the radius ratio

The radius ratio is just the cation's radius divided by the anion's radius, r_cation / r_anion, a single number between 0 and 1 that captures how big the little marble is compared with the big one. It is the ionic model's cleverest trick: from this one ratio you can often predict, with nothing but geometry, how many anions will fit around the cation and therefore what kind of crystal it builds.

Picture the cation nestled among touching anions. If the cation is too small for the cage, the anions clatter against each other and the arrangement is unstable, so the cation drops to a smaller cage with fewer neighbours; if it is large enough to hold the anions apart, a bigger cage is stable. Running the geometry gives clean thresholds: a ratio from 0.225 to 0.414 gives four-fold (tetrahedral) coordination, 0.414 to 0.732 gives six-fold (octahedral), and above 0.732 gives eight-fold (cubic). Take MgO: Mg2+ is 0.072 nm and O2- is 0.140 nm, ratio 0.51, squarely in the octahedral band, and indeed each magnesium has six oxygen neighbours.

The radius ratio is a beautiful first guess, and for strongly ionic oxides it works impressively. But it is only a guide: it assumes hard, spherical, purely ionic ions, so it fails when bonds are covalent (silicon should be triangular by some radius sets yet is always tetrahedral) or when ions are very polarizable. Counting successes across simple compounds, it predicts the right coordination roughly two times in three, brilliant for a rule this simple, but not a law.