Trapped on an Island? No, Islanded on a Trap. We have gained a fundamental understanding of one means to exploit the unusual features of quasicrystal surfaces. Frequently, deposition of material on a surface leads to the self-assembly of nanoscale clusters of atoms, a process called nucleation and growth. The problem is that nucleation and growth of these so-called islands occurs randomly, so control of island size and location is difficult. However, on quasicrystals, the atomic structure forms an array of "traps" at the surface. This is a highly-unusual feature. We have designed and made key measurements, then constructed a model built on the idea of surface traps, to fit the measurements. The model can predict ways to manipulate self-assembly of solids on these unusual surfaces, which in turn may lead to nano-islands, or arrays of nano-islands, with unusual electronic, magnetic, or catalytic properties.