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This volume contains papers presented at the Twelfth Taniguchi Symposium on the Theory of Condensed Matter, which was held at Kashikojima (in Ise Shima National Park), Japan, November 14-19, 1989. The general purpose of the Taniguchi Symposia is to encourage important developing, rather than established, fields in condensed matter theory. The topic of the present sym posium, Quasicrystais, is quite typical. In 1984, Shechtman, Blech, Gratias and Cahn discovered the icosahedral symmetry of a diffraction pattern and Levine and Steinhardt independently presented the notion of quasicrystals. Before these discoveries, Roger Penrose of Oxford University had invented a space-filling non-periodic tiling, now called Penrose tiling. These factors form a new field that had become mathematically viable by the end of 1984, and many important new ideas are still being created. In standard textbooks of solid-state science, the first chapter used to be devoted to symmetry and periodicity in crystals. Now, the textbooks should be revised; quasi-periodicity and its physical properties should be added in several chapters and almost all standard conceptions should be reconsidered. However, the facts that are known about quasiperiodicity are not enough to complete even an introductory chapter of a textbook. Revision should be extended to generalized crystallography, defects, crystal growth, electronic structure, spectral theory and localization, electron transport, spin statistics, etc. These are all topics treated in this volume.

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I Introduction.- to Quasicrystals.- II Symmetry and Generalized Crystallography.- Chains, Flowers, Rings and Peanuts: Graphical Geodesic Lines and Their Application to Penrose Tiling.- The Ehrenfest Process, a Rotational Representation of Spin Operators and Explicit Forms of Rotation Operators.- Generalized Crystallography in Two and Three Dimensions.- III Atomic Structures.- Progress on the Atomic Structure of Quasicrystals.- From Crystal Approximants to Quasicrystals.- Ideal Structures of the Icosahedral Al-Mn and Al-Cu-Li Quasicrystals.- High-Resolution Electron Microscopy and Atomic Arrangement of an Al-Li-Cu Quasicrystal.- IV Quasiperiodic Lattices and Random Tiling.- Icosahedral and Decagonal Quasicrystals in Al-Ni-M and Al-Cu-M (M=Transition Metal) Systems.- Growing Perfect Quasicrystals.- The Alternation Condition and 2D Quasicrystals.- Random Tiling Model of Quasicrystalline Order.- Phason Freezing.- Elastic Instability Induced by Phasons in Icosahedral Quasicrystals.- V Electronic Structure.- Electronic States in a Quasicrystal.- Scaling of Structures, Spectra and Wave Functions.- Multifractal Method for Spectra and Wave Functions of Quasiperiodic Systems.- Localization in One-Dimensional Quasiperiodic Systems.- Quasiperiodic Systems with Long-Range Hierarchical Interactions.- Conductance of a Penrose Lattice.- Electronic Structure in Three-Dimensional Quasicrystals.- VI Spin Statistics.- Monte Carlo Study of Spin Statistics in Two- and Three-Dimensional Quasicrystals.- Index of Contributors.

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