Aperiodic Order: Volume 2, Crystallography and Almost Periodicity (Encyclopedia of Mathematics and its Applications)

Quasicrystals are non-periodic solids that were discovered in 1982 by Dan Shechtman, Nobel Prize Laureate in Chemistry 2011. The mathematics that underlies this discovery or that proceeded from it, known as the theory of Aperiodic Order, is the subject of this comprehensive multi-volume series. This second volume begins to develop the theory in more depth. A collection of leading experts, among them Robert V. Moody, cover various aspects of crystallography, generalising appropriately from the classical case to the setting of aperiodically ordered structures. A strong focus is placed upon almost periodicity, a central concept of crystallography that captures the coherent repetition of local motifs or patterns, and its close links to Fourier analysis. The book opens with a foreword by Jeffrey C. Lagarias on the wider mathematical perspective and closes with an epilogue on the emergence of quasicrystals, written by Peter Kramer, one of the founders of the field. Contents......Page 6 Contributors......Page 10 Foreword......Page 12 Preface......Page 20 1.1. A simple inflation tiling without FLC......Page 22 1.2 One-parameter families of inflation rules......Page 25 1.3. A tiling with non-unique decomposition......Page 26 1.4. Uberpinwheel......Page 27 1.5. Tile orientations with distinct frequencies......Page 30 1.7. Cyclotomic rhombus tilings......Page 33 1.8. Infinitely many prototiles......Page 37 1.9. Inflations with an empty supertile......Page 38 1.10. Overlapping tiles......Page 40 1.11. Tiles from automorphisms of the free group......Page 44 1.12. Mixed inflations......Page 49 1.13. Fusion tilings......Page 53 References......Page 56 2.1. Introduction......Page 60 2.2. Basic notions of discrete tomography......Page 62 2.3. Algorithmic issues in discrete tomography......Page 63 2.4. Computational complexity of discrete tomography......Page 67 2.5. Discrete tomography of model sets......Page 72 2.6. Uniqueness in discrete tomography......Page 78 References......Page 90 3.1. Introduction......Page 94 3.2. Preliminaries on lattices......Page 97 3.3. A hierarchy of planar lattice enumeration problems......Page 99 3.4. Algebraic and analytic tools......Page 104 3.5. Similar sublattices......Page 108 3.6. Similar submodules......Page 127 3.7. Coincidence site lattices and modules......Page 133 3.8. (M)CSMs of planar modules with N-fold symmetry......Page 145 3.9. The cubic lattices......Page 150 3.10. The four-dimensional hypercubic lattices......Page 158 3.11. More on the icosian ring......Page 169 3.12. Multiple CSLs of the cubic lattices......Page 176 3.13. Results in higher dimensions......Page 186 References......Page 188 4.1. Introduction......Page 194 4.2. Topological background......Page 202 4.3. Almost periodic functions......Page 205 4.4. Weak topologies and consequences......Page 213 4.5. Means......Page 238 4.6. The Eberlein convolution......Page 245 4.7. WAP = SAP ⊕ WAP0......Page 251 4.8. Fourier transform of finite measures......Page 256 4.9. Fourier transformable measures......Page 262 4.10. Almost periodic measures......Page 277 4.11. Positive definite measures......Page 283 References......Page 289 5.1. Introduction......Page 292 5.2. The Baake–Moody construction of a CPS......Page 296 5.3. Almost periodic measures......Page 307 5.4. Dense weighted model combs......Page 312 5.5. Continuous weighted model combs......Page 316 5.6. On ε-dual characters......Page 319 5.7. Almost lattices......Page 326 5.8. WAP measures with Meyer set support......Page 333 5.9. Diffraction of weighted Dirac combs on Meyer sets......Page 337 5.10. More on Bragg spectra of Meyer sets......Page 353 5.11. Concluding remarks......Page 358 5.A. Appendix. Harmonious sets......Page 359 References......Page 361 6.1. Introduction......Page 364 6.2. Preliminaries and general setting......Page 365 6.3. Averaging periodic functions......Page 370 6.4. Averaging almost periodic functions......Page 373 6.5. Further directions and extensions......Page 380 References......Page 382 E.1. Classical periodic crystallography......Page 384 E.2. Point symmetry: Das Pentagramma macht Dir Pein?......Page 386 E.4. Aperiodic tilings of the plane......Page 388 E.6. Quasiperiodicity and Fourier modules......Page 390 E.7. Scaling and the square lattice......Page 391 E.9. Incommensurate and modulated crystals......Page 394 E.10. The quasiperiodic Penrose pattern......Page 395 E.11. Icosahedral tilings in three dimensions......Page 396 E.12. Discovery of iscosahedral quasicrystals......Page 397 E.13. Postscriptum......Page 398 References......Page 399 Index......Page 402 Quasicrystals are non-periodic solids that were discovered in 1982 by Dan Shechtman, Nobel Laureate in Chemistry 2011. The mathematics that underlies this discovery or was stimulated by it, which is known as the theory of Aperiodic Order, is the subject of this comprehensive multi-volume series. This second volume begins to develop the theory in more depth. A collection of leading experts in the field, among them Robert V. Moody, introduce and review important aspects of this rapidly-expanding field. The volume covers various aspects of crystallography, generalising appropriately from the classical case to the setting of aperiodically ordered structures. A strong focus is placed upon almost periodicity, a central concept of crystallography that captures the coherent repetition of local motifs or patterns, and its close links to Fourier analysis, which is one of the main tools available to characterise such structures. The book opens with a foreword by Jeffrey C. Lagarias on the wider mathematical perspective and closes with an epilogue on the emergence of quasicrystals from the point of view of physical sciences, written by Peter Kramer, one of the founders of the field on the side of theoretical and mathematical physics. -- from back cover Quasicrystals Are Non-periodic Solids That Were Discovered In 1982 By Dan Shechtman, Nobel Prize Laureate In Chemistry 2011. The Underlying Mathematics, Known As The Theory Of Aperiodic Order, Is The Subject Of This Comprehensive Multi-volume Series. This First Volume Provides A Graduate-level Introduction To The Many Facets Of This Relatively New Area Of Mathematics. Special Attention Is Given To Methods From Algebra, Discrete Geometry And Harmonic Analysis, While The Main Focus Is On Topics Motivated By Physics And Crystallography. In Particular, The Authors Provide A Systematic Exposition Of The Mathematical Theory Of Kinematic Diffraction. Numerous Illustrations And Worked-out Examples Help The Reader To Bridge The Gap Between Theory And Application. The Authors Also Point To More Advanced Topics To Show How The Theory Interacts With Other Areas Of Pure And Applied Mathematics--publisher Description. Volume 1. A Mathematical Invitation -- Volume 2. Crystallography And Almost Periodicity. Michael Baake, Universität Bielefeld, Germany, Uwe Grimm, The Open University, Milton Keynes. Includes Bibliographical References (pages 489-516) And Index.