Recently it was experimentally demonstrated that sputtering under normal incidence leads to the formation of spatially ordered uniform nanoscale islands or holes. Here we show that these nanostructures have inherently nonlinear origin, first appearing when the nonlinear terms start to dominate the surface dynamics. Depending on the sign of the nonlinear terms, determined by the shape of the collision cascade, the surface can develop regular islands or holes with identical dynamical features, and while the size of these nanostructures is independent of flux and temperature, it can be modified by tuning the ion energy.PACS numbers:68.55. -a,68.65.+g,05.45.-a The fabrication and physical properties of quantum dots (QDs) are topics of high current interest due to their applications in optical devices and as potential construction blocks of novel computer architectures [1]. However, despite the high interest in the subject, the methods available for the fabrication of such dots are rather limited. Lithographic techniques, while undergoing rapid improvements in resolution [2], still cannot produce small and dense enough dots necessary for device applications. While much research has focused on self-assembled QD formation [3], that generates dots through the unique combination of strain and growth kinetics, these techniques offered relatively uniform islands only for a few material combinations and still have to find their way into devices. Consequently, there is continued high demand for alternative methods that would allow low cost and efficient mass fabrication of QDs. In the light of these technological driving forces, the recent demonstration that sputter erosion can lead to uniform nanoscale islands, that exhibit quantum confinement, will undoubtly capture the interest of the scientific community [4]. Ion beam sputtering has long been a leading candidate for surface patterning. While ripple formation on sputter eroded surfaces has been observed already in the 70s [5], in the last decade much work has been devoted to understand both the experimental and theoretical aspects of this fascinating self-organized phenomena. However, most experiments have focused on off-normal incidence, that, by breaking the symmetry along the surface, leads to anisotropic structures, such as ripples. While theoretically it was expected that under normal incidence the ripples should be replaced by some periodic cellular structures, such surface features have not been observed experimentally. Recently, two groups have obtained simultaneous advances in this direction. Facsko et al., investigating low-energy normal incident Ar + sputtering of GaSb (100) surfaces [4], observed that as erosion proceeds, nanoscale islands appear on the surface, that are remarkably well ordered, and have a uniform size distribution. On the other hand, recent experiments of Ar + sputtering of Cu(110), and Ne + sputtering of Ag (001) under normal incidence lead to relatively uniform depressions or holes [6]. These experiments, while represent significant tech...