Twentieth Century
The next step toward a solution was taken by Hungarian mathematician László Fejes Tóth. Fejes Tóth (1953) showed that the problem of determining the maximum density of all arrangements (regular and irregular) could be reduced to a finite (but very large) number of calculations. This meant that a proof by exhaustion was, in principle, possible. As Fejes Tóth realised, a fast enough computer could turn this theoretical result into a practical approach to the problem.
Meanwhile, attempts were made to find an upper bound for the maximum density of any possible arrangement of spheres. English mathematician Claude Ambrose Rogers (1958) established an upper bound value of about 78%, and subsequent efforts by other mathematicians reduced this value slightly, but this was still much larger than the cubic close packing density of about 74%.
Hsiang (1993, 2001) claimed to prove the Kepler conjecture using geometric methods. However Gábor Fejes Tóth (the son of László Fejes Tóth) stated in his review of the paper "As far as details are concerned, my opinion is that many of the key statements have no acceptable proofs." Hales (1994) gave a detailed criticism of Hsiang's work, to which Hsiang (1995) responded. The current consensus is that Hsiang's proof is incomplete.
Read more about this topic: Kepler Conjecture
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