  Significant advances in numerical simulations of black-hole binaries have
recently been achieved using the puncture method. We examine how and why this
method works by evolving a single black hole. The coordinate singularity and
hence the geometry at the puncture are found to change during evolution, from
representing an asymptotically flat end to being a cylinder. We construct an
analytic solution for the stationary state of a black hole in spherical
symmetry that matches the numerical result and demonstrates that the evolution
is not dominated by artefacts at the puncture but indeed finds the analytical
result.
