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Second, he claims—also without justification—that the tightest packing is achieved when twelve seeds arrange themselves around a central seed in a regular manner. That is why, so he claimed without further ado, that when they are pressed together, the central seed is deformed into a twelve-faced rhomboid, and not a dodecahedron, as Hales had thought. So, here it is. On pages 9 and 10 of Kepler’s little treatise we find the formulation of his famed conjecture: A sphere surrounded in a certain way by twelve other spheres represents the tightest possible packing.

We recognize the FCC. Now on to the hexagonal base. Here six spheres in a plane surround the central sphere. In between these seven spheres there remain six holes—or interstices. Spheres are placed into every second interstice to form the next layer. The same is done in the layer below. And so, again, a total of twelve spheres touch the central sphere: six in the plane, three above, and three below. We now have the HCP, the hexagonal close packing. Now Kepler points out something truly remarkable: the two arrangements are equivalent.

The young man’s name was Irenée Du Pont. The rest is history. 44 KEPLER’S CONJECTURE the residence of his father to convey to him the republic’s congratulations on the achievements of his son. After Napoleon came to power, Lagrange was named officer of the Légion d’Honneur and made a Count of the Empire. On April 3, 1813, one week before his death, he was awarded the Ordre Impérial de la Réunion. He is considered the foremost mathematician of the eighteenth century. 5 In 1773, while still at King Frederick’s court, Lagrange wrote a treatise entitled “Recherche d’arithmétique,” which was published in Nouveaux mémoires de l’Académie royale des Sciences et Belles-Lettres de Berlin.

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