When Hilbert introduced his famous list
of 23 problems, he said a test of the
perfection of a mathematical problem is
whether it can be explained to the
first person in the street. Even after
a full century, Hilbert's problems have
never been thoroughly tested. Who has
ever chatted with a telemarketer about
the Riemann hypothesis or discussed
general reciprocity laws with the
family physician?
Last year, a journalist from Plymouth,
New Zealand, decided to put Hilbert's
18th problem to the test, and took it
to the street. Part of that problem can
be phrased, Is there a better stacking
of oranges than the pyramids found at
the fruit stand? In pyramids, the
oranges fill just over 74% of space
(Figure 1). Can a different packing do
better?
The greengrocers in Plymouth were not
impressed. ``My dad showed me how to
stack oranges when I was about four
years old,'' said a grocer named Allen.
Told that mathematicians have solved
the problem after 400 years, Allen was
asked how hard it was for him to find
the best packing. ``You just put one on
top of the other,'' he said. ``It took
about two seconds.''
Not long after I announced a solution
to the problem, calls came from the Ann
Arbor farmers market. ``We need you
down here right away. We can stack the
oranges, but we're having trouble with
the artichokes.''
To me as a discrete geometer, there is
a serious question behind the
flippancy. Why is the gulf so large
between intuition and proof? Geometry
taunts and defies us. For example, what
about stacking tin cans? Can anyone
doubt that parallel rows of upright
cans give the best arrangement? Could
some disordered heap of cans waste less
space? We say certainly not, but the
proof escapes us. What is the shape of
the cluster of three, four, or five
soap bubbles of equal volume that
minimizes total surface area? We blow
bubbles and soon discover the answer,
but cannot prove it. Or what about the
bee's honeycomb? The threedimensional
design of the honeycomb used by the bee
is not the most efficient possible.
What is the most efficient
design?
This article will describe some recent
theorems that might have been proved
centuries ago, if only our toolbag had
matched our intuition in power. This
article will describe the proof that
the pyramid stacking of oranges is the
best possible. But first, I explain a
few terms. A sphere packing always
refers to a packing by solid balls.
(The subject of sphere packings should
more properly be called ball packings.)
Density is the fraction of a region of
space filled by the solid balls. If
this region is bounded, this fraction
is the ratio of the volume of the balls
to the volume of the region. If any
ball crosses the boundary of the
region, only the part of the ball
inside the region is used. If the
region is unbounded, the density of the
intersection of the region with a ball
of radius R is calculated, and the
density of the full region is defined
as the lim sup over R.
