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WHAT IS…

Aperiodic Order?Michael Baake, David Damanik and Uwe Grimm

Communicated by Cesar E. Silva

With their regular shape andpronounced faceting, crystalshave fascinated humans for ages. About a century ago,the understanding of the internal structure increasedconsiderably through the work of Max von Laue (NobelPrize in Physics, 1914) and William Henry Bragg andWilliam Lawrence Bragg (father and son, joint Nobel Prizein Physics, 1915). They developed X-ray crystallographyand used it to show that a lattice-periodic array of atomslies at the heart of the matter. This became the acceptedmodel for solids with pure point diffraction, which waslater extended in various ways.

In 1982, the materials scientist Dan Shechtman dis-covered a perfectly diffractive solid with a noncrystal-lographic (icosahedral) symmetry; see Figure 1 for aqualitatively similar experimental diffraction image. Thisdiscovery, for which he received the 2011 Nobel Prizein Chemistry, was initially met with disbelief and heavycriticism, although such structures could have been ex-pected on the basis of Harald Bohr’s work on almostperiodic functions. In fact, the situation is a classic caseof a “missed opportunity.” Let us try to illustrate this alittle further and thus explain some facets of what is nowknown as the theory of aperiodic order.

Let us consider a uniformly discrete point set Λin Euclidean space, where the points are viewed as

Michael Baake is professor of mathematics at Bielefeld University.His email address is mbaake@math.uni-bielefeld.de.

David Damanik is Robert L. Moody Professor and Chair of mathe-matics at Rice University. His email address is damanik@rice.edu.

Uwe Grimm is professor and head of the Department of Mathe-matics and Statistics at the Open University outside London. Hisemail address is uwe.grimm@open.ac.uk.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1394

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Figure 1. Dan Shechtman received the 2011 NobelPrize in Chemistry for his discovery of “impossible”symmetries, such as the tenfold symmetry in thiselectron diffraction image of an AlMnPd alloy(intensity inverted).

idealizations of atomic positions.Much of the terminologyfor such point sets was developed by Jeffrey C. Lagarias.Placing unit point measures at each position in Λ leads tothe associated Dirac comb

𝛿Λ = ∑𝑥∈Λ

𝛿𝑥.

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L A S E R

O S

Figure 2. Schematic representation of an opticaldiffraction experiment. The object (O) is illuminatedby a coherent light source, and the diffractedintensity is collected on a screen (S), with sharpintensity peaks arising from constructiveinterference of scattered waves.

A diffraction experiment measures the correlation be-tween atomic locations. Mathematically, this is expressedthrough the diffraction measure, which is the Fouriertransform �̂�Λ of the autocorrelation

𝛾Λ = lim𝑅→∞

𝛿Λ∩𝐵𝑅 ∗ 𝛿−Λ∩𝐵𝑅vol(𝐵𝑅)

,

provided this limit exists in the vague topology, in whicha sequence of measures {𝜇𝑛} converges to 𝜇 if and only if𝜇𝑛(𝑓) → 𝜇(𝑓) for all continuous functions 𝑓 with compactsupport. This approach goes back to Hof (1995). Here,�̂�Λ describes the outcome of a (kinematic) diffractionexperiment, such as that of Figure 2 with an opticalbench, which should be available in most physics labs forexperimentation.

For a lattice periodic point set, the diffraction measureis supported on a discrete point set, namely, the duallattice. This implies that a tenfold symmetric diffractiondiagram, such as the one of Figure 1, cannot be producedby a lattice periodic structure, as lattices in two or threedimensions can only have two-, three-, four- or sixfoldrotational symmetry by the crystallographic restriction.This raises the question of what types of point setscan generate such new kinds of diffraction measures,which are pure point but display noncrystallographicsymmetries.

Let us begin with an example. Already in 1974, RogerPenrose constructed an aperiodic tiling of the entire plane,equivalent to the one shown in Figure 3. Taking this finitepatch and considering the set of vertices as the point setΛ, the diffraction measure is the one shown in Figure 4,which resembles what Alan L. Mackay observed whenhe performed an optical diffraction experiment with anassembly of small disks centered at the vertex pointsof a rhombic Penrose tiling shortly before the discoveryof quasicrystals. For an infinite tiling, the diffractionmeasure is pure point and tenfold symmetric.

