R. Weaire was writing a book on sphere
packings when I finished the proof of
the Kepler conjecture, and we began to
correspond. Under his influence, I
turned to the planar version of the
foam problem. This problem goes back
over 2000 years. What is the most
efficient partition of the plane into
equal areas? The honeycomb conjecture
asserts that the answer is the regular
hexagonal honeycomb.
Pappus
Around 36 BC, the Roman scholar Marcus
Terentius Varro wrote a book on
agriculture in which he discusses the
hexagonal form of the bee's honeycomb.
There were two competing theories of
the hexagonal structure. One theory
held that the hexagons better
accommodated the bee's six feet. The
other theory, supported by the
mathematicians of the day, was that the
structure was explained by the
isoperimetric property of the hexagonal
honeycomb. Varro writes, ``Does not the
chamber in the comb have six angles
¼ The geometricians prove that
this hexagon inscribed in a circular
figure encloses the greatest amount of
space.''
This ancient proof has been lost,
unless it was the proof presented a few
centuries later by Pappus of Alexandria
in the preface to his fifth book. The
argument in Pappus is incomplete. In
fact it involves nothing more than a
comparison of three suggestive cases.
It was known to the Pythagoreans that
only three regular polygons tile the
plane: the triangle, the square, and
the hexagon. Pappus states that if the
same quantity of material is used for
the constructions of these figures, it
is the hexagon that will be able to
hold more honey. Pappus's reason for
restricting his attention to the three
regular polygons that tile are not
mathematical (bees avoid dissimilar
figures). He also excludes gaps between
the cells of the honeycomb without
mathematical argument. If the cells are
not contiguous, ``foreign matter could
enter the interstices between them and
so defile the purity of their
produce.''\footnote"*"% {T. Heath, A
history of Greek mathematics, Vol II,
Oxford, 1921, page 390.}
In 1943, L. Fejes T\'oth gave a proof
of the honeycomb conjecture under the
hypothesis that all the cells are
convex polygons. He stated that the
general conjecture had ``resisted all
attempts at proving it.'' In 1999, I
found the first general
proof.
Theorem (Honeycomb conjecture).
Any partition of the plane into regions
of equal area has perimeter at least
that of the regular hexagonal honeycomb
tiling.
The main ingredient of the proof is a
new isoperimetric inequality, which has
the regular hexagon as its unique
minimum.
My expectations of mathematics have
been shaped by the Kepler conjecture. I
have come to expect every theorem to be
a monumental effort. I was
psychologically unprepared for the
light 20page proof of the honeycomb
conjecture. It makes no significant use
of computers and took less than six
months to complete. In contrast with
the years of forced labor that gave the
proof of the Kepler conjecture, I felt
as if I had won a lottery.
