The rule of four: anomalous stoichiometries of inorganic compounds

Why are materials with specific characteristics more abundant than others? This is a fundamental question in materials science and one that is traditionally difficult to tackle, given the vastness of compositional and configurational space. We highlight here the anomalous abundance of inorganic compounds whose primitive unit cell contains a number of atoms that is a multiple of four. This occurrence - named the 'Rule of Four' - has to our knowledge not previously been reported or studied rigorously. Here, we first highlight the rule's existence, especially notable when restricting oneself to experimentally known compounds, and explore its possible relationship with established descriptors of crystal structures, from symmetries to energies. We then investigate this relative abundance by looking at structural descriptors, both of global (packing configurations) and local (the smooth overlap of atomic positions) nature. Contrary to intuition, this overabundance does not correlate with low-energy or high-symmetry structures; in fact, structures which obey the 'Rule of Four' are characterized by low symmetries and loosely packed arrangements maximizing the free volume. We are able to correlate this abundance with local structural symmetries, and visualize the results using a hybrid supervised-unsupervised machine learning method.