The Penrose tiling is a particularly prominent exampleof a large class of point sets with pure point diffraction.Such sets, which were first introduced by Yves Meyer andare nowadays called model sets, are constructed from a

Figure 3. Fivefold symmetric patch of the rhombicPenrose tiling, which is equivalent to many otherversions. The arrow markings represent local rulesthat enforce aperiodicity.

cut and project scheme (CPS)

ℝ𝑑 𝜋←⎯⎯⎯⎯⎯⎯⎯⎯⎯ ℝ𝑑 ×𝐻 𝜋int⎯⎯⎯⎯⎯⎯⎯⎯→ 𝐻∪ ∪ ∪ dense

𝜋(ℒ) 1−1←⎯⎯⎯⎯⎯⎯⎯⎯⎯ ℒ ⎯⎯⎯⎯⎯⎯⎯→ 𝜋int(ℒ)‖ ‖𝐿 ⋆⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ 𝐿⋆

where ℒ is a lattice in ℝ𝑑 × 𝐻 whose projection 𝜋int(ℒ)to “internal space” 𝐻 is dense. The projection 𝜋 to“physical space” ℝ𝑑 is required to be injective on ℒso that 𝜋 is a bijection between ℒ and 𝐿 = 𝜋(ℒ).This provides a well-defined mapping ⋆ ∶ 𝐿 ⟶ 𝐻 with𝑥 ↦ 𝑥⋆ = 𝜋int((𝜋|ℒ)−1(𝑥)). The internal space𝐻 is often aEuclidean space, but the general theory works for locallycompact Abelian groups. A model set is obtained bychoosing a suitable “window” 𝑊 ⊂ 𝐻 and defining thepoint set

Λ = {𝑥 ∈ 𝐿 ∣ 𝑥⋆ ∈ 𝑊} ⊂ ℝ𝑑.The Penrose point set arises in this framework with atwo-dimensional Euclidean internal space and the rootlattice 𝐴4, while the generalisation to model sets withicosahedral symmetrywas first discussed by Peter Kramerin 1984. It was Robert V. Moody who recognized theconnections toMeyer’s abstract concepts and championedtheir application in the theory of aperiodic order and theirfurther development.

For a nonempty, compact window that is the closureof its interior and whose boundary has Haar measure 0,the resulting model set has a diffraction measure that issupported on the projection𝜋(ℒ∗) of the dual lattice and

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Figure 4. Central part of the (intensity inverted)diffraction image of the patch of Figure 3, with pointmeasures placed on all vertex points of the rhombictiling.

hence is a pure point measure. The diffraction intensitiescan be calculated explicitly in this setting.

While a pure point diffraction detects order, it is nottrue that all ordered structures havepure point diffraction.As an example, consider the bi-infinite sequence

…0110100110010110|0110100110010110…and notice that it is invariant under the square of theThue–Morse substitution 0 ↦ 01, 1 ↦ 10. Now, take onlythose 𝑥 ∈ ℤ that correspond to the positions of 1sin this sequence. The diffraction measure of this one-dimensional point set is not pure point, as it contains anontrivial singular continuous component.

There is a very useful connection between the diffrac-tion measure and the spectral measures of an associateddynamical system. Starting from a point set Λ ⊂ ℝ𝑑 offinite local complexity (which means that, up to transla-tions, there are only finitely many patches for any givensize), we define its hull

𝕏(Λ) = {𝑥 + Λ ∣ 𝑥 ∈ ℝ𝑑}where the closure is taken in the local topology (closenessforces coincidence on a large ball around the origin upto a small translation). ℝ𝑑 acts on 𝕏(Λ) by translations.Select an invariant probability measure 𝜇, which is alwayspossible. Then, one gets an action of ℝ𝑑 on 𝐿2(𝕏,𝜇)via unitary operators. It is a fundamental result thatthe diffraction measure is pure point if and only if allspectral measures are pure point. More generally, thespectral measures correspond to diffraction measures

associated with elements of certain topological factors ofthe dynamical system (𝕏,ℝ𝑑, 𝜇).

For example, the diffraction measure of the Thue–Morse point set consists of the pure point part 𝛿ℤ and anontrivial singular continuous component. The nontrivialpoint part of the dynamical spectrum, which is ℤ[ 1

2], istherefore not fully detected by the diffraction measureof the Thue–Morse point set and shows up only in thediffraction measure of a global 2-to-1 factor, which is amodel set with 𝐻 = ℤ2, the 2-adic numbers.

