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:
“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)
“Film is more than the twentieth-century art. It’s another part of the twentieth-century mind. It’s the world seen from inside. We’ve come to a certain point in the history of film. If a thing can be filmed, the film is implied in the thing itself. This is where we are. The twentieth century is on film.... You have to ask yourself if there’s anything about us more important than the fact that we’re constantly on film, constantly watching ourselves.”
—Don Delillo (b. 1926)
“Advertising is the greatest art form of the twentieth century.”
—Marshall McLuhan (1911–1980)
“If the twentieth century is to be better than the nineteenth, it will be because there are among us men who walk in Priestley’s footsteps....To all eternity, the sum of truth and right will have been increased by their means; to all eternity, falsehoods and injustice will be the weaker because they have lived.”
—Thomas Henry Huxley (1825–95)
“... the nineteenth century believed in science but the twentieth century does not. Not.”
—Gertrude Stein (1874–1946)