Pucker Up

The world took notice when some old mathematical conjectures  were finally solved in our own lifetimes:  Fermat’s Last “Theorem”, and the Poincaré Conjecture.   To somewhat less fanfare, a conjecture concerning sphere packing, dating back to Johannes Kepler in the seventeenth century (yes, that Kepler -- the planet guy) was finally solved in the twenty-first.   A breakthrough came in 1998, but a formal proof was not recognized until this very year.

George Szpiro wrote an unusually readable account in his book Kepler’s Conjecture (2003).   It is an example of a sphere-packing problem,  which is a global problem, with applications stretching from the fruiterer’s tray to error-correcting codes. Among the related matters is the kissing number problem, which is a local problem:  how many spheres (or, in two dimensions, discs) can you pack around a given central sphere.  The answer is the Newton number or kissing number for that dimension.   

In two dimensions, the answer is six, as you can verify by ringing a penny with six of its fellows;  that no further coin could butt in, can be seen “by inspection”.  (That, incidentally, is the only known use for pennies, in our own day.)   For three dimensions (the everday space we live in), the answer turns out to be 12.  That is more difficult to see, though you can accomplish it using a baker’s dozen of beachballs and a dozen undergraduates.

Somewhat surprisingly, for dimensions four, five, six and seven, the answer is not known exactly, only upper limits.  Very surprisingly, we then suddenly get a break :

In eight dimensions, all of a sudden the Newton number is known exactly:  240 white balls can kiss the black ball in the center.  Why is that number known precisely?  In this case the upper bound came out to be 240;  but a certain well-known lattice arrangement, called E₈, also allows 240 balls to touch the central sphere.   Since the actual kissing number of E₈ coincides with the theoretical upper bound, 240 must be the highest kissing number. -- From dimensions nine to twenty-three, again, only bounds are known.   (p. 96)

What is so gratifying about this is that what is in essence a simply, concretely conceptualizable problem  should be solved by the testimony of so fiercely abstract an algebraic object as E_8.  (Szpiro calls it “well-known”;  depends what circles you move in.  I just asked the fellow next to me in the pub, and the poor fool confused it with the Leech lattice.)  I first learned about this enormous structure from a 2007 article in the New York Times.   Or rather, learned virtually nothing about it at all, beyond its mere name, since the journalist, and all hands asked, concurred that the thing is indescribable.  A sketch of that encounter, back from when we were shilling for Platonism, can be read here.

Language note:  This metaphor of ‘kissing’ (for tangency) is used elsewhere in mathematics: osculating plane, osculating circle.