  Macroscopic properties of materials stem from fundamental atomic-scale
details, yet for insulators, resolving surface structures remains a challenge.
The basal (0001) plane of ${\alpha}$-Al$_{2}$O$_{3}$ was imaged with noncontact
atomic force microscopy with an atomically-defined tip apex. The surface forms
a complex $({\sqrt31} {\times} {\sqrt31})R{\pm}9{\deg}$ reconstruction. The
lateral positions of the individual O and Al surface atoms come directly from
experiment; how these connect to the underlying crystal bulk was determined
based on computational modeling. Before the restructuring, the surface Al atoms
assume an unfavorable, threefold planar coordination; the reconstruction allows
a rehybridization with subsurface O that leads to a substantial energy gain.
The reconstructed surface remains stoichiometric, Al$_{2}$O$_{3}$.
