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Acta Cryst. (1993). A49, 791

Mathematical crystallography - an introduction to the mathematical foundations of crystallography. Reviews in mineralogy, Vol. 15 (revised).

By M. B. Boisen and G. V. Gibbs

Pp. xii + 460. Washington, DC: The Mineralogical Society of America, 1992
Price (paper) US $20.00. ISBN 0-939950-26-X.

The scope of this monograph, which is a collaboration between a mathematician and a mineralogist, is much narrower than its very general title would seem to imply and considerably deeper than is suggested by the words `an introduction to ...' in its subtitle. It is, in fact, a comprehensive treatment of group theory as it is applied to the symmetry of three-dimensional periodic structures. It begins by discussing the concept of symmetry in molecules and crystals and then introduces some mathematical tools for describing symmetry, including vector spaces and basis sets but not, curiously, in view of their fundamental importance, matrices, which are consigned to an appendix. Subsequent chapters discuss geometrical aspects of crystals and point isometries. The final four chapters give derivations from first principles of the 32 crystallographic point groups, the 14 Bravais lattices and the 230 space groups. A set of appendices includes discussion in greater detail of various group-theoretical concepts. Many excellent illustrations appear throughout the volume and there are numerous exercises, with answers added in this second edition.

I have great difficulty imagining what audience this book is intended to serve. A foreword states that it was first introduced as a short course in conjunction with annual meetings of the Mineralogical Society of America and the Geological Society of America, which suggests that it is aimed primarily at earth scientists whose backgrounds do not include structural crystallography, but it includes detail that has little utility for someone who wants to learn structure techniques and omits much that I would classify as mathematical crystallography and that is vitally important for such a person. For example, it gives derivations for many of the results that appear in Volume I of the older series and Volume A of the newer series of International Tables for Crystallography, but it gives little guidance for finding and interpreting the information that appears in these standard references. Furthermore, although it discusses the reciprocal lattice and mentions X-ray diffraction, it doesn't discuss the implications of glide planes and screw axes for systematic space-group absences, and there is no mention of Fourier transforms, Patterson functions, structure factors, refinement techniques or methods of phase determination.

The first edition of this book was published when the revolution in the technology of publication was just beginning, while the second, which differs from the first mainly through the addition of a large set of answers to the exercises in the text, appeared when the revolution was well advanced. This has the ironic result that the quality of reproduction of the answers is vastly superior to that of the main text, which is a typescript that is unattractive in appearance and not at all easy to read.

For someone who is interested in the mathematical underpinnings of the theory of symmetry, this book could be a useful self-study text. There is little in it that would interest anyone else.

Edward Prince

Reactor Radiation Division
National Institute of Standards and Technology
MD 20899

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