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.

In Japanese culture the concept of katachi has special significance, connoting relationships and connectedness. Although katachi cannot be translated precisely, it corresponds most closely to "form," "shape," "pattern," or "Gestalt." The contemporary study of katachi is interdisciplinary and encompasses virtually all scientific and aesthetic endeavors. Katachi research seeks to bridge the gap between cultures - whether the "two cultures" of C.P. Snow or the contrasting cultures of East and West. To help achieve this aim and to foster international cooperation, the interdisciplinary symposium titled "Katachi "U" Symmetry" was convened in Tsukuba, Japan, November 21 - 25, 1994. With many participants from differing backgrounds and cultural perspectives, the symposium was the culmination of 15 years of work in the field. Like-minded researchers and philosophers came together from two movements in interdisciplinary studies of katachi and symmetry that arose in the 1980s, one in Japan, the other in Hungary. The proceedings of the symposium will stimulate and provoke the interest of scientists and mathematicians, engineers and architects, philosophers and semioticians - indeed, all those with a lively sense of curiosity and a wide-ranging intellect.