If we turn to the next page after the
Kepler conjecture in Kepler's
SixCornered Snowflake, we find a
discussion of the structure of the
bee's honeycomb. The rhombic
dodecahedron was discovered by Kepler
through close observation of the
honeycomb. The honeycomb is a sixsided
prism sealed at one end by three
rhombi. By sealing the other end with
three additional rhombi, the honeycomb
cell is transformed into the rhombic
dodecahedron.
Figure 8 

During the 18th century, mathematicians
made extensive studies of the
isoperimetric properties of the
honeycomb and believed the honeycomb to
be the most efficient design possible.
For example, in 1743, C. MacLaurin in
his investigation of the rhombic bases
of the honeycomb concluded, ``The
cells, by being hexagonal, are the most
capacious, in proportion to their
surface, of any regular figures that
leave no interstices between them, and
at the same time admit of the most
perfect bases.'' However, the obvious
answer provided by the honeybee turned
out incorrect. Upsetting the prevailing
opinion, L. Fejes T\'oth discovered
that the threedimensional honeycomb
cell is not the most economical (Figure
8). The most economical form has never
been determined.
The cannonball packing of balls leads
to honeycomb cells. It is also related
to more general foam problems. If we
tile space with hollow rhombic
dodecahedra, and imagine that each has
walls made of a flexible soap film, we
have an example of a foam.
Kelvin
The problem of foams, first raised by
Lord Kelvin, is easy to state and hard
to solve. How can space be divided into
cavities of equal volume so as to
minimize the surface area of the
boundary? The rhombic dodecahedral
example is far from optimal. Lord
Kelvin proposed the following solution.
Truncated octahedra fill space (see
Figure 9). In fact, their cross sections
are regular octagons, and the octagons
tile the plane except for square holes.
Each truncated octahedron contains
square plugs, so that the next layer of
octahedra plugs the square holes of the
previous layer.
Lord Kelvin found that by warping the
faces of the truncated octahedra ever
so slightly, he could obtain a foam
with smaller surface area than the
cells of the truncated octahedra. This
was Lord Kelvin's proposed solution. It
satisfies the conditions Plateau
discovered more than a century ago for
minimal soap bubbles. Everyone seemed
satisfied with Kelvin's solution; only
a proof of optimality was
missing.
Kelvin and his supporters were wrong,
as the physicists D. Phelan and R.
Weaire showed in 1994. They produced a
foam with cavities of equal volume with
a considerably smaller surface area
than the Kelvin foam. The PhelanWeaire
foam contains two different types of
cavities, one with 14 sides, the other
with 12. (Figure 10).
A finite version of the problem might
be more tractable. What shape minimizes
surface area if the foam contains only
finitely many bubbles of equal volume?
If there is a single bubble, the
problem is the classical isoperimetric
problem. The sphere uniquely minimizes
the area of a surface enclosing a given
volume. The problem of two bubbles,
known as the double bubble conjecture,
was solved only recently by J. Hass, M.
Hutchings, and R. Schlafy.(The
double bubble conjecture, Electron.
Res. Announc. Amer. Math. Soc. 1
(1995), no. 3, 98102.) The problem for
more than two bubbles is still
unsolved.
The Kepler conjecture and the Kelvin
problem are both special cases of a
more general foam problem. Phelan and
Weaire ask us to imagine that the soapy
film walls have a measurable thickness.
We interpolate between the Kepler and
Kelvin problems with a parameter w
(measuring the wetness of the film)
that gives the fraction of space filled
by the thick film walls, and 1w is
the fraction filled by the cavities. If
the foam is perfectly dry, then w = 0,
and the film walls are surfaces. The
Kelvin problem asks for the most
efficient design. When the foam becomes
sufficiently wet, w is close to 1,
and the cavities of the foam can be
independently molded. The isoperimetric
inequality dictates that they minimize
surfaces area by forming into perfect
spheres. The Kepler problem asks for
the smallest value of w for which
every cavity is a perfect
sphere.
At this time of renewed interest in
Hilbert's problems, many mathematicians
are proposing new lists of problems. My
submission to Hilbert's millennial list
is the Kelvin problem. It has a rich
history. Its solution will require new
ideas from geometric measure theory.
Frank Morgan predicts that the Kelvin
problem might take a century to be
solved. For starters: does an optimal
solution exist?
