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"Does the proof stack up?" by George Szpiro. Nature, 3 July 2003, pages
13-14.
"The 24-dimensional greengrocer" by Ian Stewart. Nature, 21 August 2003, pages
895-896.
"To Prove the Optimal Packing," by Doron Zeilberger. Science, 29 August 2003.
"The proof of packing": Review of Kepler's Conjecture: How Some of the
Greatest Minds in History Helped Solve One of the Oldest
Math Problems in the World by George G. Szpiro. Reviewed by Neil Sloane.
Nature,
11 September 2003, pages 126-127.
It has been shown that the most efficient way to arrange circles in a plane is in a "honeycomb." The subject of these articles is the problem of the most efficient way to stack spheres in three dimensions and, more generally, how to stack the higher-dimensional analogs of spheres in higher-dimensional spaces. Stewart talks about work by Henry Cohn and Noam Elkies ("New upper bounds on sphere packings I.," published in Annals of Mathematics, Volume 157, pages 689-714) related to the higher-diimensional cases, especially dimensions 8 and 24. He gives some history of the problem as well as the bounds for dimensions 8 and 24 arrived at by Cohn and Elkies. Szpiro talks about work in three dimensions. About five years ago Thomas Hales claimed that he had proved the result for sphere packing. The proof fills 250 pages and involves so much computer checking that it has taken referees more than four years to review. Now the referees "believe the proof is correct, but are are so exhausted with the verification process that they cannot definitively rule out any errors." The Annals of Mathematics will publish the proof accompanied by an introduction saying that proofs requiring computer-checking of a huge number of statements may be impossible to completely review. Hales has invited mathematicians to use their computers to completely check the proof. He needs about ten volunteers for this project, called Flyspeck. There is more information about the project here.
--- Mike Breen
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