I've been having some email problems, but despite them I've been following
the discussion of kissing numbers and how hard it is to establish them. In
particular, I'm wondering whether we may not be missing some possible
low-hanging fruit that might result from the sort of greedy packing
procedure that Jud McCranie sketched, and about which John Conway expressed
skepticism.
The central problem is the size of the configuration space that must be
explored: 24 or 25 spheres jostling around, each contributing 3 degrees of
freedom to the configuration space. JM proposed, if I read him right, to
prune this space by considering only configurations of a certain special
kind, in which the configuration may be built by adding each new sphere in
the 'socket' formed by three previously-placed spheres. (This is hard to
visualize, but in four-space, you need three other spheres to form a
well-defined socket with no degrees of freedom.) Enumerating such
configurations would be much easier than searching an enormous
75-dimensional configuration space.
JHC expresses doubt that this program could work, because of the
possibility of losing the baby with the bathwater. It might be possible
for 25 non-overlapping 4-spheres to kiss a central one, but only in
configurations that are _not_ in the restricted class JM proposes. Let us
call these JM-type configurations _tight_ configurations; forgive me if I
do not give a rigorous definition -- I'll be willing to ante up later, but
for the moment I think we all know what we're talking about.
Obviously, the JM program depends on an unstated conjecture, one that JHC
believes to be false. If forced to place bets at this point, I'd wager
that JHC is right ... unless JHC offered me attractive odds. Because I'm
not yet morally certain he's right. The conjecture, which I will call the
Tightening Conjecture, is: if there exists ANY configuration of n
non-overlapping radius-R k-spheres kissing a given unit k-sphere, then
there exists a TIGHT such configuration.
JHC puts forward what I believe is intended to be a counterexample to this
conjecture, but I claim it isn't really a counterexample. In the case he
presents, k=3, n=8, and R=(sqrt3+1)/2. He points out that it works to
position each of the 8 peripheral spheres at the corners of a cube of side
2R, whose center is the center of the unit sphere. This configuration is
_not_ tight, in the sense given above, and JHC implies by his use of the
phrase "the correct configuration" that it is the only one. If this were
true, it would certainly follow that a search confined to tight
configurations would fail to find one, and if we believed the Tightening
Conjecture we would incorrectly conclude that the 8 peripheral spheres
simply would not fit.
However, the configuration described by JHC is _not_ the only one that
works. Begin with the cubical configuration described by JHC. Run a plane
through the center of the central sphere, so that all eight peripheral
spheres are tangent to that plane. Now it is clear that four peripheral
spheres on one side of the plane (call it the north side) can be rotated
independently of the other four, and after they have been rotated 45
degrees, they can move south a little bit, across the equator, and settle
into the four northward-facing sockets offered between the four southern
spheres. The resulting crownlike configuration still has one degree of
freedom, but I am reasonably sure that this extra flex can be expunged by
puckering the square formed by the four southern spheres. Eventually
something will bump, at which point the configuration will be tight.
The Tightening Conjecture may still be false, but JHC's cubical example
does not kill it outright. My guess is that JHC has a stronger
counterexample in his arsenal, that _will_ disprove the TC; my only real
point in this message is that the offered example doesn't do the job.
-A
