Field ion microscopy combined with video techniques and chemical probing reveals the existence of catalytic oscillatory patterns at the nanoscale. This is the case when a rhodium nanosized crystal--conditioned as a field emitter tip--is exposed to hydrogen and oxygen. Here, we show that these nonequilibrium oscillatory patterns find their origin in the different catalytic properties of all of the nanofacets that are simultaneously exposed at the tip's surface. These results suggest that the underlying surface anisotropy, rather than a standard reaction-diffusion mechanism, plays a major role in determining the self-organizational behavior of multifaceted nanostructured surfaces. Surprisingly, this nanoreactor, composed of the tip crystal and a constant molecular flow of reactants, is large enough for the emergence of regular oscillations from the molecular fluctuations.