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 phenomenon of nature is more splendid than the daily events of nature, certainly, so then the twentieth century is splendid.”
—Gertrude Stein (1874–1946)
“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)
“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)
“War is bestowed like electroshock on the depressive nation; thousands of volts jolting the system, an artificial galvanizing, one effect of which is loss of memory. War comes at the end of the twentieth century as absolute failure of imagination, scientific and political. That a war can be represented as helping a people to ‘feel good’ about themselves, their country, is a measure of that failure.”
—Adrienne Rich (b. 1929)
“The real passion of the twentieth century is servitude.”
—Albert Camus (1913–1960)