The pyramid stacking of oranges is
known to chemists as the
facecentered cubic packing. It is
also known as the cannonball packing,
because it is commonly used for that
purpose at war memorials. The oldest
example I have seen is the pyramid of
cannonballs from the 16th century that
rests in front of the City Museum of
Munich. Formulas for the number of
cannonballs stacked in mounds have been
known this long. In the 16th century,
Sir Walter Raleigh gave his
mathematical assistant, Thomas Harriot,
the task of finding the formula.
Harriot did this without
difficulty.
As Harriot grew in reputation as a
scientist, spheres became a favorite
topic of his. To him, atoms were
spheres. To understand how they stack
together is to understand nature.
Numbers were spheres. In the tradition
of Pythagoras, triangular numbers are
stacked like billiard balls set in a
triangle. Harriot draws Pascal's
triangle, but with a sphere packing
with that many spheres replacing each
number.
In 1606, Kepler complained that his
recent book on optics was based
entirely on theology. He turned for
help to Harriot, who had been
conducting experiments in optics for
years. In fact, Harriot's knowledge of
optics was so advanced that he had
discovered Snell's law  20 years
before Snell and 40 years before
Descartes. Harriot supplied Kepler with
valuable data in optics, but he also
tried to persuade Kepler that the
deeper mysteries of optics would be
unfolded through atomism. Unlike
Kepler, Harriot was an ardent atomist,
believing that the secrets of the
universe were to be revealed through
the patterns and packings of small,
spherical atoms. Kepler was skeptical.
Nature abhors a vacuum, and between the
atoms lies the void.
Harriot persisted. Kepler relented. In
1611, Kepler wrote a little booklet,
The SixCornered Snowflake, that
influenced the direction of
crystallography for the next two
centuries. This slender essay was the
``first recorded step towards a
mathematical theory of the genesis of
inorganic or organic form. (L.L. Whyte,
in The SixCornered Snowflake,
Oxford Clarendon Press, Oxford, 1966.)
In a discussion of sphere packings, he
constructed the facecentered cubic
packing. Kepler asserted that it would
be ``the tightest possible, so that in
no other arrangement could more pellets
be stuffed into the same container.''
This assertion has come to be known as
the Kepler conjecture. It went without
a proof for nearly 400 years, until
August 1998, when I gave a proof with
the help of a graduate student, Samuel
P. Ferguson.
Theorem (Kepler conjecture). No
packing of balls of the same radius in
three dimensions has density greater
than the facecentered cubic packing.
The facecentered cubic packing is
constructed by setting one layer of
balls upon another. Each layer is a
regular pattern of balls in a grid of
equilateral triangles. But there are
other sphere packings that have exactly
the same density (p/Ö[18] » 0.74) as the facecentered cubic
packing. The best known alternative is
the hexagonal closepacking. Its
individual layers are identical to
those of the facecentered cubic. But
the layers are staggered to produce a
different global packing with the same
density (Figure 2a and 2b). There is no
simple list of all packings of density
p/Ö[18], but there are many
other possibilities. The density of the
packing in space is defined as a limit;
and removing one, two, or a hundred
balls from the packing will not affect
the limiting density. Nor will removing
an entire infinite layer of
balls.
This article gives the broad outlines
of the proof of the Kepler conjecture
in the most elementary possible terms.
The proof is long (282 pages), and
every aspect of it is based on even
longer computer calculations. A jury of
12 referees has been deliberating on
the proof since September 1998. Nobody
has raised doubts about the overall
correctness of the proof. And yet, to
my knowledge, no one has made a
thorough independent check of the
computer code.
