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:
“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)
“The phenomenon of nature is more splendid than the daily events of nature, certainly, so then the twentieth century is splendid.”
—Gertrude Stein (1874–1946)
“In the middle of the next century, when the literary establishment will reflect the multicultural makeup of this country and not be dominated by assimiliationists with similar tastes, from similar backgrounds, and of similar pretensions, Langston Hughes will be to the twentieth century what Walt Whitman was to the nineteenth.”
—Ishmael Reed (b. 1938)
“Predatory capitalism created a complex industrial system and an advanced technology; it permitted a considerable extension of democratic practice and fostered certain liberal values, but within limits that are now being pressed and must be overcome. It is not a fit system for the mid- twentieth century.”
—Noam Chomsky (b. 1928)
“As the twentieth century ends, commerce and culture are coming closer together. The distinction between life and art has been eroded by fifty years of enhanced communications, ever-improving reproduction technologies and increasing wealth.”
—Stephen Bayley (b. 1951)