With the promise of greater safety and adaptability, modular reconfigurable uncrewed air vehicles have been proposed as unique, versatile platforms holding the potential to replace multiple types of monolithic vehicles at once. State-of-the-art rigidly assembled modular vehicles are generally two-dimensional configurations in which the rotors are coplanar and assume the shape of a "flight array". We introduce the Dodecacopter, a new type of modular rotorcraft where all modules take the shape of a regular dodecahedron, allowing the creation of richer sets of configurations beyond flight arrays. In particular, we show how the chosen module design can be used to create three-dimensional and fully actuated configurations. We justify the relevance of these types of configurations in terms of their structural and actuation properties with various performance indicators. Given the broad range of configurations and capabilities that can be achieved with our proposed design, we formulate tractable optimization programs to find optimal configurations given structural and actuation constraints. Finally, a prototype of such a vehicle is presented along with results of performed flights in multiple configurations.