Box orbits in triaxial potentials are generically thin, that is, they lie close in phase space to a resonant orbit satisfying a relation of the form lω_1 +mω_2+nω_3=0 between the three fundamental frequencies. Resonant orbits are confined to a membrane; they play roughly the same role, in three dimensions, that closed orbits play in two. Stable resonant orbits avoid the center of the potential; orbits that are thick enough to pass near the center are typically stochastic. Very near the center, where the gravitational potential is dominated by the black hole, resonant orbits continue to exist, including at least one family whose elongation is parallel to the long axes of the triaxial figure.

20 Latex pages, 11 Postscript figures. Submitted to The Astronomical Journal