Directions in Mathematical Quasicrystals
Directions in Mathematical Quasicrystals

Author: Michael Baake

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 389

ISBN-13: 0821826298

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This volume includes twelve solicited articles which survey the current state of knowledge and some of the open questions on the mathematics of aperiodic order. A number of the articles deal with the sophisticated mathematical ideas that are being developed from physical motivations. Many prominent mathematical aspects of the subject are presented, including the geometry of aperiodic point sets and their diffractive properties, self-affine tilings, the role of $C*$-algebras in tiling theory, and the interconnections between symmetry and aperiodic point sets. Also discussed are the question of pure point diffraction of general model sets, the arithmetic of shelling icosahedral quasicrystals, and the study of self-similar measures on model sets. From the physical perspective, articles reflect approaches to the mathematics of quasicrystal growth and the Wulff shape, recent results on the spectral nature of aperiodic Schrödinger operators with implications to transport theory, the characterization of spectra through gap-labelling, and the mathematics of planar dimer models. A selective bibliography with comments is also provided to assist the reader in getting an overview of the field. The book will serve as a comprehensive guide and an inspiration to those interested in learning more about this intriguing subject.

Introduction to the Mathematics of Quasicrystals
Introduction to the Mathematics of Quasicrystals

Author: Marko V. Jaric

Publisher: Elsevier

Published: 2012-12-02

Total Pages: 238

ISBN-13: 0323159478

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Introduction to the Mathematics of Quasicrystals provides a pedagogical introduction to mathematical concepts and results necessary for a quantitative description or analysis of quasicrystals. This book is organized into five chapters that cover the three mathematical areas most relevant to quasicrystals, namely, the theory of almost periodic functions, the theory of aperiodic tilings, and group theory. Chapter 1 describes the aspects of the theory of tiling in two- and three-dimensional space that are important for understanding some of the ways in which “classical mathematical crystallography is being generalized; this process is to include possible models for aperiodic crystals. Chapter 2 examines the non-local nature of assembly “mistakes that might have significance to the quasicrystals growth. This chapter also describes how closely a physical quasicrystal might be able to approximate a three-dimensional version of tilings. Chapter 3 discusses the theoretical background and concepts of group theory of icosahedral quasicrystals. Chapter 4 presents the local properties of the three-dimensional Penrose tilings and their global construction is described through the projection method. This chapter emphasizes the relationship between quasiperiodic sets of points and quasiperiodic tiling. Chapter 5 explores the analysis of defects in quasicrystals and their kinetics, as well as some properties of the perfect system. This book is of great value to physicists, crystallographers, metallurgists, and beginners in the field of quasicrystals.

Quasicrystals and Geometry
Quasicrystals and Geometry

Author: Marjorie Senechal

Publisher: CUP Archive

Published: 1996-09-26

Total Pages: 310

ISBN-13: 9780521575416

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This first-ever detailed account of quasicrystal geometry will be of great value to mathematicians at all levels with an interest in quasicrystals and geometry, and will also be of interest to graduate students and researchers in solid state physics, crystallography and materials science.

Quasicrystals and Discrete Geometry
Quasicrystals and Discrete Geometry

Author: Jiri Patera

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 303

ISBN-13: 0821806823

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Comprising the proceedings of the fall 1995 semester program arranged by The Fields Institute at the U. of Toronto, Ontario, Canada, this volume contains eleven contributions which address ordered aperiodic systems realized either as point sets with the Delone property or as tilings of a Euclidean space. This collection of articles aims to bring into the mainstream of mathematics and mathematical physics this developing field of study integrating algebra, geometry, Fourier analysis, number theory, crystallography, and theoretical physics. Annotation copyrighted by Book News, Inc., Portland, OR

Introduction to the Mathematics of Quasicrystals
Introduction to the Mathematics of Quasicrystals

Author: Marko V. Jarić

Publisher:

Published: 1989

Total Pages: 226

ISBN-13:

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1. A brief introduction to tilings / Marjorie Senechal--2. Tilings and quasicrystals : a non-local growth problem? / R. Penrose--3. Group theory of icosohedral quasicrystals / P. Kramer and R.W. Haase--4. Some local properties of the three-dimensional Penrose tilings / Andre Katz--5. Defects in quasicrystals / J. Bohsung and H.-R. Trebin.

