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:
“Advertising is the greatest art form of the twentieth century.”
—Marshall McLuhan (1911–1980)
“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)
“The nineteenth century planted the words which the twentieth ripened into the atrocities of Stalin and Hitler. There is hardly an atrocity committed in the twentieth century that was not foreshadowed or even advocated by some noble man of words in the nineteenth.”
—Eric Hoffer (1902–1983)
“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)
“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)