cass 1 Packing Pennies in the Plane An illustrated proof of Kepler's conjecture in 2D by Bill Casselman NOTE: This month's contribution contains several Java applets. They may not work on your particular computer, for any of various reasons. If you do not have Java enabled in your browser, for example, you will see only static images representing the animated applets. If you have trouble with viewing the applets even though Java is enabled, or if you want to print out this note, you should disable Java. If Java is enabled and you still have trouble viewing the applets, please let Bill Casselman know about it. 1. "Kepler's Conjecture" This and the other image nearby are from Kepler's pamphlet on snowflakes. Contrary to what one might think at first. they are not of two dimensional objects, but rather an attempt to render on the page three dimensional packings of spheres.  In his book De nive sexangula (`On the sixsided snowflake') of 1611, Kepler asserted that the packing in three dimensions made familiar to us by fruit stands (called the facecentred cubic packing by crystallographers) was the tightest possible: Coaptatio fiet arctissima: ut nullo praetera ordine plures globuli in idem vas compingi queant. He didn't elaborate much, and his statement lacks precision. It is almost certain that he had no idea that this assertion required rigorous proof. At any rate, this claim came to be known as Kepler's conjecture, and it turned out to be extremely difficult to verify.  Kepler quite likely would have thought that the analogous assertion about the hexagonal packing in 2D was even more obvious. However, it took about 300 years before it was proven, by the Norwegian mathematician Axel Thue. It is arguable that it took that long just to understand that such an `obvious' assertion required proof. It took another century before a proof of the much more difficult claim about 3D was found, by Tom Hales.  Both images are from photographs taken of the copy of the original edition of Kepler's pamphlet now located at the Thomas L. Fisher Library at the University of Toronto.   Bill Casselman University of British Columbia

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