Coverings of Discrete Quasiperiodic Sets: Theory and Applications to Quasicrystals (Springer Tracts in Modern Physics, 180)

In this up-to-date review and guide to most recent literature, the expert authors develop concepts related to quasiperiodic coverings and describe results. The text describes specific systems in 2 and 3 dimensions with many illustrations, and analyzes the atomic positions in quasicrystals. front-matter.pdf......Page 1 Convering of Discrete Quasiperiodic Sets......Page 3 Preface......Page 6 List of Contributors......Page 7 Contents......Page 9 Packing, Tiling, and Covering......Page 14 Aperiodic and Quasiperiodic Systems with Long-Range Order......Page 19 Fibonacci Tiling and Klotz Construction......Page 21 Quasiperiodic functions on a parallel line section of E2......Page 22 Fundamental Domain Compatible with a Tiling......Page 23 Decagonal Voronoi Clusters and Covering of the Penrose Tiling......Page 24 Covering and Cluster Density in 2D Systems......Page 27 Covering of Atomic Positions in Icosahedral Quasicrystals......Page 28 Clusters in Quasicrystals......Page 29 Perspectives on the Theory of Covering for Discrete Quasiperiodic Sets......Page 30 References......Page 31 Index......Page 0 2.1 Introduction......Page 35 2.2 Bergman and Mackay Clusters......Page 37 2.3 The Al-Cu-Fe/Al-Pd-Mn Models......Page 38 2.4 Local Environments......Page 44 Convex Polyhedra.......Page 45 Unions of Convex Polyhedra.......Page 47 Computing Local Environments.......Page 48 2.4.2 Atomic Clusters......Page 49 2.4.3 B and B' Clusters......Page 50 2.4.4 M and M' Clusters......Page 53 2.5 Atomic Clusters and Chemical Decoration of i-AlPdMn......Page 58 2.6 Covering Clusters: the XB Cluster......Page 63 References......Page 72 Introduction......Page 75 Important Concepts and Tools......Page 79 Perfect Decagon Coverings......Page 80 Random Decagon Coverings......Page 83 Cluster Density Maximization......Page 85 Entropy Density......Page 87 Couplings Between Clusters......Page 88 An Atomic Cluster Enforcing the Relaxed Overlap Rules......Page 90 The Alternation Condition......Page 91 An Atomic Model for Octagonal Mn-Si-Al......Page 95 The Socolar Tiling......Page 96 Cluster Covering and Cluster Densities......Page 99 The Shield Tiling......Page 100 Voronoi and Delone Clusters......Page 103 References......Page 105 Introduction......Page 108 Lattices in En, Cells, Sections and Quasiperiodic Functions......Page 109 Fundamental Domains and Coverings for Quasiperiodic Tilings......Page 110 Voronoi and Delone Clusters in 2D Quasiperiodic Tilings......Page 111 V- and D-clusters in Dual Canonical Icosahedral Tilings......Page 112 Voronoi and Delone Polytopes and Dual Boundaries......Page 113 Dual Tilings and Their Windows......Page 114 Fundamental Domains and Spaces of Functions......Page 115 Delone Clusters and Their Windows......Page 117 Covering by Delone Clusters......Page 119 The Lattice A4 and the Triangle Tiling......Page 120 Standard Positions of Dual 2-Boundaries......Page 123 Delone Clusters D(a,j) and Their Windows......Page 124 The Window for a Fixed Orientation and Hole Class (a,1).......Page 125 All Windows of Hole Class (a,1) for a Fixed Orientation.......Page 126 Total Window for All Orientations and Hole Class (a,1).......Page 127 Delone Clusters D(b,j) and their Windows......Page 128 The Window for a Fixed Orientation and Hole Class (b,3).......Page 129 All Windows for a Fixed Orientation and Hole Class (b,3).......Page 130 Total Window for all Orientations and Hole Class (b,3).......Page 131 Covering of Vertices and Tiles......Page 132 Thickness of the Covering......Page 133 Fundamental Domains in the Triangle Tiling......Page 135 Linkage of Delone Clusters in (T*, A4)......Page 136 6D Lattices and the Icosahedral Coxeter Group......Page 141 Lattices D6, P and Their Holes......Page 142 Point Groups and Icosahedral Symmetry......Page 144 Scaling Symmetry in Icosahedral Lattices......Page 148 The Icosahedral Tiling (T*, D6)......Page 151 