Aste T., Weaire D. Pursuit of perfect packing (IOP 2000)(147s).pdf

36 Proof positive
from sophistication to more elementary methods was one secret of his success.
Nevertheless his preprint ran to about 100 pages and was not easily digested even
by those hungry for information. As his colleagues and competitors picked over
the details, some errors became apparent.
This is not unusual. Another recent claim, of an even more dramatic result—
the proof of Fermat’s Last Theorem—has required some running repairs, but is
still considered roadworthy and indeed prizeworthy. However, Hsiang did not
immediately succeed in rehabilitating his paper.
Exchanges with his critics failed to reach a resolution. A broadside was
eventually launched at Hsiang by Thomas Hales in the pages of the splendidly
entertaining Mathematical Intelligencer. Hales’ piece lies at the serious end of
that excellent magazine’s spectrum but nevertheless it grips the reader with its
layers of implication and irony, most unusual in a debate on a piece of inscrutable
academic reasoning.
Despite the inclusion of some conciliatory gestures (‘promising programme’,
‘improves the method’) the overall effect is that of a Gatling gun, apparently puncturing
the supposed proof in many places. Hales begins with the statement that
‘many of the experts in the field have come to the conclusion that [Hsiang’s]
work does not merit serious consideration’ and ends with a demand that the claim
should be withdrawn: ‘Mathematicians can easily spot the difference between
hand-waving and proof’.
Hsiang replied at length in the same magazine, protesting against the use of a
‘fake counter example’. Meanwhile Hales and others were themselves engaged in
defining programmes for further work, as the explorer stocks supplies and makes
sketch-maps for a hopeful expedition. Indeed he was already at base camp.
A comment by Kantor on Hilbert’s 18th problem
Hilbert’s text gives the impression that he did not anticipate the
success and the developments this problem would have.
The hexagonal packing in plane is the densest (proof by Thue
in 1892, completed by Fejes in 1940). In space, the problem
is still not solved. Although there is very recent progress by
Hales. For spheres whose centres lie on a lattice, the problem
is solved in up to eight dimensions. The subject has various
ramifications: applications to the geometry of numbers, deep
relations between coding theory and sphere-packing theory, the
very rich geometry of the densest known lattices. (Kantor J M
1996 Math. Intelligencer Winter, p 27)