Densest Packing of Identical Circles

Theorem
The densest packing of identical circles in the plane obtains a density of $\dfrac \pi {2 \sqrt 3} = \dfrac \pi {\sqrt {12} }$:
 * $\dfrac \pi {2 \sqrt 3} = 0 \cdotp 90689 \, 9968 \ldots$

This happens when they are packed together in a hexagonal array, with each circle touching $6$ others.