Sphere packing and kissing numbers have intrigued me recently. I've included this topic in a manipulatives course I've developed for middle and high school teachers.
In 2-D, the sphere packing problem is easy to demonstrate and establish: what is the strategy for maximizing the number of circles that can be packed into a finite plane. This can be modeled with pennies or other manipulatives. The most dense packing covers pi/sqrt(12) per cent of the plane (this is not so easy to see or establish).
For 3-D, the problem can be modeled with tennis balls (or cannon balls). I mention cannon balls because the optimal packing strategy looks like a stack of cannon balls on the court house square. While the most dense packing known fills pi/sqrt(18) per cent of the space, it remains to be proven that this _is_ the densest packing. W.-Y. Hsiang of Cal-Berkeley offered a proof in 1991 but it has not been accepted as correct.
The kissing number is an analoguous problem. In 2D, what is the maximum number of circles that can touch (kiss, as in billiards) another circle. In 3D, it's what is the maximum number of spheres that can touch a (central) sphere. In the late 1600's Newton argued 12 but Gregory said 13. Who's right? (Proving this is another story.) The kissing number for 4D spheres is an unsolved problem. "Most mathematicians believe, and all physicists know" the kissing number in 4D is 24, the only accepted proof to date establishes that the number is either 24 or 25.
Sphere packing in all dimensions has applications in telecommunications and other areas.
I have found these problems to be accessible to 2nd graders while their solutions challenge the best living mathematicians. (I can't resist recounting my own 2nd grader's response to the question of what is the kissing number for 4D spheres? -- without hesitation he recited: 1-800-KISSING)
More information on this topic can be found in Barry Cipra's brief article "Music of the Spheres" in _Science_, 1 March 1991, p. 1028 and a thorough coverage in Conway and Sloane's _Sphere Packings, Lattices, and Groups_, Springer-Verlag, 1993.
> On Tue, 25 Apr 1995, Ed Wall wrote: > > > I was reviewing a video of the Fermat Fest held to celebrate Andrew Wyles > (sp?) > > solution of Fermat's Last Theorem to see if it was reasonable for my > > Algebra > > class to view (I've been doing a little with Diophantine problems so it > > probably is). Andrew was talking about when he was ten of trying it for the > > first time. I can remember when I 'tried' also (with less result :) ) and > > probably others can remember also. It was an accessible problem and > > definitely > > a challenge. > > > > Some of the panel members were talking about this accessibility and the > > fact > > that some major mathematics was being done later in one's career because of > > the need to gain more background. I began wondering what were the new > > accessible challenges for our 'teens'. Are there any more accessible, > > easily > > stated, unproven mathematical statements or are those days over? > > > > Ed Wall > > > > > >