robinson triangles by starting with a finite subset of the tiling. To generate new iterations of the Penrose tiling, we inflate each triangle prototile by and then subdivide it, expanding the tiling to cover the plane. The rules for subdividing each of the four inflated base prototiles are shown on the next slide.

10 Inflate and Subdivide Substitution Method Note: Triangles are not drawn to scale.

11 Substitution Rules Penrose Tiling Substitution Method Rule Shared Edge External Adjacencies adj P 3 B 3 1 P1 3 B3 2, B1 1 P1 2 B 2 W 2 P 2 P 1 Y 1 B 1 2 B 1 adj P 1 P 2 1 B2 2, Y 3 1 W 3 2 P 1 Y 1 B 1 B 2 W 2 P 2 3 Y 3 adj W 3 Y 2 3 W 2 4 P 3 Y 3 W 4 B 4

12 Substitution Method

13 Substitution Method Using MatLab to generate finite iterations We created a function genpentiling.m that generates finite iterations of the Penrose tiling using this substitution method.

14 Substitution Method MatLab Code genpentiling.m Inputs oldtiles Creates children tiles and adjacencies Uses the substitution method to generate the next iteration Outputs 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

15 Definition (Sheng) The Laplacian matrix of a finite graph G is Definition (Chung) (G) = D(G) A(G). In a graph G, let u and v be vertices and d v the degree of vertex v d v if u = v, (u, v) = 1 if u and v are adjacent, 0 otherwise. Definition The spectrum of is the set of eigenvalues of.

16 Definition Let f : V (G) R be a function that assigns a real value to each vertex of the graph G. The Laplacian operator is a function acting on f, defined by f (v) = f (v) f (w) w:d(v,w)=1 which sums the difference of the real values of adjacent nodes. Remark The above definition does not require that G has finitely many vertices.

17 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)

18 D(G) = A(G) =

19 (G) = σ( ) = {0, , , 3, , }

20 MatLab Code tilinglaplacian.m Inputs newtiles Returns Robinson triangles to Penrose prototiles Constructs adjacency and degree matrices Forms Laplacian matrix Outputs eigenvalues Remark The Laplacian matrix is constructed using the dual graph of the Penrose tiling. Each tile s unique id number corresponds to a vertex.

21 Demonstration newtiles = genpentiling( oldtiles ); A =

22 Demonstration newtiles = genpentiling( oldtiles ); A = , A =

23 Demonstration newtiles = genpentiling( oldtiles ); AJ =

24 Demonstration (G) = σ( ) = {0, , , , }

25 MatLab Code plotmat.m Inputs eigenvalues Plots eigenvalues by iteration Outputs plot

26 Iteration of Penrose Tiling 8 Spectrum of Finite Iterations of Penrose Tiling Eigenvalues

27 Definition The cumulative distribution function gives the probability that a random eigenvalue is less than or equal to a given real-valued number. MatLab Code cdf.m Inputs eigenvalues The probability increases by 1 n Outputs CDF at every eigenvalue

28 1 Cumulative Distribution Function 0.8 Probability Real Numbers

29 Definition Let X R n be nonempty, and define U := sup { x y : x, y U}. We say {U i } is a δ-cover of X if 1 X U i i=1 2 0 U i δ

30 Definition (Falconer, Equations 2.1, 2.2) Let X R n be nonempty and s be any nonnegative real number. For δ > 0 define { } Hδ s (X ) := inf U i s : U i is a δ-cover of X. i=1 The s-dimensional Hausdorff measure of X, denoted H s (X ) is lim H δ s δ 0 (X ).

31 Definition Let X R n be nonempty and s be any nonnegative real number. For δ > 0 define dim H (X ) := inf{s : H s (X ) = 0} = sup{s : H s (X ) = }. We call dim H (X ) the Hausdorff dimension of X.

32 Hausdorff dimension H s (X ) 0 dim H (X ) s

33 Let us divide the interval A = [0, 1] into n closed subintervals of equal length. Let U i := [ i n, i+1 ] n, where i = {0, 1,..., n 1}. Therefore {U i } is a δ-cover of A. Let ( ) 1 s Ui s = n = n be an approximation of H s δ (A). ( ) 1 s 1 n

34 As δ 0, that is n we have 0 if s > 1, H s (A) = 1 if s = 1 otherwise. This indicates that the Hausdorff dimension of A is 1 which agrees with our intuition about the topological dimension of A.

35 H s (X ) 0 1 for [0, 1] with δ = 1 n gleaned from Falconer s

36 MatLab Code ndstable.m Inputs eigenvalues and s [0, 1] Partitions [0, 8] into n intervals Approximates δ 0 by 1 2 i for i {1, 2,..., 10} Outputs columns of nδ s values

37 MatLab Code ndstable.m Inputs eigenvalues and s [0, 1] Partitions [0, 8] into n intervals Approximates δ 0 by 1 2 i for i {1, 2,..., 10} Outputs columns of nδ s values

38 Table: Values of nδ s as Functions of i and s s i Remark As δ 0 (down the column), Hδ s = 0, or. If the values increase (resp. decrease), the estimated s is not the Hausdorff dimension, as the values are going to (resp. 0). The values of s in which the columns neither clearly decrease nor increase provide an interval for the Hausdorff dimension.

39 We know that the spectrum of the Penrose Laplacian is bounded by eight, which is twice the highest degree of any vertex (see, for example, Spielman). The results on the first seven iterations illustrate this. We estimate the Hausdorff dimension of the Penrose Laplacian spectrum to be between.85 and.99. This implies that the spectrum of the Penrose Laplacian has fractal structure.

40 Future Work We would like to explore ways to generate more iterations of the Penrose tiling to improve the estimate of the Hausdorff dimension. We hope to use the methods established here to generalize these results to other aperiodic tilings.

41 We would like to thank Dr. May Mei and Drew Zemke for their help with this project. We would also like to thank the Summer Math Institute and the Mathematics Department at Cornell University for the use of their resources. This work was supported by NSF grant DMS

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