How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

The next layer is going to look just the same as this one, but cunningly placed so that each seed sits in the little triangular divot formed by three seeds below it. Then just keep adding more layers in the same way. It’s best to be a little careful here: only half the divots are going to support spheres in the next layer up, and at each stage you have a choice of which half of the divots you want to fill. The most customary choice, called the face-centered cubic lattice, has the nice property that every layer has the spheres placed directly over the spheres three layers below. According to Kepler, there is no denser way to pack spheres in space. And in the face-centered cubic lattice, each sphere touches exactly twelve others. As the pomegranate seeds grew, Kepler reasoned, each one would press against its twelve neighbors, flattening its surface near the point of contact and producing the twelve-sided figures he observed. Whether Kepler was right about pomegranates I have no idea,* but his claim that the face-centered cubic lattice was the densest possible sphere packing became a topic of intense mathematical interest for centuries. Kepler offered no proof of his statement; apparently it just seemed right to him that the face-centered cubic lattice couldn’t be beat. Generations of grocers, who stack oranges in a face-centered cubic configuration without any worry as to whether their method is the absolute best possible, agree