Foreword

I was delighted to be asked to write a preface to this beautiful and outstandingly original book. It is the unique treatise on its subject, it fills a serious gap in the literature and it covers the theory and the huge range of applications in a masterly way.

The authors are right to distinguish Voronoi diagrams and Delone tessellations. The Delone construction decomposes a Euclidean space of m dimensions, containing a given set of points, into non-overlapping space-filling simplexes (not, of course, all of the same shape and size), so that it tessellates the space using tiles that are identical with one another up to linear transformations. The Voronoi construction also splits up the space into polyhedral cells, but now they are much less uniform in character - the number of faces will vary from one cell to another, so that it would be wrong to call the result a tessellation.

These mutually dual procedures give fascinating but different insights into the structure of a set of points in m dimensions, and they have found numerous applications. At the time of writing there is a new application on the largest of all possible scales which throws light on the structure of the Universe as we see it. This will be seen as a particularly interesting development when one recalls that most of the earlier applications (for example, to the study of the structure of metallic composites, and other such aggregates) were on the microscopic scale. The reader of this book is strongly urged to look at a review paper just published by Icke and van de Weygaert (Quart. Journal, Royal Astronomical Society, 32, 85-112). There it is shown that the Voronoi construction not only gives insight into the distribution of galaxies, but also permits a new approach to the dynamics that mould the shape of the universe we live in.

My own contributions have been in the Delone tradition, and are concerned (for example) with the way in which high-dimensional Delone simplexes pack together around a common vertex. Thus in 15 dimensions the number of such locally associated simplexes turns out to be of the order of 44 million million. This implies a related statement about the Voronoi polyhedra, and there tells us something about the number of faces of an individual cell. It seems likely that the huge number of Delone simplexes in such a local 'fan' can be roughly partitioned into a moderate number of 'chunky' simplexes (substantial faces in the Voronoi case), and a vast number of 'needle-like' ones (tiny faces), but we have no precise information on this matter at the moment.

It is a great pleasure to welcome this book to the Wiley series.

David Kendall


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