The Icosahedral Tiling (T*, P)......Page 154 Filling of Delone Clusters in (T*, D6)......Page 155 The Window and Filling for Da......Page 157 The Window and Filling for Db......Page 158 Details of the Filling of Delone clusters......Page 159 Delone Clusters in the Icosahedral Tiling (T*, D6)......Page 163 Delone Clusters in the Tiling (T*, P)......Page 165 Fundamental Domains and Icosahedral Tilings......Page 168 Tiling (T*, D6)......Page 170 Tiling (T*, D6)......Page 171 Towards Complete Covering: Coloring......Page 172 Conclusion......Page 173 References......Page 174 Introduction......Page 177 Local Derivations of Tilings and Coverings Containing Pentagonal Prototiles from the Tiling T*(A4)......Page 178 Local Derivation of the Tiling T*(z): T*(A4)-3muT*(z)......Page 179 Local Derivation of the Covering CkT*(A4): T*(z)-3muCkT*(A4)......Page 181 Local Derivation of the Partly Random Penrose Tiling T*(p1)r: T*(z)-3muT*(p1)r......Page 183 Local Derivation of the Partly Random Niizeki Tiling T*(nr): T*(z)-3muT*(nr)......Page 184 Delone Clusters of the Icosahedral Tiling T*(D6) and Their Codings......Page 186 Delone Covering CT*(D6) of the Tiling T*(D6)......Page 189 Decking Fractions, Thickness of the Covering CT*(D6)......Page 192 Conclusion and Perspectives......Page 194 References......Page 195 Introduction......Page 197 Notation......Page 199 Lattices and Crystals......Page 200 Quasilattices......Page 203 Cut-and-Project Schemes......Page 204 An Example......Page 205 Subquasilattices and Quotients......Page 208 Quasilattices from Quadratic Fields......Page 210 Modules over K......Page 211 Dual Modules......Page 213 10-Fold Quasilattices......Page 214 8-Fold Quasilattices......Page 220 12-Fold Quasilattices......Page 222 Icosians......Page 225 1-Dimensional Submodules......Page 227 2-Dimensional Submodules......Page 229 Window Statistics......Page 230 Relation to Root Lattices......Page 232 Quasilattices from Higher-Degree Fields......Page 233 14-Fold Quasilattices......Page 234 Summary......Page 235 References......Page 237 Introduction......Page 238 Basic Concepts......Page 239 Experimental Procedures......Page 243 Superlattice Ordering......Page 244 Phason Fluctuations......Page 253 Summary......Page 264 References......Page 265 Introduction......Page 268 Initial Results......Page 269 Higher-Resolution Studies......Page 271 Quasi-Unit-Cell Covering Model......Page 273 Experimentally Derived Covering......Page 276 References......Page 278 back-matter.pdf......Page 280 Coverings Are Efficient Ways To Exhaust Euclidean N-space With Congruent Geometric Objects. Discrete Quasiperiodic Systems Are Exemplified By The Atomic Structure Of Quasicrystals. The Subject Of Coverings Of Discrete Quasiperiodic Sets Emerged In 1995. The Theory Of These Coverings Provides A New And Fascinating Perspective Of Order Down To The Atomic Level. The Authors Develop Concepts Related To Quasiperiodic Coverings And Describe Results. Specific Systems In 2 And 3 Dimensions Are Described With Many Illustrations. The Atomic Positions In Quasicrystals Are Analyzed. Covering Of Discrete Quasiperiodic Sets (kramer) -- Atomic Clusters And Covering In Icosahedral Quasicrystals (duneau, Gratias) -- Generation Of Quasiperiodic Order By Maximal Cluster Covering (gähler, Gummelt, Ben-abraham) -- Ammann Grid Decorations Of Covering Clusters In Quasiperiodic Tilings (lück, Scheffer) -- Voronoi And Delone Clusters In Dual Quasiperiodic Tilings (kramer) -- The Efficiency Of The Delone-coverings Of Canonical Tilings (papadopolos, Kasner) -- Covering Presentation And Colouring Of Dual Canonical Tilings (kramer, Papadopolos, Kasner) -- Lines And Planes In 2 And 3 Dimensional Quasicrystals (pleasants) -- Thermally Induced Tile Rearrangements In Decagonal Quasicrystals - Superlattice Ordering And Phason Fluctuations (edagawa) -- Experimentally Derived Tilings Of Quasicrystal Surfaces (mcgrath, Diehl, Ledieu). Peter Kramer, Zorka Papadopolos (eds.). Includes Bibliographical References And Index.