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APERIODICORDER Volume2:CrystallographyandAlmostPeriodicity Quasicrystalsarenon-periodicsolidsthatwerediscoveredin1982byDan Shechtman,NobelLaureateinChemistry2011.Themathematicsthatunderliesthis discoveryorwasstimulatedbyit,whichisknownasthetheoryofAperiodicOrder, isthesubjectofthiscomprehensivemulti-volumeseries. Thissecondvolumebeginstodevelopthetheoryinmoredepth.Acollectionof leadingexpertsinthefield,amongthemRobertV.Moody,introduceandreview importantaspectsofthisrapidly-expandingfield. Thevolumecoversvariousaspectsofcrystallography,generalisingappropriately fromtheclassicalcasetothesettingofaperiodicallyorderedstructures.Astrong focusisplaceduponalmostperiodicity,acentralconceptofcrystallographythat capturesthecoherentrepetitionoflocalmotifsorpatterns,anditscloselinksto Fourieranalysis,whichisoneofthemaintoolsavailabletocharacterisesuch structures.ThebookopenswithaforewordbyJeffreyC.Lagariasonthewider mathematicalperspectiveandcloseswithanepilogueontheemergenceof quasicrystalsfromthepointofviewofphysicalsciences,writtenbyPeterKramer, oneofthefoundersofthefieldonthesideoftheoreticalandmathematicalphysics. EncyclopediaofMathematicsandItsApplications Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationin mathematicsormathematicalscienceandforwhichadetaileddevelopmentofthe abstracttheoryislessimportantthanathoroughandconcreteexplorationofthe implicationsandapplications. BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir subjectscomprehensively.Lessimportantresultsmaybesummarisedasexercises attheendsofchapters.Fortechnicalities,readerscanbereferredtothe bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare encyclopaedicreferencesormanageableguidestomajorsubjects. EncyclopediaofMathematicsanditsApplications AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge UniversityPress.Foracompleteserieslistingvisit www.cambridge.org/mathematics. 117 R.Glowinski,J.-L.LionsandJ.HeExactandApproximateControllabilityforDistributedParameter Systems 118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks 119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs 120 T.NishiuraAbsoluteMeasurableSpaces 121 M.PrestPurity,SpectraandLocalisation 122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions 123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity 124 F.W.KingHilbertTransformsI 125 F.W.KingHilbertTransformsII 126 O.CalinandD.-C.ChangSub-RiemannianGeometry 127 M.Grabischetal.AggregationFunctions 128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopologicalGraph Theory 129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata 130 T.G.FaticoniModulesoverEndomorphismRings 131 H.MorimotoStochasticControlandMathematicalModeling 132 G.SchmidtRelationalMathematics 133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic 134 Y.CramaandP.L.Hammer(eds.)BooleanModelsandMethodsinMathematics,ComputerScience,and Engineering 135 V.BertheándM.Rigo(eds.)Combinatorics,AutomataandNumberTheory 136 A.Krista´ly,V.D.Ra˘dulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,Geometry,and Economics 137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications 138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic 139 M.FiedlerMatricesandGraphsinGeometry 140 N.VakilRealAnalysisthroughModernInfinitesimals 141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation 142 Y.CramaandP.L.HammerBooleanFunctions 143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses 144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets 145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegraland DifferentialEquations 146 G.DassiosEllipsoidalHarmonics 147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.OellermannTopicsinStructuralGraphTheory 148 L.Berlyand,A.G.KolpakovandA.NovikovIntroductiontotheNetworkApproximationMethodfor MaterialsModeling 149 M.BaakeandU.GrimmAperiodicOrderI:AMathematicalInvitation 150 J.Borweinetal.LatticeSumsThenandNow 151 R.SchneiderConvexBodies:TheBrunn–MinkowskiTheory(SecondEdition) 152 G.DaPratoandJ.ZabczykStochasticEquationsinInfiniteDimensions(SecondEdition) 153 D.Hofmann,G.J.SealandW.Tholen(eds.)MonoidalTopology 154 M.CabreraGarc´ıaandA´.Rodr´ıguezPalaciosNon-AssociativeNormedAlgebrasI:TheVidav–Palmerand Gelfand–NaimarkTheorems 155 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables(SecondEdition) 156 L.W.BeinekeandR.J.Wilson(eds.)withB.ToftTopicsinChromaticGraphTheory 157 T.MoraSolvingPolynomialEquationSystemsIII:AlgebraicSolving 158 T.MoraSolvingPolynomialEquationSystemsIV:BuchbergerTheoryandBeyond 159 V.BertheándM.Rigo(eds.)Combinatorics,WordsandSymbolicDynamics 160 B.RubinIntroductiontoRadonTransforms:WithElementsofFractionalCalculusandHarmonicAnalysis 161 M.GherguandS.D.TaliaferroIsolatedSingularitiesinPartialDifferentialInequalities 162 G.MolicaBisci,V.RadulescuandR.ServadeiVariationalMethodsforNonlocalFractionalProblems 163 S.WagonTheBanach–TarskiParadox(SecondEdition) 164 K.BroughanEquivalentsoftheRiemannHypothesisI:ArithmeticEquivalents 165 K.BroughanEquivalentsoftheRiemannHypothesisII:AnalyticEquivalents 166 M.BaakeandU.Grimm(eds.)AperiodicOrderII:CrystallographyandAlmostPeriodicity Encyclopedia of