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(08-03-2020 07:56 AM)EdS2 Wrote:  … Nice links found - thanks for sharing - but this quote > The classical sphere packing problem, still unsolved even today has become out of date. Kepler's conjecture has been proved, some ten years after this book, and then formally proven some 16 years later …

An excerpt from the preface of the third edition, ISBN 978-1-4757-6568-7 (eBook), © 1999, 1998, 1993 Springer Science+Business Media New York.

" Notes on Chapter 1: Sphere Packings and Kissing Numbers Hales [Hal92], [Hal97], [Hal97a], [Hal97b] (see also Ferguson [Ferg97] and Ferguson and Hales [FeHa97]) has described a series of steps that may well succeed in proving the long-standing conjecture (the so-called "Kepler conjecture") that no packing of three-dimensional spheres can have a greater density than that of the face-centered cubic lattice. In fact, on August 9, 1998, just as this book was going to press, Hales announced [Hal98] that the final step in the proof has been completed: the Kepler conjecture is now a theorem.
     The previous best upper bound known on the density of a three dimensional
packing was due to Muder [Mude93], who showed that the density cannot exceed 0.773055 ... (compared with :n'Jv'18 = 0.74048 ... for the f. c. c. lattice).
     A paper by W.-Y. Hsiang [Hsi93] (see also [Hsi93a], [Hsi93b]) claiming to prove the Kepler conjecture contains serious flaws. G. Fejes T6th, reviewing the paper for Math. Reviews [Fej95], states: "If I am asked whether the paper fulfills what it promises in its title, namely a proof of Kepler's conjecture, my answer is: no. I hope that Hsiang will fill in the details, but I feel that the greater part of the work has yet to be done." Hsiang [Hsi93b] also claims to have a proof that no more than 24 spheres can touch an equal sphere in four dimensions. For further discussion see [CoHMS], [Hal94], [Hsi95]. "

A significant history of dialog, with some similarity to the early years of astronomy, is encompassed in those 16 years. Like a detective-murder novel, a complete read has its' merits.

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