*Acta
Cryst.* (1993). A**49**,
791

**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
Gaithersburg
MD
20899
USA
*

**Copyright © 1997 International Union of Crystallography**