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
These mutually dual procedures give fascinating but different insights into
the structure of a set of points 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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