As we increase the radius ratio, we can still maintain the touching of the cations and the anions, which is necessary. And what we begin to do is to increase the distance between the anions. And this can continue on until we reach another critical value and that critical value occurs when we go from a coordination number of three to a coordination number of four. And looking at that coordination number of four, what we have is the geometry of tetrahedron and we can easily identify the tetrahedron, with respect to the fact that a tetrahedron in which we have all the faces being the same, so it's a regular tetrahedron, lies inside of a cube. And I've indicated the positions of the anions by the black circle. And the position of the cation as the center position. And what has to happen in order to establish that minimum radius ratio, what we need to begin to do is to look at the dimensions of the face diagonals and the body diagonals. And when we look at the face diagonal, the distance between those two black spheres at the top of the face of the cube, that distance is the distance a times square root of 2 where a represents the length of the unit cell or the unit cube. So that's that dimension. And what's happening is as those anions become larger and larger, they wind up touching right at a critical position along that body diagonal. Now if we look at the positioning with respect to the cations and anions along the body diagonal, as those cations get larger and the anion gets larger, those become closer and closer to one another and what we can then see is that the distance along that body diagonal is related to the edge of the cube by the square root of three. So now we have the two sides that we can use to identify the triangle where the anions touch and where the anions touch the cation. And what we'll then do is to be able to come up with that critical radius ratio. And that critical radius ratio for just touching along the body diagonal of the cube and the faces then represents the distance 0.225 which is the critical ratio with respect to the cation and the anion for a coordination number of 4. So what we'll look at are the different coordination numbers that we can expect to see in the ionic structures. We've talked about two, with respect to a daisy chain of cations and anions, and the relationship between the radius ratio. So that radius ratio will result as long as, or that coordination number will result as long as the radius ratio lies between 0 and 0.155. If that radius ratio increases, then what we do is we go to the equilateral triangles which have a coordination number of 3. And that critical radius where that critical radius ratio occurs when the values become equal to 0.155. And as we continue on with an increase in the radius ratio, we see that ultimately we wind up having the critical radius for a coordination number of 4 as 0.225. And once we get to that value we have the coordination number of 4 which is illustrated on the right. When we go to a coordination number of 6, what we see is we can identify that lower radius ratio as 0.414. And I would leave this up to you to work through a problem that would give you the radius ratio critical for the coordination number of 6. And, likewise, if we look at the coordination number of eight, that minimum radius ratio, or that critical value, is .732 and it goes up to a value of one. And when we reach one then we're at this situation where the radius of the cation and the radius of the anion are equal to one another giving us that value of 1. Thank you.