I have looked at the cation anion ratio of cubic, octahedral and tetrahedral arrangments on an internet site. By a mathematical derivation they find the minimum value for the cation anion ratios for cubic, octahedral and tetrahedral arrangments. My problem is that even though I get the mathematics I dont get why the derivations show that there is an interval of values that the cation anion ratios for the cubic, octahedral and tetrahedral arrangments can have. I only get that they can have the values that they derive which is cubic: 0.732 octahedral: 0.414 tetrahedral: 0.225 I need a mathematical justification on why it is an interval of values for the arrangments that in fact are cubic: 0.732 to 0.999 octahedral: 0.414 to 0.732 tetrahedral: 0.225 to 0.414 I have uploaded a derivation online that says mathematically how the minimum values are found. In addition the question about how the intervals are found are posed there as well The answer is not mathematical. Can anyone help me in finding a mathematical answer to the intervals? The intervals are also in the attachment from an inorganic chemistry book.