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.
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.: https://mathworld.wolfram.com/HexagonalClosePacking.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)