THE PURSUIT OF PERFECT PACKING
T. Aste & D. Weaire, Institute of Physics Publishing, 2000; pp: 132, ISBN 0-7503-0648-3 (pbk), Price: $29.00
SUMMARY: The alliterative title sets the tone for this largely historical, sometimes witty discussion of attempts to find the densest arrangements of various objects. There are some mathematical details but minimal technical background is needed. There is also gossip.
I recommend the book to anyone, specialist or not, interested in the subject.
DETAILED REVIEW: Packing problems range from squeezing items into refrigerators to David Hilbert's 18th problem: finding the densest packing of an infinite number of equal solids.
Chapter 2 proves that a triangular pattern gives the densest arrangement of circular coins on a flat surface. The packing fraction, (area covered)/(total area), is greatest for identical, regular polygonal patterns. Using discs of different sizes introduces disorder.
Chapter 3 examines packing of spheres, especially the work of Desmond Bernal and John Finney, who discovered the densest random packing of hard spheres.
Osborne Reynolds showed that--paradoxical as it may seem—stepping on wet sand expands it. Crystallographers sought the least dense rigid packing of hard spheres. Johannes Kepler asserted that a face-centered cubic arrangement produced the greatest density of hard spheres. This statement lacked rigorous proof for centuries.
Chapter 5 discusses packing of soft objects. Stephen Hales experimented with water-soaked peas. Earlier, Kepler experimented with pomegranate seeds. Other studies involved biological cells, lead shot, soap bubbles, and foam. The authors challenge the reader to find four states that meet at a point. I think Zambia, Zimbabwe, Botswana, and Namibia qualify.
Chapter 6 mentions honeycombs, which are hexagonal, and a more conservative pattern that bees occasionally make.
Chapter 7 discusses soap bubbles. Lord Kelvin decided that the optimum packing is the tetrakaidecahedron or Kelvin cell. Robert Phelan and Denis Weaire found a still denser pattern.
Chapter 8 covers packing of atoms in crystals. The 1984 discovery of quasicrystals opened research anticipated by Kepler. "Amorphous" solids like glass are actually made of crystalline grains, despite the non-crystalline classification.
Chapter 9 discusses packing of objects of various sizes, as in concrete. So-called Apollonian packing of circles or spheres can, in the limit, fill available space.
Chapter 10 discusses Ireland's Giant's Causeway, which is not as uniformly hexagonal as some people claimed.
Chapter 11 discusses soccer balls, golf balls, buckyballs, the J.J. Thomson problem of arranging electric charges on a sphere for minimum energy, pollen grains, and helices.
Chapter 12 indicates that as the number of dimensions increases, maximum density decreases. The "kissing" problem—finding the maximum number of equal spheres that can touch an equal central sphere—appears in Chapter 12. Solutions are known only for Dimensions 1, 2, 3, 8, and 24.
Chapter 13 covers situations from parking cars to Cinderella's separating of lentils from ashes.
Chapter 14 mentions a close packing of tetrahedral containers that was not examined much because a bus did not stop long enough.
Here are samples of gossip: Pages 22 and 23 mention rumours that Desmond Bernal had a great sexual appetite, and page 33 suggests that Kepler was a nerd.
One annoyance: References to footnotes sometimes appear as superscripts on variables, making them look like exponents.
I like this book, and recommend it to any educated layman.
Dr. David P. Maroun
Chilliwack, BC