The concept of a new piezoelectric actuator design is the focus of an ongoing smart structures research at DLR. It envisages actuator geometries, that have their template in nature and are aimed at optimally transfering their mechanical power to the structure in which they are integrated. The second aspect can only be achieved in case of an impedance matching. We found out, that these very specific designs have highly interesting differential geometry descriptions and that these geometries can be realized in a congenial combined manufacturing procedure: we start with 3D-printed filigree lightweight structures and continue with self-organizing processes. The results are simply, doubly and triply periodic piezoelectric membranes, grids and truss elements. One of their essential properties is their almost perfect stress distribution under mechanical loads. Any notch stresses are prevented. In this paper the first example is the modified catenoid piezoactuator. We continue with a doubly periodic honeycomb structure and finish with triply periodic modified gyroid and diamond-like actuators.