Answers on the back of a beer mat

One of the main dichotomies in modern mathematics is that between the continuous subjects - such as calculus, analysis and differential geometry, which model the real world as we experience it - and the discrete subjects - such as algebra and combinatorics, which govern the symbols that we (and, increasingly, our computers) manipulate in order to study that world. In many cases, we know powerful theorems about continuous phenomena, yet our understanding of them is not sufficiently concrete to allow us to perform explicit calculations. A classic example of this is complex function theory, which has a rich history dating back to the 19th century: here, results such as the Riemann mapping theorem establish strong links between geometric and analytic phenomena, yet there are very few cases where we can make these connections explicit.

Kenneth Stephenson's book explores an ambitious and imaginative attempt to bridge this gap by using essentially discrete structures called circle packings. Complex function theory is largely concerned with analytic functions between two-dimensional domains. These can be fiendishly difficult to work with, and some 20 years ago William Thurston proposed the use of circle packings to imitate the complex structures on these domains and the mappings between them. As with several other crucial concepts from this fertile mind, others were left to fill in the details. Stephenson, who has made significant contributions, both theoretical and practical, to this project, is well placed to write the first detailed account of its achievements. This he has done with an impressive combination of style and mathematical rigour, resulting in a beautiful book that should give pleasure and stimulation to any mathematician from advanced undergraduate to Fields medallist.

I imagine that most of us have explored the patterns that can be formed by covering a table with mutually touching round beer mats. Circle packings represent a generalisation of this, where the surface may be curved rather than flat, and the circles may have different radii. There are a few simple rules (for instance touching circles must be exterior to each other), but these are sufficiently natural and weak to allow a wide variety of circle packings. This book starts with a guided tour of a menagerie of examples, beautifully illustrated with diagrams produced by the author's software package, CirclePack (available for Unix systems from his University of Tennessee website). Some of these are reinterpretations of classical objects, such as the tessellations associated with the modular group or Klein's quartic curve, whereas others are entirely new.

A circle packing is determined by its combinatorics (which circles touch each other) and the radii of the circles. Part two is devoted to rigidity concepts, the main result being that any sensible set of combinatorial data corresponds to a circle packing. Part three, by contrast, is about flexibility, showing that any set of boundary conditions on radii determines an essentially unique circle packing in a domain. In part four, Stephenson proves the Rodin-Sullivan theorem and various generalisations, verifying Thurston's conjecture that mappings between circle packings can be used to approximate analytic functions arbitrarily closely.

The book closes with brief accounts of a number of interesting applications and connections with other areas, including Alexander Grothendieck's theory of dessins d'enfants . Perhaps the most intriguing application is brain flattening, where circle-packing techniques allow neuroscientists to study activity on the cerebral cortex, the highly convoluted surface of the human brain, by using discrete complex analysis to flatten it out. This must surely lead to other beneficial applications.

Gareth Jones is professor of mathematics, Southampton University.