 So the way we describe the symmetry of an object in complete detail is to describe the various different symmetry operations that leave it completely unchanged or unmodified. So over here I've got a reminder of the different symmetry operators that are important that we've talked about. So let's work a few examples of seeing if we can identify all of the operations that leave something unchanged. So sticking with capital letters for now, let's take this example of a capital letter Q. So that's not a particularly symmetric looking letter and we can ask ourselves what are the symmetry, better define what symmetry elements are before I ask you for them, what are the operations that leave this element, this letter unchanged? So if we just go through this list and ask ourselves if I operate with this operator on this element, does it leave it modified or unmodified? If I do the identity operator on Q, then I get back the same letter Q and it's not any different than it was. So certainly E is an operation that leaves the letter Q unchanged. How about rotations? Are there any rotations of this molecule, sorry, this letter that I can do that leave it unchanged? If I imagine a vertical axis, if I spin this letter around by any amount around this vertical axis, I can spin it by 180 degrees but then the tail ends up on the other side. That's no good. If I spin it around a horizontal axis, the tail will end up somewhere, if I spin it by 180, for example, the tail will end up on the top. So all those rotations are changing the molecule. If I spin it around this axis, down the center of the Q, then the tail spins around and it's not until I do a full rotation that I get back to the same Q that I had originally. There is, if I look in the right way, I can find a C2 axis that's not one of the Cartesian XYZ axes. If I point directly along the Q, if I spin the letter by 180 degrees around that axis, then it leaves the letter unmodified. So essentially this curve of the Q becomes this curve of the Q after I spin it. So there is a single C2 rotation axis. What else do we have? Are there any reflection planes? Again we can consider, if I make a plane bisecting the molecule in this way, it's not a symmetry element. It doesn't leave the molecule unmodified because, again, the tail flips through the mirror to the other side. Likewise, the horizontal planes are no good. There is a plane in the plane of the board. So if I imagine this Q as a three-dimensional object, if I bisect it in this way, essentially the plane of this glass board, then if I flip the molecule through that mirror, then it remains unchanged. So there is a sigma axis in the plane of the board. There's also a sigma plane that contains the C2 axis. If I have a plane perpendicular to the board that contains the C2 axis, the top right and the bottom left are mirror images of one another. So I actually have two different reflection planes that could leave this Q unmodified. If I move on to inversion, so the center of the letter is right there. If I look for things that I can rotate the molecule and then invert it, sorry, not invert it. I move the molecule and then reflect it through a bisecting plane. I don't find any of those other than this original C2 axis. If I rotate around this C2 and then reflect, no, even that one is no good. So there are no S, there's no improper rotations for this letter. So there's no inversions, no improper rotations. So this list, identity, a C2 and two sigmas, that would be the list of what we call symmetry elements for this letter, capital letter Q, at least in the way I've drawn it. So symmetry elements, what we mean by symmetry elements is the set, the list of all the symmetry operations that leave an object unmodified or unchanged. And the convention is, since we're talking about a set, a list of objects, the elements of the set or the symmetry elements, because they're elements of a set rather than operations that I'm intending to apply to something else, we list those elements of the set without the hats on top. So that just helps us distinguish often between whether we're talking about an operator, the identity operator, or this symmetry element as an element of a set that's describing this particular object. So we have, I'll do one more example, none of the examples we've worked so far have showed a very good example of this improper rotation. So as an example, actually no, even this next example I will do won't have a good example of an improper rotation. We don't have any capital letters that provide terribly good examples of that. But let's work one more example just so we get used to this idea of applying these symmetry operators to objects. So let's consider a capital letter H. And again, I've drawn this in a particular way. For me, the cross bar of the H is halfway up the H. So if I try to list briefly the symmetry elements of the capital letter H, again, just go down this list and ask yourself, are there any of these objects that leave the letter unchanged? Certainly the identity leaves the H unchanged. If I do nothing to the H, it's the same as it was before. Identity is always going to be one of the symmetry elements for every object you consider. C2s, are there any C2 or in fact C3s or C4s or any other rotations? I can find a C2 axis right there. If I spin the molecule around that axis, spin the letter around that axis, it looks the same. There's another C2 axis right there. If I spin the molecule, the letter around that axis, top becomes bottom and it looks the same. I've also got a C2 axis straight down the center of the molecule pointing out of the screen towards you. If I rotate the molecule by 180 degrees around that axis, it remains unchanged. So there's actually three different C2 axes that I can find for that letter H. How about reflection planes? Are there any reflection planes? Those are somewhat similar for this case. I can bisect the molecule letter left to right. The left half is equal to the right half. If I reflect through that mirror plane, it's the same as before the reflection. I can bisect the letter top bottom. Reflecting it top to bottom gives the same as it was before. And I can also reflect it in the plane of the board. If I cut the molecule in half this way, the letter in half this way, the forward and the backward halves are identical. So there's three different reflection planes that I can find for that letter. Inversions, there is in fact an inversion center. The center of the molecule where these two reflection planes cross. If I invert the letter through that inversion center, then as usual top left becomes bottom right and so on. And everywhere there's a point in the molecule when I invert it through the center, it lands on another point of the existing letter. So there is an inversion center that can only ever be one inversion center because there's only one center of the molecule itself. And now the point about improper rotations. You might be tempted to say that there's an improper rotation for this molecule, but in fact we don't allow there to be an improper rotation for this molecule. This axis that we've identified as a C2 axis. If I were to rotate this molecule on the C2 axis, it looks the same as it did before. If I then reflect it through the perpendicular mirror plane top to bottom, it will look the same as it did before. That's the definition of an improper rotation. Rotate it and then reflect. That S2 that I've just described has left the capital letter H unchanged, but we don't include the S2 as a symmetry element because in fact both the C2 itself and the reflection plane already exist as symmetry elements of the molecule. So we only include the improper rotations when the constituent actions, the C2 or the C whatever and the reflection plane are not themselves already symmetry elements. So we'll see a better example of that when we consider these actions applied to real molecules. And in fact that's what we'll do next. We're not really interested as chemists in the symmetry elements of letters. In fact, the fact that they are embedded in the plane of this board actually tends to make some features of this a little more confusing. We're much more interested in the symmetry properties of real molecules. So that's the next step is to see if we can identify symmetry elements for molecules.