Inorganic crystalline materials promise solutions to some of our major global problems. Thermoelectric materials, for example, are expected to increase the energy efficiency of many electric devices or production sites to lower the global need for energy. To arrive at materials with properties relevant for thermoelectric applications, a better understanding of structure-property relationships could play a key role. Linus Pauling’s famous five rules on the stability of crystal structures will serve as a starting point for such an improved understanding because they describe the relation between structure and stability based on coordination polyhedra and their connections. They will be assessed for oxides and other chemistries. This will be based on the Materials Project database and a recent study of the statistics of coordination environments in oxides from the host institute. Next, a graph representation of crystal structures based on coordination polyhedra and their connections will be established to go beyond Pauling’s rules. The graphs will then be analysed and classified. Beyond stability, there is a recent design principle relating coordination polyhedra to lattice thermal conductivity–an essential property for thermoelectrics. It will be applied to search for oxides as potential candidates for thermoelectric applications. The design principle is based on the instability of small cations in octahedral coordination environments that is connected to low thermal conductivity. The recent development of one of the first databases of phonon computations at the host institute offers a great opportunity to link phonon and especially soft modes to coordination environments. The best candidates from the search will be synthesized and analysed by a cooperation partner of the host institute. In sum, the project is expected to lead to a better understanding of coordination polyhedra, their connections, and their relation to the stability and other properties of crystals.