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The Pursuit of Perfect Packing

Tomaso Aste and Denis Weaire
Taylor and Francis
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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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 ( 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.

How Many Sweets in the Jar? 
Loose Change and Tight Packing 
A Teasing but Tractable Problem
A Handful of Coins
Order and Disorder
Hard Problems with Hard Spheres
The Greengrocer’s Dilemma
Ordered Close Packing—The Kepler Problem
The Kepler Conjecture
Marvelous Clarity, Neurotic Anxiety: The Life of Kepler
Progress by Leaps and Bounds?
Proof Positive?
News from the Western Front
The Programme of Thomas Hales
At Last?
Polishing Off the Programme
The Acceptance of Proof
The Flyspeck Project
Who Cares?
The Power of Thought
Disordered Packings
Balls in Bags
A New Way of Looking
How Many Balls in the Bag?
Is the Bernal Close Packing Well Defined?
Bernal’s Long-Running Ball Game
Tomography Takes Over
Sands and Grains 
Granular Materials
Osborne Reynolds: A Footprint in the Sand
Major Bagnold’s Desert Drive
Order from Shaking
Granular Temperature
Divide and Conquer: Tiling Space 
Packing and Tiling
The Voronoï Construction
The Dual Construction of Delaunay
Vertices in Tilings
Three Dimensions
Regular and Semiregular Packings
Peas and Pips 
Vegetable Staticks
Stephen Hales
Pomegranate Pips
Biological Cells, Lead Shot, Rubber Balls, and Soap Bubbles: Plus ça Change 
Enthusiastic Admiration: The Honeycomb
The Honeycomb Problem
What the Bees Do Not Know
A Search for Structure 
A Voice in the Wilderness
The Two-Dimensional Soap Froth
The Rules of the Game
In a Cambridge Garden
Toils and Troubles with Bubbles 
Playing with Bubbles
A Blind Man in the Kingdom of the Sighted
Proving Plateau
Foam and Ether
The Kelvin Cell
Most Beautiful and Regular
The Twinkling of an Eye
Simulated Soap
A Discovery in Dublin
Crystals of Small Bubbles
Bubbles in Beijing 
An Olympian Vision
Fun and Fit for Purpose?
A Flexible Friend?
The Architecture of the World of Atoms 
Molecular Tactics
Atoms and Molecules: Begging the Question
Atoms as Points
Playing Hardball
Modern Crystallography
Crystalline Packings
Packing Tetrahedra
Changed Utterly: Quasicrystals
Amorphous Solids
Crystal Nonsense
Apollonius and Concrete 
Mixing Concrete
Apollonian Packing
Packing Fraction and Fractal Dimension
Packing Fraction in Granular Aggregates
Packings and Kisses in High Dimensions
Packing in Many Dimensions
A Kissing Competition
Kissing the Neighbors in Higher Dimensions
Will Disorder Win in the End?
The Sweets in the Jar, the Pebbles on the Beach 
Those Sweets
Hey, What Shape Do You Want Your Ice Cubes?
Another Walk on the Beach
The Giant’s Causeway
Worth Seeing?
Idealization Oversteps Again
The First Official Report
Mallett’s Model
A Modern View
The Last Word?
Lost City?
Finite Packings and Tessellations, from Soccer to Sausages 
The Challenge of a Finite Suitcase
Soccer Balls
Golf Balls
Buckminster Fuller
The Thomson Problem
Packing Points on a Disk
The Tammes Problem
Universal Optimal Configurations
Helical Packings
Virus Architecture
Stuffing Sausages
Filling Boxes
The Malfatti Problem
Odds and Ends 
Ordered Loose Packings
Parking Cars
Goldberg Variations
Packing Regular Pentagons
Dodecahedral Packing and Curved Spaces
Microspheres and Opals
Protein Folding
The Tetra Pak Story
Packing Regular Tetrahedra
Nature and Geometry
Appendix A: The Best Packing in Two Dimensions
Appendix B: Turning Down the Heat: Simulated Annealing