Mathematics and its Applications Aperiodic Order Volume 2: Crystallography and Almost Periodicity Editedby MICHAEL BAAKE Universita¨tBielefeld,Germany UWE GRIMM TheOpenUniversity,MiltonKeynes UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 4843/24,2ndFloor,AnsariRoad,Daryaganj,Delhi–110002,India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9780521869928 DOI:10.1017/9781139033862 ©CambridgeUniversityPress2017 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2017 PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-0-521-86992-8Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Contents Contributors ix Foreword by Jeffrey C. Lagarias xi Preface xix Chapter 1. More Inflation Tilings Dirk Frettlo¨h 1 1.1. A simple inflation tiling without FLC 1 1.2. One-parameter families of inflation rules 4 1.3. A tiling with non-unique decomposition 5 1.4. U¨berpinwheel 6 1.5. Tile orientations with distinct frequencies 9 1.6. Tilings inspired by girih tiles 12 1.7. Cyclotomic rhombus tilings 12 1.8. Infinitely many prototiles 16 1.9. Inflations with an empty supertile 17 1.10. Overlapping tiles 19 1.11. Tiles from automorphisms of the free group 23 1.12. Mixed inflations 28 1.13. Fusion tilings 32 References 35 Chapter 2. Discrete Tomography of Model Sets: Reconstruction and Uniqueness Uwe Grimm, Peter Gritzmann and Christian Huck 39 2.1. Introduction 39 2.2. Basic notions of discrete tomography 41 2.3. Algorithmic issues in discrete tomography 42 2.4. Computational complexity of discrete tomography 46 2.5. Discrete tomography of model sets 51 2.6. Uniqueness in discrete tomography 57 References 69 vi CONTENTS Chapter 3. Geometric Enumeration Problems for Lattices and Embedded Z-Modules Michael Baake and Peter Zeiner 73 3.1. Introduction 73 3.2. Preliminaries on lattices 76 3.3. A hierarchy of planar lattice enumeration problems 78 3.4. Algebraic and analytic tools 83 3.5. Similar sublattices 87 3.6. Similar submodules 106 3.7. Coincidence site lattices and modules 112 3.8. (M)CSMs of planar modules with N-fold symmetry 124 3.9. The cubic lattices 129 3.10. The four-dimensional hypercubic lattices 137 3.11. More on the icosian ring 148 3.12. Multiple CSLs of the cubic lattices 155 3.13. Results in higher dimensions 165 References 167 Chapter 4. Almost Periodic Measures and their Fourier Transforms Robert V. Moody and Nicolae Strungaru 173 4.1. Introduction 173 4.2. Topological background 181 4.3. Almost periodic functions 184 4.4. Weak topologies and consequences 192 4.5. Means 217 4.6. The Eberlein convolution 224 4.7. WAP=SAP⊕WAP 230 0 4.8. Fourier transform of finite measures 235 4.9. Fourier transformable measures 241 4.10. Almost periodic measures 256 4.11. Positive definite measures 262 References 268 Chapter 5. Almost Periodic Pure Point Measures Nicolae Strungaru 271 5.1. Introduction 271 5.2. The Baake–Moody construction of a CPS 275 5.3. Almost periodic measures 286 5.4. Dense weighted model combs 291 5.5. Continuous weighted model combs 295 5.6. On ε-dual characters 298 5.7. Almost lattices 305 CONTENTS vii 5.8. WAP measures with Meyer set support 312 5.9. Diffraction of weighted Dirac combs on Meyer sets 316 5.10. More on Bragg spectra of Meyer sets 332 5.11. Concluding remarks 337 5.A. Appendix. Harmonious sets 338 References 340 Chapter 6. Averaging Almost Periodic Functions along Exponential Sequences Michael Baake, Alan Haynes and Daniel Lenz 343 6.1. Introduction 343 6.2. Preliminaries and general setting 344 6.3. Averaging periodic functions 349 6.4. Averaging almost periodic functions 352 6.5. Further directions and extensions 359 References 361 Epilogue. Gateways Towards Quasicrystals Peter Kramer 363 E.1. Classical periodic crystallography 363 E.2. Point symmetry: Das Pentagramma macht Dir Pein? 365 E.3. Mathematical crystallography in higher dimensions 367 E.4. Aperiodic tilings of the plane 367 E.5. Cells and diffraction of the Penrose pattern 369 E.6. Quasiperiodicity and Fourier modules 369 E.7. Scaling and the square lattice 370 E.8. Harmonious sets 373 E.9. Incommensurate and modulated crystals 373 E.10. The quasiperiodic Penrose pattern 374 E.11. Icosahedral tilings in three dimensions 375 E.12. Discovery of iscosahedral quasicrystals 376 E.13. Postscriptum 377 References 378 Index 381 Contributors Michael Baake Dirk Frettlo¨h Fakultät fu¨r Mathematik Technische Fakultät UniversitätBielefeld, Germany UniversitätBielefeld, Germany Uwe Grimm Peter Gritzmann SchoolofMathematicsandStatistics Zentrum Mathematik The Open University Technische UniversitätMu¨nchen MiltonKeynes, UK Garching,Germany Alan Haynes Christian Huck Department of Mathematics Fakultät fu¨r Mathematik Universityof Houston, USA UniversitätBielefeld, Germany Peter Kramer Jeffrey C. Lagarias Institut fu¨r Theoretische Physik Department of Mathematics UniversitätTu¨bingen Universityof Michigan Germany Ann Arbor, USA Daniel Lenz Robert V. Moody Institut fu¨r Mathematik Dept.ofMathematicsandStatistics UniversitätJena, Germany Universityof Victoria,Canada Nicolae Strungaru Peter Zeiner Dept. of Mathematicsand Statistics Fakultät fu¨r Mathematik MacEwanUniversity UniversitätBielefeld, Germany Edmonton,Canada