 So let's explore this idea a little further of what we mean when we say an object is symmetric. And we'll avoid molecules for now and deal with something that's a little simpler, easier to visualize. And I'll use letters for example. Suppose I were to write a capital letter A and I were to ask you, is this letter symmetric or isn't it? You would probably tell me, yes it is symmetric and if I asked you to back that statement up, you could tell me it's symmetric because it has some left-right symmetry. The left half of the letter is the same as the right half of the letter. If I cut the molecule, cut the letter in half and then pretend that line is a mirror and reflect the letter in the mirror. If I had drawn it more cleanly, we'd see that the left half and the right half are exactly the same. So it has some left-right symmetry, some reflection symmetry. So now let's say I ask you, let me try to draw this one a bit more symmetrically. If I draw a capital letter S and I ask you, is that letter symmetric? You'd likely say, yeah it still has some symmetry and no longer has left-right symmetry. If I cut the molecule in half and exchange the two halves, then if I flip that letter in a mirror, it's not going to look the same. A backwards S doesn't look the same as a forward S. So what is the symmetry that we find in this letter? That's a rotational symmetry. If I, or at least one of the possible symmetries is a rotational symmetry. If I rotate the molecule by 180 degrees in that way, so I rotate it about that point in the middle of the molecule and spin it by 180 degrees, then it looks exactly the same. So now it's not a left-right asymmetry, it's a rotational symmetry. So there's clearly different types of symmetry and we can define a number of operations like reflecting in a mirror or rotating about an axis that help us identify what types of symmetry a letter has or soon enough a molecule has. So let's consider those a few at a time. So the first type of symmetry we discussed was this mirror reflection symmetry. So mirror reflection, so I'll go ahead and give that a name. When we're talking about mirror reflection, the name we give to that usually is a sigma, lowercase Greek letter sigma, with a hat on top of it. So this is an example of a symmetry operator. So sigma hat is just a variable we give to name this behavior. So I'll use a bunch of examples over on this side of the board. If I take the sigma mirror reflection operation and I do it to this letter A, that's just what we were talking about in this example. If I reflect this letter A through this vertical mirror, then the result of that is just the same letter, same image that I started with. So that's why we decided this letter was fairly symmetric because it has this mirror reflection symmetry. So I'll point out that the hat on top of this operator. This operator behaves in much the same way that the operators behaved when we talked about operators in quantum mechanics. It itself is just an operator. I can't say anything about what the mirror reflection itself is. It acts upon something else. So mirror reflection acting on the letter A gives me one result. The mirror reflection acting on a letter S would give me a backwards letter S. So the operators act on something else and this hat is here for the same reason it was in quantum mechanics to remind us that it's not an object itself. It's an operator that acts on something else. So this mirror reflection, what it does is it reflects the object through some planes, through some flat surface. And we have to specify, if the context doesn't make it clear, exactly which plane we're reflecting through. In this case, we've been talking about this vertical plane. I could say that the reflection through a vertical mirror plane gives me this letter back the same way it started, but I can also talk about other planes. If I talk about reflecting this letter A through a horizontal plane, so this would be my horizontal reflection plane. If I reflect the molecule through that plane instead, not the molecule, the letter A, then what I'll get is an upside down letter A as a result of doing that reflection. So we'll get different results if we reflect through different planes so it's necessary to make sure it's clear which reflection we're talking about. The other symmetry operation we've described so far is this rotation operation. So I'll describe that as a higher one on this list. I want this list to appear in a particular order. So there's a rotation operation that we can give a symbol to. That rotation we call C and we'll rotate about some axis. Notice though that I had to specify not just that we're rotating this molecule about some axis, the axis through the center of the letter, but I had to tell you rotate it by 180 degrees. We could also have rotated it by 90 degrees or 10 degrees or some different amount. So we'll identify the different types of rotations with the subscript. If I want to rotate a letter or a molecule by 180 degrees, I call that a C2 rotation because it takes me two of those rotations to finish a full circle. A 90 degree rotation would be a C4 rotation because it would take four of them to get around a full circle. So a CN rotation would rotate by an amount 2 pi over N about some axis that I specify. And just like the mirror reflection, I have to specify what the axis is. If I use S as my example, if I do a C2 rotation of the letter S through, let's say, well, the axis we've already talked about through that axis that points out of the board through the center of the molecule, then as we've seen I get back this same letter S. If I were to do that C2 rotation around, let's say, so that would be another way of drawing that axis would be, I could say, rotate around that axis. If I instead rotate my letter S around, let's say, this axis. So rotate with a C2 around that axis, then that's going to flip the molecule by 180 degrees around this axis and I would get something that looks like an upside down or a backwards S. Upside down in this particular case because I turned it upside down is literally the rotation that I've done. So I have to specify not just the amount but also the axis about which I'm rotating the molecule or the letter. So there are other different types of symmetry operations besides just reflection and rotation that we can define. In fact, let me, as one of my examples, let me