*Acta
Cryst.* (1993). A**49**,
214-215

**Pp. xiii + 314. New
York, Weinheim, Cambridge: VCH Publishers,
1990
Price £55.00. ISBN 0-89573-723-X**

This expensive but attractive book is a collection of 19 short specialist essays, connected by the single theme of fivefold symmetry.

Fivefold symmetry will always have a certain fascination
for crystallographers. It is the symmetry governing two of
the five regular polyhedra of Plato and it is a symmetry
extremely common in biology, yet it is the first `forbidden'
symmetry for crystals. Only in the past ten or so years has
this situation rather suddenly changed, for we now know
that fivefold symmetry *can *fill space, only not periodically.

This change began quietly enough around the l950s, on the far periphery of crystallography, in the mathematics of tiling (two-dimensional space filling), and advanced most notably when Penrose's paper of 1974 on `The role of aesthetics in mathematical research' (a discussion with no apparent relevance to crystals) produced the paradigm `Penrose tiling'. All this was abstract mathematics, even when Mackay, in 1981, generalized this nonperiodic tiling into three dimensions. But when Schectman and his colleagues in the US Bureau of Standards published electron diffraction pictures of fivefold symmetry obtained from micrometre-size grains of an aluminium alloy, crystallographers had to take notice. That was in 1984. Since then, about a thousand research papers have followed `directly from the stimulus of that discovery' (Mackay's comment).

Not everything has changed, of course. The great majority
of solids remain periodic, but the incidence of fivefold
symmetry in certain special cases, forming `regular' if not
periodic solids, able to give high-quality diffraction, is
important. These solids are now properly called quasicrystals. Hence the title of this book. This development has
enormously spurred theoretical work on the mathematics
of lattices in general - for 5-, 7-, 9- and 12-fold symmetries,
in *N*-dimensional spaces, and so on - during the past
decade. It is clear that good-quality quasicrystals are not
easy to produce. They require quite special shapes of the
packing polyhedra, they come only from certain special
alloys, they require long annealing to be more than micrometre sized and they come with compositions such as
Al_{63.5}Cu_{24}Fe_{12.5}. These restrictions and isolated cases of
apparent fivefold symmetry observed since the earliest days
of crystallography have led some researchers, notably and
most recently Pauling, to argue that they could be explained
by specific types of twinning associated with an unusually
large cubic unit cell. Pauling's explanation has been very
carefully tested by Dunlap and his collaborators at Dalhousie University and their findings (reported in detail in
Chapter 6 of this book) show that it is definitely inferior
to analyses based on true quasicrystallinity. We may take
it that quasicrystals are real; they are also exciting,
especially to theorists.

For crystallographers who wish to get up to date with these developments, Professor Hargittai of Budapest has provided this album of specialist essays. Hargittai himself has contributed only the six-sentence Preface; but he has assembled no fewer than 37 distinguished authors (from 9 countries) who have supplied 19 separate articles to cover as much as possible of this exotic field.

The first 12 essays focus on quasiperiodicity: tilings of
all sorts, with fivefold and other symmetries; whereas the
next 6 are concerned with fivefold symmetry in isolated
molecules, principally the new icosahedral molecule C_{60}
(which is the subject of four essays on its history, energy
levels and vibrational modes). The final chapter is a short
essay on the fusion of five-membered carbon rings - essentially an essay in organic chemistry and somewhat removed
in tone from the rest of the book.

Each one of these essays is quite brief. There is little unity among them other than that they are all involved with fivefold symmetry of one sort or another; their order in the book does follow their subject orientation. A great variety of tilings and elaborate theoretical discussion are much in evidence, both in the text and in the numerous diagrams. A highlight is the essay by E. J. W. Whittaker and his coauthor R. M. Whittaker on fivefold symmetry in higher-dimensional spaces, for which there are six colour plates showing, stereoscopically, stick models that are three-dimensional projections of various four-dimensional structures. The principal highlight for this reviewer is the chapter exhibiting real quasicrystal specimens. This is a 14 page article by Denoyer, Heger & Lambert, of Université de Paris-Sud and CEN-Saclay. It shows a visibly pentagonal dodecahedral quasicrystal of a few tenths of a millimetre in diameter and another of triacontahedral symmetry, glued to a glass fibre, plus superb pictures of precession-type X-ray diffraction by these same specimens.

The delightful essay by Mackay, of Birkbeck College,
University of London, is well worth reading in its own right
and, considering the role that he has played in the development of this subject, it is fitting that he should have the
honour of writing the introductory chapter. With broad
coverage and innumerable scholarly asides, Mackay succeeds in mentioning magic, determinism, chaos, religion,
the date of Easter, Olber's paradox and the legal profession,
as well as the Babylonians and the Greeks. His article also
contains a charming example of the absence of cohesion
between the contributors, characteristic of this kind of book:
Mackay refers to the synthesis of the `Platonic' molecule
(CH)_{20}, as an `Everest of alicyclic chemistry', hoping (p. 15)
that it could be crystallized and its structure determined;
Kuck, of Universität Bielefeld, in his article at the end of
the book (p. 290), casually mentions this same molecule as
`of course... well established...', giving the reference to
its structure determination, which was published in 1986!

**J. H. Robertson**

*
School of Chemistry
University of Leeds
Leeds LS2 9JT
England
*

**Copyright © 1997 International Union of Crystallography**