Kepler's Conjecture

Theorem
The densest packing of identical spheres in space is obtained when the spheres are arranged with their centers at the points of a face-centered cubic lattice.

This obtains a density of $\dfrac \pi {3 \sqrt 2} = \dfrac \pi {\sqrt {18} }$:
 * $\dfrac \pi {\sqrt {18} } = 0 \cdotp 74048 \ldots$