remind you, we talked briefly about rotating by amounts other than 180 degrees. If I do a C4 rotation, that's rotating by one-fourth of a full circle, one-fourth of 2 pi or just 90 degrees. So if I do, let's say, a C4 rotation on the letter W through an axis down the center of the molecule, the center of the letter pointing out of the board, then if I do that rotation clockwise from my point of view, then it's going to end up looking like that. So there's a 90-degree rotation of a letter W. So let's point out, so if I were to do that C4 rotation again, it would be upside down again and again. I'd return it back to where it started. Likewise, for these C2 rotations, if I rotate the molecule by 180 degrees, I flipped it upside down. If I rotate it by another 180 degrees, it's back to right side up again. So if I do a C2 rotation twice, that's the same as doing a C4 rotation four times, going full circle and getting the letter or the object back to where it started. That would be the same as doing any reflection operation twice. So all these symmetry operations, if I keep doing them over and over, I'll get back to where I started. We give a name, actually, to this operation of doing nothing, taking a molecule or an object from its configuration to the exact same configuration. And that's called the identity operation, and we give the letter E to that operation. So that's a nice simple operator. The E operator, the identity operator, does nothing to an object. So if I operate with the identity operator on a letter, I always get back the same letter. If I operate with the identity operator on a molecule, I always get back the same molecule in the exact same orientation. So that seems like a relatively boring operator. It is a trivial operator, but there's an important reason to include it in the list along with all the rest of these. And we'll get to that soon enough when we start talking about this full collection of symmetry operators. There's two more symmetry operators I want to discuss. One of them, the next one, is the inverse operator, or the inversion operator, I suppose I should call it. And we'll call that one lowercase i for an inversion, and i lowercase i with a hat on it. And the inversion operator does what the name says, it inverts the molecule, or object or letter. So we find the center of the object and flip all the points of the object through the center. So for example, let me take the inversion operator acting on the letter F. So the center of the letter is whatever where the center of the letter is. So the parts of the letter that are in the top left, I flip them through the center out the other side, and they end up in the bottom right. The top right flips through and becomes the bottom left, vice versa. So the molecule just collapses on itself and turns inside out and comes out the other side. So what I get if I do that to the letter F, this top right, sorry, top left corner ends up in the bottom right. The bottom becomes the top, and simultaneously the left becomes the right. So that's what the inverted letter F looks like. Top left became bottom right, top right became bottom left, bottom left became top right, and so on. So that can be difficult to visualize, especially once we start talking about these objects in three dimensions, rather than just the flat two dimensions of a blackboard. But it's a relatively simple operation to describe. The last one that we'll discuss is another flavor of rotation, one that we call an improper rotation. So rather than a rotation or a proper rotation that we give the label C, an improper rotation we give the label S. And it's a rotation just like a proper rotation is. So the N tells us by how much we're rotating. So this operation says we rotate by an amount 2 pi over N, just like with a proper rotation. And also we reflect that object through a reflection plane that's perpendicular to the rotation axis. So as an example of that, let's do an S2 rotation on, let's see, we've already used S. Let me, so this is a numeral 2, let me do an S2 rotation, improper rotation on the capital letter Z. So, and in particular, we'll make our axis, we'll make this the S2 axis. So we're going to do a rotation around this axis, and then we're going to reflect through a horizontal plane that bisects the molecule perpendicular to that axis. So there's two steps. The first step, I need to rotate that molecule so that Z, after I rotate it by 180 degrees, becomes a backwards looking Z, which I have a hard time drawing. So there's my rotated Z, and then I reflect that molecule in my reflection plane. So this is like a C2 rotation. The reflection plane, if I reflect it through this plane that bisects the letter in the middle, top becomes bottom, bottom becomes top without any rotation. And so this corner reflects down to the bottom, this corner reflects up to the top, and my Z looks like that. So this is the end result of doing the S2 rotation, which is a combination of doing a C2 rotation followed by a perpendicular reflection. So in this case, notice that the S2 turned the molecule back into itself. Neither the C2 rotation of the molecule, the letter, neither the C2 rotation of this letter, nor a reflection of this letter through a horizontal plane leaves the molecule unchanged, but this composite operation of doing a rotation and a reflection at the same time does leave the letter unchanged. So that's the main reason we need to be able to talk about these improper rotations is there are molecules for which, as we will see, if we do a rotation and a reflection at the same time, the molecule is unchanged even though it doesn't have a corresponding C-axis or a mirror reflection plane. So there are the five important symmetry operators, operations that we need to know about as chemists. We could certainly define more. You can invent a symmetry operation that does whatever you want to a molecule or to an object, but the symmetry operations that are important for molecules, as we'll see, are the five that we've listed here. So the next step will be to go back to this idea of recognizing whether something is or isn't symmetric or how symmetric it is or isn't based on which of these symmetry operations leave the object unchanged, like a reflection left the A unchanged and a rotation left the S unchanged. So that'll introduce the idea of symmetry elements, which is the next topic.