Today, a thought about commerce, cannonballs and
M&Ms. The University of Houston presents this
series about the machines that make our
civilization run, and the people whose ingenuity
created them.
The other day at the candy
counter, I got malt balls and my wife picked up
M&Ms. Now an article in Science magazine tells
about packaging objects. A picture shows groups of
smooth spheres and groups of ellipsoids. They look
just like M&Ms and malt balls.
The article talks about creating the densest
packages -- the smallest ones for a given number of
items -- candy, grain or, for that matter, cannon
balls. First, spherical objects:
You might arrange ten rows of ten balls in the
bottom of a square box. Then lay in identical
layers, up to the top. That is very poor packaging.
Only 52 percent of the space gets used. But if you
shake the box, the spheres will find a much closer
packing -- 64 percent. (Notice how dry cereal often
settles, so the package looks only half full when
you open it.)
Now the catch: 64 percent is far from the tightest
packing for spheres. Eighteenth-century sailors did
much better when they stacked cannonballs in
pyramids, nesting them within one another. That
calls up an urban legend: On the old warships,
stacks of cannonballs were held in place by frames
called brass monkeys.
They were brass, since cannonballs rusted and stuck
to an iron frame. The story says that, since brass
contracts more than iron, balls could be dislodged
in very cold weather. Hence the saying that "It's
cold enough to freeze the ..." Well, you know the
rest. (And if you don't, I should not be the one to
tell you.)
It's an unlikely story. What is true is
that, in that kind of hexagonal nest, each sphere
occupies a space shaped as a dodecahedron -- a
figure with twelve equal sides. That arrangement is
74 percent full. (A lot more malt balls in the
package.)
Kepler suggested that optimal packing for spheres,
four hundred years ago. Ever since, mathematicians
have been trying to prove that you can do no better
in packing spheres. Only now, in the early
twenty-first century, are they succeeding.
But come back to those boxes of candy. Shake a box
of malt balls and they won't reach Kepler's
optimum. Shake a box of M&Ms and they
will. That's because you can push a sphere
only along a line through its center. Push on an
M&M, and you can exert a torque. The M&M
can be twisted about, but the sphere cannot.
Shaking a box of M&Ms, or grains of sand, will
nudge particles in far more ways. They'll find
their optimal packing. That's probably the reason
M&Ms are shaped like little flying saucers.
So my wife got more candy in her box than I did in
mine. To get the maximum number of malt balls in a
box, you'd have to stack them manually -- the way
sailors once stacked cannonballs.
This may sound like a wedding of frivolity with
arcane math. But think about our vast traffic in
small objects -- ball bearings, rice, gravel,
oranges. The people who manage all this commerce
think long and hard about the weight and volume of
moving produce. In the end, we've found a place far
beyond just math or malt balls.
I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
work.
(Theme music)
David A. Weitz, Packing in the Spheres. Science
Magazine, Vol. 303, 13 February, 2004, pp. 968-969.
You'll find many websites on this subject. See
e.g.:
http://mathworld.wolfram.com/HexagonalClosePacking.html
http://www.georgehart.com/virtual-polyhedra/kepler.html
Another related matter is that of walking on wet
sand. See Episode 1529.
Optimal packing -- the same form as used for
cannonballs
(photo by John
Lienhard)
The Engines of Our Ingenuity is
Copyright © 1988-2004 by John H.
Lienhard.
