Designers of the National Aquatics Center in Beijing (the “Water Cube”), built for the 2008 Olympics, used the Weaire-Phelan structure (named in part for one of the authors of this book) to create a building incorporating bubbles via thick slices of the Weaire-Phelan foam. The path to this foam started with Archimedes, gained considerable momentum with Plateau and Kelvin, and culminated with Weaire and Phelan. The object was to determine what partitioning of space into equal volumes minimizes their surface area.
The Pursuit of Perfect Packing explores this and dozens more of packing problems and their applications. There is not a formal theorem or proof in sight. Instead, this is an eclectic collection of stories mostly about packing and tiling in two and three dimensions. One of the major themes is frustration — the conflict between global and local requirements in establishing an optimal solution. Triangular packing of equal-radius coins in a plane is optimal; here the best local packing can be extended without variation to the whole structure. When coins of mixed radii are considered, the problem becomes much harder.
In nineteen chapters we see an amazing variety of packing problems. Perhaps one of the best known is the Kepler problem for optimal packing of identical hard spheres (where ten years ago Tom Hales established that face-centered cubic packing is indeed optimal). The authors’ approach here is representative of their efforts throughout. They describe the packing question, provide specific examples of situations where it occurs, illustrate these with good drawings and photographs, discuss the history of the problem, and then describe solutions or progress toward solutions. As with the Kepler problem, there are often fascinating controversies or side stories.
The closest the authors get to describing analytical tools for packing is a brief section on the Voronoi construction and Delaunay decomposition. For the most part, the authors limit themselves to brief descriptions of packing applications. These include highly organized packings (including crystal forms), highly disordered packings, packing with in-between order (quasi-crystals) and arrangements of granular material. There is a nice short chapter on the honeycomb structure and Hales’ proof that it is optimal among cellular structures.
Among other packing applications treated are granular aggregrations and recursive packing (using the right mixture of particle sizes, for example, to make good concrete), finite packings and tesselations, packings in higher dimensions, parking cars and folding proteins.
Although this book is entertaining and easy to read, it is often frustrating because the vignettes are so short and they often end just when you want to know more. There is no bibliography but there are some references in footnotes.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.