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
Famous quotes related to twentieth century:
“The real passion of the twentieth century is servitude.”
—Albert Camus (1913–1960)
“The descendants of Holy Roman Empire monarchies became feeble-minded in the twentieth century, and after World War I had been done in by the democracies; some were kept on to entertain the tourists, like the one they have in England.”
—Ishmael Reed (b. 1938)
“A writer is in danger of allowing his talent to dull who lets more than a year go past without finding himself in his rightful place of composition, the small single unluxurious ‘retreat’ of the twentieth century, the hotel bedroom.”
—Cyril Connolly (1903–1974)
“One of the peculiar sins of the twentieth century which we’ve developed to a very high level is the sin of credulity. It has been said that when human beings stop believing in God they believe in nothing. The truth is much worse: they believe in anything.”
—Malcolm Muggeridge (1903–1990)
“In the twentieth century, death terrifies men less than the absence of real life. All these dead, mechanized, specialized actions, stealing a little bit of life a thousand times a day until the mind and body are exhausted, until that death which is not the end of life but the final saturation with absence.”
—Raoul Vaneigem (b. 1934)