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 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.

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

This 2006 book presents the geometrical ideas of structure at the atomic level that are being developed and integrated into materials science. Emphasis is placed on the intuitive understanding of geometrical principles through illustrations not detailed computation. This book will appeal to those working in crystallography, solid-state science and materials science.

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.

From tilings to quasicrystal structures and from surfaces to the n-dimensional approach, this book gives a full, self-contained in-depth description of the crystallography of quasicrystals. It aims not only at conveying the concepts and a precise picture of the structures of quasicrystals, but it also enables the interested reader to enter the field of quasicrystal structure analysis. Going beyond metallic quasicrystals, it also describes the new, dynamically growing field of photonic quasicrystals. The readership will be graduate students and researchers in crystallography, solid-state physics, materials science, solid- state chemistry and applied mathematics.

Designed for precollege teachers by a collaborative of teachers, educators, and mathematicians, Fractions, Tilings, and Geometry is based on a course offered in the Summer School Teacher Program at the Park City Mathematics Institute. The overall goal of the course is an introduction to non-periodic tilings in two dimensions and space-filling polyhedra. While the course does not address quasicrystals, it provides the underlying mathematics that is used in their study. Because of this goal, the course explores Penrose tilings, the irrationality of the golden ratio, the connections between tessellations and packing problems, and Voronoi diagrams in 2 and 3 dimensions. These topics all connect to precollege mathematics, either as core ideas (irrational numbers) or enrichment for standard topics in geometry (polygons, angles, and constructions). But this book isn't a “course” in the traditional sense. It consists of a carefully sequenced collection of problem sets designed to develop several interconnected mathematical themes. These materials provide participants with the opportunity for authentic mathematical discovery—participants build mathematical structures by investigating patterns, use reasoning to test and formalize their ideas, offer and negotiate mathematical definitions, and apply their theories and mathematical machinery to solve problems. Fractions, Tilings, and Geometry is a volume of the book series “IAS/PCMI—The Teacher Program Series” published by the American Mathematical Society. Each volume in this series covers the content of one Summer School Teacher Program year and is independent of the rest.