Tilings and Patterns [antikvár]

PREFACEThe increasing popularity of puzzles and games based on the interplay of shapes and positions illustrates the attraction that geometric forms and their relations hold for many people. Since space and motion are of great importance to most animals and all primitive peoples, this attraction probably has deep evolutionary roots. In addition, modern society increasingly demands that we cope with intricately shaped objects, follow the mazes of huge buildings and transportation systems, and orient ourselves on land and water and in the air (not to mention the problems of dealing with submarine or interplanetary space).It is curious that almost all aspects of geometry relevant to the "man in the street" are ignored by our educational systems. Geometry has been almost squeezed out of school and university syllabuses, and what little remains is rarely of any use to people who wish to apply geometric ideas in their workengineers, scientists, architects, artists, and the like. There are two causes of this state of affairs. At high-school level it has long been traditional to use geometry as a vehicle for teaching logical reasoning and the deductive method, without much regard for the geometric content. At the research level geometry has become no more than a specialized branch of algebra or analysis.In each case the essence of the subjectits visual appealhas been completely submerged in technicalities and abstractions. Other branches of mathematics have also suffered from this neglect of geometry, since many basic ideas in topology, analysis, measure theory and so on owe their origin to geometric intuition, and the workers in those fields would probably profit from some knowledge of geometry.Each of us has long deplored this situation, and so, when we met at a conference in 1975, we decided to write a book on "visual geometry"a rigorous book, but one that would also encourage geometric appreciation by the use of "pure" geometric reasoning. To cover many branches of the subject turned out to be too big a task. Eventually we decided on the study of tilings and patterns as a first step in carrying out the program we had in mind.Perhaps our biggest surprise when we started collecting material for the present work was that so little about tilings and patterns is known. We thought, naively as it turned out, that the two millenia of development of plane geometry would leave little room for new ideas. Not only were we unable to find anywhere a meaningful definition of a pattern, but we also discovered that some of the most exciting developments is this area (such as the phenomenon of aperiodicity for tilings) are not more than twenty years old.We have written this book with three main groups of readers in mindstudents, professional mathematicians and non-mathematicians whose interests include patterns and shapes (such as artists, architects, crystal-lographers and others). Each of these groups will find some parts of the book more interesting than others, but we hope that we have struck a reasonable balance in both content and form. The whole book should be accessible to any reader attracted to geometry, regardless of (or even in spite of) his previous mathematical education. Our presentation is somewhat informal (but nevertheless precise) and we have used diagrams and illustrations in profusion. We have invested great effort to make these attractive since we feel that any visual