Quasicrystals
Quasicrystals

Author: Hans-Rainer Trebin

Publisher: John Wiley & Sons

Published: 2006-05-12

Total Pages: 668

ISBN-13: 3527606785

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Quasicrystals form a new state of solid matter beside the crystalline and the amorphous. The positions of the atoms are ordered, but with noncrystallographic rotational symmetries and in a nonperiodic way. The new structure induces unusual physical properties, promising interesting applications. This book provides a comprehensive and up-to-date review and presents most recent research results, achieved by a collaboration of physicists, chemists, material scientists and mathematicians within the Priority Programme "Quasicrystals: Structure and Physical Properties" of the Deutsche Forschungsgemeinschaft (DFG). Starting from metallurgy, synthesis and characterization, the authors carry on with structure and mathematical modelling. On this basis electronic, magnetic, thermal, dynamic and mechanical properties are dealt with and finally surfaces and thin films.

The Physics of Quasicrystals
The Physics of Quasicrystals

Author: Paul J. Steinhardt

Publisher: World Scientific

Published: 1987

Total Pages: 792

ISBN-13: 9789971502270

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This book comprises an introductory lecture outlining the basic concepts and challenges in the field. This is followed by a collection of reprinted articles which are important in understanding the subject. The book will focus mainly on mathematical and physical foundations of the subject rather than experimental progress. By concentrating on theoretical topics, this volume has long-lasting as well as immediate value to physicists, crystallographers, metallurgists and mathematicians.

Author:

Publisher: IOS Press

Published:

Total Pages: 6097

ISBN-13:

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A New Direction in Mathematics for Materials Science
A New Direction in Mathematics for Materials Science

Author: Susumu Ikeda

Publisher: Springer

Published: 2015-12-08

Total Pages: 93

ISBN-13: 4431558640

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This book is the first volume of the SpringerBriefs in the Mathematics of Materials and provides a comprehensive guide to the interaction of mathematics with materials science. The anterior part of the book describes a selected history of materials science as well as the interaction between mathematics and materials in history. The emergence of materials science was itself a result of an interdisciplinary movement in the 1950s and 1960s. Materials science was formed by the integration of metallurgy, polymer science, ceramics, solid state physics, and related disciplines. We believe that such historical background helps readers to understand the importance of interdisciplinary interaction such as mathematics–materials science collaboration. The middle part of the book describes mathematical ideas and methods that can be applied to materials problems and introduces some examples of specific studies—for example, computational homology applied to structural analysis of glassy materials, stochastic models for the formation process of materials, new geometric measures for finite carbon nanotube molecules, mathematical technique predicting a molecular magnet, and network analysis of nanoporous materials. The details of these works will be shown in the subsequent volumes of this SpringerBriefs in the Mathematics of Materials series by the individual authors. The posterior section of the book presents how breakthroughs based on mathematics–materials science collaborations can emerge. The authors' argument is supported by the experiences at the Advanced Institute for Materials Research (AIMR), where many researchers from various fields gathered and tackled interdisciplinary research.

Quasicrystals
Quasicrystals

Author: D. P. DiVincenzo

Publisher: World Scientific

Published: 1999

Total Pages: 634

ISBN-13: 9789810241568

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Quasicrystals: The State of the Art has proven to be a useful introduction to quasicrystals for mathematicians, physicists, materials scientists, and students. The original intent was for the book to be a progress report on recent developments in the field. However, the authors took care to adopt a broad, pedagogical approach focusing on points of lasting value. Many subtle and beautiful aspects of quasicrystals are explained in this book (and nowhere else) in a way that is useful for both the expert and the student. In this second edition, some authors have appended short notes updating their essays. Two new chapters have been added. Chapter 16, by Goldman and Thiel, reviews the experimental progress since the first edition (1991) in making quasicrystals, determining their structure, and finding applications. In Chapter 17, Steinhardt discusses the quasi-unit cell picture, a promising, new approach for describing the structure and growth of quasicrystals in terms of a single, repeating, overlapping cluster of atoms.