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Using the Ammann-Beenker Tiling to Model QuasicrystalsBrittany Livsey, Jason Mifsud, and Francesca Romano
Cornell University Department of Mathematics Summer Math Institute
IntroductionIt was shown in [6] that a two-dimensional quasicrystal with aneightfold symmetry could be produced in a laboratory. “Thecorresponding high-resolution electron microscopic image agreeswell with the 2D eightfold quasilattice consisting of squaresand 45◦ rhombi” [6]. This particular quasicrystal is modeledmathematically as an aperiodic octagonal tiling, known as theAmmann-Beenker tiling, shown in Figure 1. The Ammann-Beenker tiling is a nonperiodic tiling with repeating subsets andeightfold symmetry named after the two mathematicians who in-dependently discovered it, Robert Ammann and F.P.M. Beenker.
Figure 1: A subset of the Ammann-Beenker tiling modified from [2]
BackgroundDefinition 1 A tiling is nonperiodic if it lacks translational sym-metry.
Definition 2 A set of tiles is aperiodic if it admits only nonperi-odic tilings.
We explored three methods to generate the Ammann-Beenkertiling: cut and project, matching, and substitution. Ammann wasable to generate iterations of the tiling through the substitutionand matching methods [1], while Beenker employed the cut andproject method [3]. In this project, we use the substitution methodto generate increasingly larger subsets of the Ammann-Beenkertiling in order to approximate the spectrum of the Laplacian op-erator.We use a rhombus with 45◦ and 135◦ angles and a square dividedinto two isosceles triangles as the prototiles. In this method, weinflate each prototile by α = 1 +
√2 and then subdivide the in-
flated tiles into prototiles of the original size, as shown in Fig-ure 2. Note that this substitution follows a set of matching rules.These rules are determined by edge markings and vertex deco-rations. The edge markings must match together to form singleor double arrows, and the vertex decorations must align so thatthey form houses. We iterate through this process to form largersubsets of the tiling.
Figure 2: The substitution process for the octagonal tiling
ResultsDefinition 3 The Laplacian ∆ = D − A, where D is the degreematrix of a graph G and A is the adjacency matrix of G. A degreematrix, D, is a zero matrix with the number of neighbors of tile iin the (i, i) entry. An adjacency matrix, A, is a zero matrix with a1 in the (i, j) entry if tile i and tile j are neighbors.
Definition 4 The spectrum of a matrix is the set of eigenvaluesof the Laplacian matrix (with multiplicity).
In order to study the spectrum of the Ammann-Beenker Lapla-cian, we need a way to generate increasingly large subsets ofthe tiling. To do this, we wrote a function in MATLAB that takesthe tiles from the current iteration and outputs the tiles of thenext iteration. We define the two isosceles triangles that corre-spond to the first substitution iteration as data structures in theterminal, and then pass these tiles to the function to generatethe next iteration. To generate subsequent iterations, we use thefunction’s output as input for the next iteration.
We plot each spectrum as shown in Figure 3. We observe thatthe spectrum is bounded by 8, which is consistent with [5], whichstates that the spectrum of any graph’s Laplacian is bounded bytwice the degree of the highest degree node in the graph.
Figure 3: Plot of the eigenvalues of the Laplacian for iterations ofthe Ammann-Beenker tiling
To further analyze the spectrum, we use a cumulative distribu-tion function on the set of eigenvalues with uniform probabilityas shown in Figure 4. It can be shown that these functions have alimiting function. The limiting function is the integrated densityof states of the spectrum of the Ammann-Beenker Laplacian. Thederivative of this function is the density of states, which providesinformation on the location of electrons in a substance.In addition, the support, which is the smallest closed set whosecomplement has probability zero, of the limiting function is thespectrum of the Ammann-Beenker Laplacian. Notice the steepslope of the fifth iteration in this plot around 4. This is a resultof a high number of eigenvalues in the fifth iteration equal to 4.Notice the flat region of the fifth iteration around 5. This corre-sponds with the gap around eigenvalues close to 5 in Figure 3.
Figure 4: Plot of the CDF of the spectrum of the Laplacian
We further analyze the spectrum through the Hausdorff dimen-sion by inputting the list of eigenvalues corresponding to ourlargest substitution iteration of the Ammann-Beenker tiling intoa MATLAB function. We want to find the s value in the tablethat corresponds to columns that are neither strictly increasingnor strictly decreasing. To do this we 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 14.35 8.59 5.15 3.09
Notice that when s = 0.75, we see a strictly increasing pattern asi increases, and when s = 0.95, we see a strictly decreasing pat-tern as i increases. However, the column between 0.75 and 0.95has neither a strictly increasing nor strictly decreasing pattern.From this, we can infer that the Hausdorff dimension of the spec-trum of the Laplacian is between 0.75 and 0.95. This implies thatthe spectrum has fractal properties, which provides informationabout the quantum diffusion rates in quasicrystals [4].
Future workWe have given an approximation of the spectrum of the Lapla-cian for the Ammann-Beenker tiling, but a better approximationcould be achieved. Due to limited computing power, we wereonly able to generate the first five substitution iterations of thetiling. In the future, a better approximation could be calculatedby using larger subsets of the infinite tiling.The same approach that we used for the Ammann-Beenker tilingcan be used to study other aperiodic tiliings that model quasicrys-tals with different symmetries.The approximation of the spectrum that we have calculated pro-vides information to physicists about the movement of the elec-trons in a quasicrystal. With the information we have provided,more research can be performed in that area as well.
References[1] R. Ammann, B. Grnbaum, and G.C. Shephard. Aperiodic tiles.
Discrete & Computational Geometry, 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 Building Blocks: A Square and a Rhombus.TH report. Eindhoven University of Technology, 1982.
[4] I. Guarneri. Spectral properties of quantum diffusion on dis-crete lattices. EPL (Europhysics Letters), 10(2):95, 1989.
[5] D. Spielman. Spectral graph theory. Lecture Notes, 2009.[6] N. Wang, H. Chen, and K. H. Kuo. Two-dimensional qua-
sicrystal with eightfold rotational symmetry. Phys. Rev. Lett.,59:1010–1013, Aug 1987.
AcknowledgmentsWe would like to thank our mentor, May Mei, for her guid-ance throughout this research project and our project TA, DrewZemke, for assistance throughout this project, especially withMATLAB. We would also like to thank the Cornell Universitymath department and the NSF for funding this project. Addition-ally, we thank Baake, Grimm, and Moody for their detailed ex-position, “What is aperiodic order?” and for inspiring the imagesused in our paper.
Contact Information• Brittany Livsey, Georgetown [email protected]
• Jason Mifsud, Binghamton [email protected]
• Francesca Romano, Siena [email protected]
Advisor: May Mei, Denison University, [email protected]
