Groupoid Analysis of Pyroxenes

The structure and space-group symmetry of modular structures can be obtained from the knowledge of the structure and subperiodic symmetry of the modules and of the structure-building operations. This leads to a space groupoid, which contains all the partial operations relating any pairs of modules. Those partial operations that possess a continuation in the whole crystal space are global operations that show up in the space group of the structure. The whole procedure is presented through the example of pyroxenes. The modular structure of these minerals has already been analyzed in the past, but the groupoid construction has only been synthetically described. In this paper, the full step-by-step process is presented, which is largely unknown in the crystallographic community.