In this lesson, we're going to continue our discussion of two dimensional symmetry. But, what we're going to do is begin to add the concept of a Lattice and a Basis which ultimately develops into a Crystal structure if we look at the array of characters that are on the figure, I've represented those as S's as turned on their side. And if you look is a square array of those vharacters distributed in space. And what we're going to do is we're going to describe what we mean by the concept first of all of the Lattice and so all we would need to do is to place a Lattice at each one of those points and we would create a space filled structure. That describes the periodicity of the structure that we're working with. So if we look at the squiggle you can see that it is translated in the A and B directions and it results in a periodic structure. And now what we're going to say is that the Lattice. Which is the square representation along with the basis, which is what is included in that red circle when we distribute that in two dimensions along the A and B directions, we're going to come up with a two dimensional crystal structure. If we look at the characters that I have indicated in this figure, we have a basis which includes two characters, one that's black and one that's blue. And what we'll do is we'll take those two figures and we will translate them in the A and B directions and what we will wind up doing is taking a Lattice, which is a square unit. And we'll take the Basis and add that to the square unit and wind up producing what we refer to as a Crystal structure. Now, we can come up with an alternative description using a different basis. Where we have two that are related to one another. But they have an additional symmetry associated with them. So we look again at the basis. And we take the basis and add it to the lattice, and we then produce the crystal structure, which is the array associated with these two bases that are distributed in space. We begin with the basis, which is in this case a periodic arrangement of circles. And what we're going to do is take that basis and superimpose a Lattice. And here's that square Lattice. And when we put them together what we do is produce the Crystal structure. Now, it turns out there are variety of ways that we can couple the Basis and the Lattice together. So, for example, what I've done is I have placed the Lattice directly over the corner positions, the center positions of all of those circles. So, that way, I have incorporated both the basis and the lattice, and they superimpose directly on top of one another. An alternative description would be that what I could easily do, is put the basis inside the center of the square Lattice points. That are indicated by that array. And now we have an alternative description of the Basis and the Lattice. And sometimes for convenience, you might want to use one distribution of the Lattice with respect to the basis or another depending upon the convenience. So what we have done then in this lesson is to look at the terminology that crystallographers use, that is describing what we mean by the basis and what we mean by the Lattice. Remember the Lattice is an abstract representation. So we'll continue on this type of discussion as we move from two dimensions into three dimensions in the next lesson. Thank you.