Quasicrystals. The state of the art. (2nd ed.) Edited by D. P. Vincenzo and P. J. Steinhardt. Series on Directions in Condensed Matter Physics, Vol. 16. Pp. xi + 618. Singapore: World Scientific, 1999. Price (hardback) USD 113, GBP 71, ISBN 981-02-4155-0; (paperback) USD 55, GBP 34, ISBN 981-02-4156-9.

Twelve years ago, the first edition of this book was published. It served for several years as a very valuable introduction to quasicrystals and as a progress report emphasizing `critical experimental developments and important theoretical breakthroughs that have occurred in the four years since, subsequent to the initial excitement surrounding the discovery of quasicrystals'. The slightly enlarged second edition appeared in 1999. Two well written chapters had been added to the 15 mostly unchanged previous ones. In one of them, A. I. Goldman and P. A. Thiel review the experimental progress between 1991 and 1998. In the other one, P. J. Steinhardt himself discusses the Gummelt-cluster approach, which allows the structure of decagonal phases to be described in terms of a single overlapping cluster.

Quasicrystals. The state of the art (2nd ed.) is still an interesting book but it no longer reflects the state-of-the art of quasicrystal research. Today the title is misleading. Too many important experimental and theor­etical discoveries have been made since 1990/91 when 15 of the 17 chapters of the book were written. This progress is reflected in the number of papers on quasicrystals that have tripled to much more than 6000 since then. New classes of stable ternary and even binary quasicrystals have been discovered, high-temperature/pressure studies have given new insight into stability and phase transformations, huge perfect single-crystalline samples have been grown and structure and physical properties of quasicrystals have been studied in great detail. Even one of the early paradigms that the negative temperature coefficient of electrical resisitivity should be an intrinsic property of quasicrystals has recently been questioned by experiments on the new binary quasicrystals.

Quasicrystals. The state of the art (2nd ed.) is definitely `a must' for everybody who is interested in the early years of quasicrystal research. Some of the theoretical chapters are of lasting value such as the one on Random tiling models by C. L. Henley. Matching rules for quasicrystalline tilings by K. Ingersent or Growth rules for quasicrystals by J. E. S. Socolar are still useful to read. However, if one wants to get an easy general introduction into the field, the textbook by C. Janot, Quasicrystals: a primer, 2nd ed. (Clarendon Press, Oxford, 1994), would be a better buy. More detailed insight into structure and properties of quasicrystals is imparted by the useful multiauthor book Physical properties of quasicrystals, edited by Z. M. Stadnik (Springer, Berlin, 1999). A more mathematical introduction into the current theory of coverings is given in the excellent book on Coverings of discrete quasiperiodic sets. Theory and application to quasicrystals, edited by P. Kramer and Z. Papadopolos (Springer, Berlin, 2003).