Okay, let's look at my favorite piece of geometry, this tetrahedral symmetry here. Tetrahedral geometry, we see it in methane, if that was carbon, four hydrogens. We see it in some polymer molecules like polyethylene, although the bond rotation threw things off a little bit. But you've still got four covalent bonds there. We see it in some ceramic structures, we can cover that later. We see it in this beautiful structure here, biggest piece of thyme and dion. This is, we would call it diamond cubic. But I certainly struggle to see what's cubic about this, right? This doesn't look cubic. I can sort of see that there's this tetrahedral unit that repeats with shared carbons, and that is true. But why do we call it diamond cubic? And the reason is because our human brains can at least mine, can digest that easier. I see this tetrahedral site here, and I struggle to make sense of how am I going to figure out how big that center is if I wanted to put a cation in amongst anions, or if I wanted to figure out that bond angle from the center out to the other ones. It's a little bit, I mean I can see the 60 degrees here, but beyond that it's a little freaky. The cube is a lot easier. You know I've got eight corners, six faces. Hey, that's all right. 90 degrees? Yeah, I like that. I can kind of work with that. So what I'd like to do is look at a cube and show how we can fit a tetrahedral interstitial site in there. That's a wonky cube. Well, it'll do. It'll do. Clean up those ones a little bit and draw in the hidden edges for you there. Really quick. Doesn't need to be perfect. But you can hopefully see I've tried to draw a cube there. I apologize it's not my best. Now what I'm going to do is I'm going to position atoms. I'll make them blue and I'll put two of them in the bottom. On the bottom face there, then they're kind of in these positions. And if you jump to the top, they're going to alternate. So they're kind of spaced out from one another as far as they can. And so I'm going to put two more up here. And those are the atoms that would form the tetrahedral site. So on the bottom I can have these two. On the top I can have these two, the top corners and the bottom corners. And then you might wonder where does the interstitial site or the central atom go. And it would go right in the very center of that cube that we've drawn. And the direction of contact, of course, between the central atom and the ones on the corners would be along these cube diagonals, just like this, like I've drawn in. And so that would be the case for dimacubic or crystalline silicon like this. If they were all a single atom, I've drawn the different. But the positions would be exactly the same. It would also be the same if it was organic like this, or if it was a cation occupying interstitial site amongst anions. And so what we want to do is we want to look at this. We'd like to figure out this bond angle here. Bond angle. Okay. And so that is this angle right here. That one. It makes that sound too. And so to do that, what we would want to do is draw a line straight down from that central one until it hits the bottom face. And then we're going to connect it like this and make a right triangle. Okay. So we've dropped this line down straight from the center. And that line would continue across. It's a face diagonal along the bottom. And if we actually continued the cube diagonal out, the one that goes from this front bottom left position to the central position here. Okay. That orange line would continue out all the way over to here. So if I said, hey, let's make this cube edge length B. Okay. This would of course also be Bb. And this would be Bb as well. And so what could we do? Well, we would say, well, in that case, this line traversing the bottom face. If the edges were B, this would be root 2B, wouldn't it? Because you got B squared plus B squared is equal to that line all squared. Take the square root you got root 2B, give you the length of that line. And so that's really nice because now we can work out an angle. And to keep it simple, I'm going to define this angle right here, phi. Okay. So let's redraw that triangle over here to keep things neat. And hopefully you can see where I got it from, at least. I use the same colors. So we keep track of what's going on. So what we'd like to do is we'd like to determine what phi is. And what we said, you know, that height was B. So if it's going halfway, say that's, if that's 1, this is root 2 times 1. Okay. And that ratio stays the same. And this angle here would be, would be phi as well. So we've got that. And that's almost all we need, right? We could say, well, with that knowledge of those edges, we could say, well, tan phi equals root 2 over 1. And so phi, the angle then, is arc tan or tan inverse of root 2, which equals 2, square root, arc tan. So that equals 54.7 degrees. And our bond angle that we're trying to determine is equal, it would be twice phi, right? That's what we said, continues across there. It's 2 phi. And so 2 phi is 109.5, 109.5 degrees. So there we go. We took that beautiful little tetrahedral geometry, put it into a arguably more beautiful cube, or at least not beautiful, but easier to understand. And we were able to work out really quickly that classic 109.5 bond angle. And we can take that same approach of putting the tetrahedral site into a cube to look at the size of the interstitial site. We can even apply it to hexagonal close packed and look at the sea of array ratio, which we'll do in another video. Okay, thanks a lot.