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in the Popular Press

"Does the proof stack up?" by George Szpiro. *Nature*, 3 July 2003, pages13-14.

"The 24-dimensional greengrocer" by Ian Stewart. *Nature*, 21 August 2003, pages895-896.

"To Prove the Optimal Packing," by Doron Zeilberger. *Science*, 29 August 2003.

"The proof of packing": Review of *Kepler's Conjecture: How Some of theGreatest Minds in History Helped Solve One of the OldestMath 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 isin a "honeycomb." The subject of these articles is the problem of the mostefficient way to stack spheres in three dimensions and, more generally, how tostack the higher-dimensional analogs of spheres in higher-dimensional spaces.Stewart talks about work by Henry Cohn and Noam Elkies ("New upper bounds onsphere packings I.," published in *Annals of Mathematics*,Volume 157, pages 689-714) related to the higher-diimensional cases, especiallydimensions 8 and 24. He gives some history of the problem as well as the boundsfor dimensions 8 and 24 arrived at by Cohn and Elkies. Szpiro talks about workin three dimensions. About five years ago Thomas Hales claimed that he hadproved the result for sphere packing. The proof fills 250 pages and involves somuch computer checking that it has taken referees more than four years toreview. Now the referees "believe the proof is correct, but are are soexhausted with the verification process that they cannot definitively rule outany errors." The *Annals of Mathematics* will publish the proofaccompanied by an introduction saying that proofs requiring computer-checkingof a huge number of statements may be impossible to completely review. Haleshas invited mathematicians to use their computers to completely check theproof. He needs about ten volunteers for this project, called Flyspeck. Thereis more information about the project here.

*--- Mike Breen*