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Three Dimensions
We return to three dimensions to
discuss the proof of the Kepler
conjecture. To avoid the unpleasant
boundary effects caused by finite
packings, we study sphere packings that
extend to all of Euclidean space.
Sphere packings are determined by a
countably infinite set of parameters,
which give the coordinates of the
center of each sphere. It was realized
in the fifties that it should be
possible to prove the Kepler conjecture
by looking at a finite number of balls
at a time. With this in mind, we
discuss finite clusters of
balls.
Voronoi
Each ball in our packings should be
painted one solid color, from a finite
color set. These colors are needed in
the details of certain constructions to
resolve degeneracies, to make piecewise
smooth functions smooth, and to keep
domains compact. These colors will help
me avoid oversimplifications in my
exposition. But by going into details
about colors, I would obscure the main
lines of the proof of the Kepler
conjecture. So now that we have
established that the balls are colored,
the reader is free to paint all the
balls black. Let t > 1 be a real
number. We define a cluster of balls to
be a set of nonoverlapping colored
balls around a fixed ball at the
origin, with the property that the ball
centers have distance at most 2t from
the origin. A cluster of n balls is
determined by the 3n coordinates of
the centers. These coordinates give a
topology on the set C = C(t) of all
clusters, making it a compact set. Two
clusters with a different number of
balls or different colorings lie in
different connected components of C.
The ball at the center of the cluster
is contained in a truncated Voronoi
cell. By definition, the Voronoi
cell is the set of all points that lie
closer to the origin than to any other
ball center in the cluster. The
truncated Voronoi cell Vt(p) is the
intersection of the Voronoi cell with a
ball of radius t at the origin. We
have seen Voronoi cells already in the
proof of Thue's theorem, without
calling them that. The regular hexagons
that appear in the proof of Thue's
theorem are the Voronoi cells of the
optimal packing. And the large disks,
sliced at times to form isosceles
triangles, are truncated Voronoi cells
(t = 2/Ö3). Truncation is purely a
matter of convenience, making the
volumes of Voronoi cells easier to
estimate.
Truncated Voronoi cells give a bound on
the density of sphere packings. We
place every ball of a packing inside
its truncated Voronoi cell. Voronoi
cells do not overlap; a point of
intersection of two closed Voronoi
cells, being equidistant from the
center of two balls, lies on the
boundary of both. The parts of space
outside all truncated Voronoi cells do
not meet any balls, and have density
0. The density of the packing is no
greater than its densest truncated
Voronoi cell. That is, the greatest
possible volume ratio of ball to
truncated Voronoi cell is an upper
bound on the density of a
packing.
 The Voronoi cells of the
face-centered cubic packing are
identical rhombic dodecahedra, as shown
in Figure 5. Let v\fcc be the volume
of the rhombic dodecahedron. The
density of the face-centered cubic
packing is the ratio of the volume of
the unit ball to v\fcc:
The most distant vertices of the
rhombic dodecahedron are Ö2
from the center. Thus, if
t ³ Ö2, then truncation has no
effect. If t < Ö2, then the
truncation cuts into the rhombic
dodecahedron and destroys the relation
between its volume and our target
p/Ö[18]. We fix the truncation
at t = Ö2, its smallest useful
value. The truncation now fixed, we
drop t from the notation and write
V(p) = Vt(p).
The minimum volume of a Voronoi cell
(either untruncated or truncated at
t = Ö2) was recently determined
by Sean McLaughlin. For this result, he
was awarded the AMS-MAA-SIAM Morgan
Prize in January 2000. It confirms a
conjecture made by L. Fejes T\'oth
nearly 60 years ago.
Theorem (McLaughlin). The volume
of the Voronoi cell of a sphere packing
of a cluster p is uniquely minimized
by a regular dodecahedron of inradius
1.
The cluster of balls that gives the
regular dodecahedron is a cluster with
one ball at the center and 12
additional balls tangent to the one at
the center, placed at the centers of
the faces of the regular
dodecahedron.
The ratio of the volume of the unit
ball to the volume of the regular
dodecahedron is an upper bound on the
density of a sphere packing. This upper
bound is about 0.75. In two
dimensions, the Voronoi cell of minimal
volume is the regular hexagon, and it
tiles the plane to form the optimal
packing. In three dimensions, the
Voronoi cell of minimal volume no
longer tiles. The locally optimal
figure, the dodecahedron, no longer
corresponds to the globally optimal
figure, the tiling by rhombic
dodecahedra. This is the source of
complications in the proof of the
Kepler conjecture.
We add correction term f to the
minimization of the volume of Voronoi
cells.
We define a continuous function f on
C, and consider the minimization
problem
We
say that f is fcc-compatible
if the minimum of \vol(V(p))+f(p) is
v\fcc, the volume of the rhombic
dodecahedron.
Let L be the set of centers of
the balls in a general packing. For
l Î L, consider the
cluster of balls centered at distance
at most 2t = 2Ö2 from l.
Translating the cluster to the origin,
we obtain a cluster pl in C.
Let LR = LÇBR be the
set of all centers within distance R
of the origin. We say that f is
transient if
Assume that f is fcc-compatible and
transient. By summing
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v\fcc £ \vol(V(p)) + f(p) |
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over
LR, we obtain
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|LR|v\fcc £ \vol(BR)+ o(R3). |
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Divide by R3 v\fcc to
get the density of a packing inside a
ball of radius R.
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|LR|
R3
|
£ |
4p/3
v\fcc
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+ o(1) = |
p
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+o(1). |
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Taking
the limit as R®¥, we obtain
the bound [(p)/(Ö[18])] on
the density of the packing.
That shows that the whole proof of the
Kepler conjecture follows if a
transient fcc-compatible function f
can be found. To establish
fcc-compatibility, an extremely
difficult nonlinear optimization
problem on C must be solved. We
select the function f with transience
in mind, so that it is automatically
satisfied.
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