  This paper applies techniques from algebraic and differential geometry to
determine how to best pack points in real projective spaces. We present a
computer-assisted proof of the optimality of a particular 6-packing in
$\mathbb{R}\mathbf{P}^3$, we introduce a linear-time constant-factor
approximation algorithm for packing in the so-called Gerzon range, and we
provide local optimality certificates for two infinite families of packings.
Finally, we present perfected versions of various putatively optimal packings
from Sloane's online database, along with a handful of infinite families they
suggest, and we prove that these packings enjoy a certain weak notion of
optimality.
