What is the Kepler Conjecture?

In a booklet published in 1611, J. Kepler described the arrangement of equal spheres into the family cannonball arrangement. He asserted that

The packing will be the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container.

This assertion has come to be known as the Kepler conjecture. It has gone centuries without rigorous proof. At the international congress of mathematicians in 1900, Hilbert proposed a list of problems to the international community. He included the Kepler conjecture in the 18th problem.

How can one arrange most densely in space an infinite number of equal solids of given form, e.g., spheres with given radii..., that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as great as possible? (Hilbert's Problem 18, part C, 1900).

There are actually infinitely many ways to pack spheres with the same density as the face-centered cubic packing. The face-centered cubic is obtained by stacking triangle plates of spheres upon each other. The plates can be shifted to produce different packings without affecting the density.

Various mathematicians have established upper bounds on the density of packings in three dimensions.

• 0.828 (Rankin, 1947)
• 0.7797 (Rogers, 1958)
• 0.77844 (Lindsey, 1987)
• 0.77836 (Muder, 1988)
• 0.7731 (Muder, 1993)