In recent years, Anisotropic Power Diagrams, a geometric structure generalizing the well-known concept of Voronoi diagrams, are increasingly used to represent grain shapes of polycrystalline materials at the microstructural scale. Anisotropic Power Diagrams can be described by a small number of parameters, and, in principle, only these parameters need to be recovered from measurements to obtain an accurate representation. Experimentalists have reported, however, that some parameters seem more important than others for describing certain families of materials. In this talk, I will discuss some of these findings, discuss their relevance and, based on recent computational results, offer some theoretical explanation. REFERENCES [1] A. Alpers, A. Brieden, P. Gritzmann, A. Lyckegaard, and H.F. Poulsen, Generalized power diagrams for 3D representations of polycrystals. Philosophical Magazine, 95(9):1016–1028, 2015. [2] A. Alpers, M. Fiedler, P. Gritzmann, and F. Klemm, Turning grain maps into diagrams. SIAM Journal on Imaging Sciences, 16(1):223–249, 2023. [3] A. Alpers, M. Fiedler, P. Gritzmann, and F. Klemm, Dynamic grain models via fast heuristics for diagram representations. Philosophical Magazine, 103(10):948–968, 2023.