  Motivated by some conjectures originating in the Physics literature, we use
Foscolo's construction of Ricci-flat Kahler metrics on K3 surfaces to locate,
with high precision, several closed geodesics and compute their index (their
length is also approximately known).
  Interestingly, the construction of these geodesics is related to an open
problem in electrostatics posed by Maxwell in 1873. Our construction is also of
interest to modern Physicists working on (supersymmetric) non-linear sigma
models with target space such a K3 surface.
