Max, Nelson, Another harmony of the spheres, Nature 355 (9 January 1992) 115-116. Stewart, Ian, Mathematical recreations: The kissing number, Scientiflc American (February 1992) 112-115.
In 1990, Wu-Yi Hsiang (University of California-Berkeley) announced a proof that face centered cubic packing is the densest packing of spheres in three dimensions, thereby settling a claim of Kepler's. Or is it settled? Hsiang's original announcement was rejected by the Bulletin of the AMS because the details of the proof had not yet been written down. Hsiang's argument depends on the classification and analysis of a large number of configurations, many of which he described originally only in qualitative terms. Now there is a draft of all the details. But will anyone "have the patience to repeat Hsiang's year-long verification that all possibilities are covered?" Hsiang's cases are not easily amenable to verification by computer, despite the fact that he used "only high-school algebra, vector identities, a little calculus and a lot of spherical geometry and spherical trigonometry," plus his TI-35+ calculator. Finally, Hsiang has recently announced that he has solved the 'kissing" problem in four dimensions, determining that the greatest number of spheres that can surround and touch another is 24.
This review was reprinted with permission from Mathematics Magazine, April 1992 issue.