We describe the fabrication of unusual classes of three-dimensional (3D) nanostructures using single step, two-photon exposures of photopolymers through elastomeric phase masks with 5-fold, Penrose quasicrystalline layouts. Confocal imaging, computational studies, and 3D reconstructions reveal the essential aspects of the flow of light through these quasicrystal masks. The resulting nanostructures show interesting features, including quasicrystalline layouts in planes parallel to the sample surfaces, with completely aperiodic variations through their depths, consistent with the optics. Spectroscopic measurements of transmission and reflection provide additional insights.