It is a somewhat surprising insight that the diffractionmeasure, which is designed to reveal as much as possibleabout the distributions of points and is thus clearly notinvariant under topological conjugacy, and the dynamicalspectrum, which is an important invariant under metricconjugation of dynamical systems and thus blind to de-tails of the representative chosen, have such an important

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Figure 5. Numerical approximation of the spectrum ofthe 1D Fibonacci Hamiltonian (top) and of theCartesian product of two of them (bottom). The 𝑥-axiscorresponds to the energy 𝐸, and the 𝑦-axis to 𝜆. Theplots illustrate the instant opening of a dense set ofgaps for the 1D model as 𝜆 is turned on, whereas forthe 2D model there are no gaps in the spectrum forall sufficiently small 𝜆.

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While the relation between dynamical and diffractionspectra is by now well understood, it continues to be anintriguing open problem to find the connection betweenthese spectra and the spectra of Schrödinger operatorsassociated with aperiodic structures. The interest inthe latter arises from quantum transport questions inaperiodically ordered solids. In particular, anomaloustransport properties have for a long time been expected(and are observed in experiments) and could recently berigorously confirmed in simple one-dimensional models.Concretely, on ℓ2(ℤ), let us consider the bounded self-adjoint operator, known as the Fibonacci Hamiltonian,

(𝐻𝜓)𝑛 = 𝜓𝑛−1 +𝜓𝑛+1 +𝜆𝑣𝑛𝜓𝑛

with potential 𝑣𝑛 = 𝜒[1−𝛼,1)(𝑛𝛼 mod 1) (with constant𝛼 = (√5− 1)/2), which alternatively could be generatedby the Fibonacci substitution 0 ↦ 1, 1 ↦ 10. For 𝜆 > 8,quantum states display anomalous transport in the sensethat they do not move ballistically or diffusively, nor dothey remain localized.

The spectrum of 𝐻, which is𝜎(𝐻) = {𝐸 ∣ (𝐻− 𝐸⋅1) is not invertible},

has a dense set of gaps; see Figure 5. All spectralmeasures associated with 𝐻 by the spectral theoremare purely singular continuous, while diffraction anddynamical spectrum are pure point in this case. Forall values of the coupling constant 𝜆, there are nowquantitative results on the local and global Hausdorffdimension of the spectrum and on the density of statesmeasure. On the other hand, similar results are currentlyentirely out of reach for Schrödinger operators associatedwith the Penrose tiling. However, there is recent progressfor higher-dimensional models obtained as a Cartesianproduct of Fibonacci Hamiltonians; see Figure 5 for anillustration (of proven properties of the spectrum).

Our above sketch is just one snapshot of a field withmany facets and new developments. Connections existwith many branches of mathematics, including discretegeometry, topology, and ergodic theory, to name but afew. Aperiodic order thus provides a versatile platformfor cooperation and proves the point that mathematics isa unified whole, not a collection of disjoint subjects.

Further Reading[1] M. Baake and U. Grimm, Aperiodic Order. Volume 1: A Math-

ematical Invitation, Cambridge University Press, Cambridge,2013. MR3136260

[2] J. Kellendonk, D. Lenz, and J. Savinien (eds.), Mathematicsof Aperiodic Order, Birkhäuser, Basel, 2015. MR3380566

ABOUT THE AUTHORS

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Michael Baake

David Damanik

Uwe Grimm

Michael Baake’s re-search interests in-clude aperiodic order, harmonic analysis, dy-namical systems and mathematical biology. He likes analogue elec-tronics, photography and cycling, with an ac-tive role in building his own gear.

David Damanik’s research in-terests lie in spectral theory, dynamical systems, and ape-riodic order. He likes music (Hip Hop, R&B, Dancehall, House) and sports (table ten-nis, lifting weights, basket-ball, baseball, chess).

Uwe Grimm studied physics and is the cur-rent Chair of the Com-mission on Aperiodic Crystals of the Inter-national Union of Crys-tallography (IUCr). His research interests in-clude aperiodic order, statistical physics and applications of math-ematics in physics or biology. He enjoys jug-gling as well as making and listening to music.

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