"In this book, we try to display the value (and joy!) of starting from a mathematically amorphous problem and combining ideas from diverse sources to produce new and significant mathematicsmathematics unforeseen from the motivating problem ... " from the Preface The common thread throughout this book is aperiodic tilings; the bestknown example is the "kite and dart" tiling. This tiling has been widely discussed, particularly since 1984 when it was adopted to model quasicrystals. The presentation uses many different areas of mathematics and physics to analyze the new features of such tilings. Although many people are aware of the existence of aperiodic tilings, and maybe even their origin in a question in logic, not everyone is familiar with their subtleties and the underlying rich mathematical theory. For the interested reader, this book fills that gap. Understanding this new type of tiling requires an unusual variety of specialties, including ergodic theory, functional analysis, group theory and ring theory from mathematics, and statistical mechanics and wave diffraction from physics. This interdisciplinary approach also leads to new mathematics seemingly unrelated to the tilings. Included are many worked examples and a large number of figures. The book's multidisciplinary approach and extensive use of illustrations make it useful for a broad mathematical audience. Readership Advanced undergraduates, graduate students, and research mathematicians. Reviews "The book serves as a solid introduction to the study of aperiodic tilings for anyone with sufficient mathematical background. I would wholeheartedly recommend this book for the library of any mathematics department. It would make a great starting point for the directed study project of a motivated senior."  MAA Online "In this short book, the author discusses several aspects of the theory of "substitution tilings" and "finite type" aperiodic tilings. The book highlights a number of relations between the longrange order properties of these tilings, properties of the translationinvariant measures on the space of tilings, and the (statistical) symmetries of these measures."  Mathematical Reviews "Reacted very positively ... lovely little book."  Palle Jorgensen Table of Contents  Introduction
 Ergodic theory
 Physics (for mathematicians)
 Order
 Symmetry
 Conclusion
 Appendix I: Geometry
 Appendix II: Algebra
 Appendix III: Analysis
 References
 List of symbols
 Subject Index
