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Topic: Accessible problems
Replies: 5   Last Post: Apr 27, 1995 9:09 PM

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Ed Dickey

Posts: 9
Registered: 12/6/04
Re: Accessible problems
Posted: Apr 27, 1995 9:58 AM
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Ed,

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.

Ed Dickey

> 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
> >
> >
> >







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