Nanometer-sized polyhedral wire-frame objects hold a wide range of potential applications both as structural scaffolds as well as a basis for synthetic nanocontainers. The utilization of DNA as basic building blocks for such structures allows the exploitation of bottom-up self-assembly in order to achieve molecular programmability through the pairing of complementary bases. In this work, we report on a hollow but rigid tetrahedron framework of 75 nm strut length constructed with the DNA origami method. Flexible hinges at each of their four joints provide a means for structural variability of the object. Through the opening of gaps along the struts, four variants can be created as confirmed by both gel electrophoresis and direct imaging techniques. The intrinsic site addressability provided by this technique allows the unique targeted attachment of dye and/or linker molecules at any point on the structure's surface, which we prove through the superresolution fluorescence microscopy technique DNA PAINT.