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:
“In the twentieth century one of the most personal relationships to have developed is that of the person and the state.... It’s become a fact of life that governments have become very intimate with people, most always to their detriment.”
—E.L. (Edgar Lawrence)
“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)
“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 phenomenon of nature is more splendid than the daily events of nature, certainly, so then the twentieth century is splendid.”
—Gertrude Stein (1874–1946)
“Doubt, it seems to me, is the central condition of a human being in the twentieth century.”
—Salman Rushdie (b. 1947)