  In this work we initiate the mathematical study of naked singularities for
the Einstein vacuum equations in $3+1$ dimensions by constructing solutions
which correspond to the exterior region of a naked singularity. A key element
is our introduction of a new type of self-similarity for the Einstein vacuum
equations. Connected to this is a new geometric twisting phenomenon which plays
the leading role in singularity formation.
  Prior to this work, the only known examples of naked singularities were the
solutions constructed by Christodoulou for the spherically symmetric
Einstein-scalar-field system, as well as other solutions explored numerically
for either the spherically symmetric Einstein equations coupled to suitable
matter models or for the Einstein equations in higher dimensions.
