The honeycomb lattice is a paradigmatic model for the study of topology in condensed-matter systems. The salient features in its band structure are two Dirac cones, each analogous to an infinitely narrow solenoid producing a half-quantum of magnetic flux. We have directly measured the Berry flux of the Dirac cones in an optical honeycomb lattice by Aharonov-Bohm interferometry with a Bose-Einstein condensate. Our broadly applicable method enables precise determination of topological invariants (Chern numbers) and provides high momentum resolution in mapping out geometric phases throughout a single band. I will touch on extensions to characterizing multi-band systems; and on prospects for engineering band structures conducive to forming topologically ordered states of interacting bosons.