 A symmetry of a geometric figure is a rearrangement of the figure that preserves the arrangements of its sides and vertices as well as distances and angles. Basically, a symmetry is just a permutation of some geometric object, maybe like a polygon or polyhedron or something, that really, you move the vertices around in such a way that it still retains the original shape it had. So for example, the set of all symmetries of the regular in-gone, so the in-gone just means it's a polygon with insides, regular means that every side has the same length and hence also every angle has the same measure. The symmetries of the regular in-gone we will denote as the set dn and this is called the dihedral group. The sets of symmetries will actually form a group structure. Let's explain why that is for a second. Well, I claim that d4, the symmetries of the square, the regular foregone, will have eight different symmetries. Now, again, symmetries themselves are just permutations of the vertices of our in-gone here. And so we know that permutation multiplication, which is really just function composition, it's a social and there is an identity and that's the first of the eight symmetries, the identity permutation. A goes to A, B goes to B, D goes to D and C goes to D. Just so you're aware how to read this right here, I'm going to have on the left hand side the original square and for convenience I've labeled the vertices of the square A, B, C, D oriented in a clockwise motion here. And then over here will be the image of set square so you can see what the permutation is, what the symmetry is. So the identity symmetry doesn't move anything. So A goes to A, B goes to B, D goes to D and C goes to C. So you get the identity that is the first of these eight symmetries. Great. In order to be a group, we do have to have inverses and we do have to make also sure that we have a binary operation that the product of two symmetries is in fact a symmetry. This isn't too hard to see, but we'll be more specific about that in a moment. The second symmetry of the square, we will call it row one for right now. Row one just corresponds to a rotation counterclockwise by 90 degrees. So A is going to go to D, D is going to go to C, C is going to go to B and B is going to go to A as I mentioned here. If we write this in a permutation notation, A goes to D like we see right here. So A is going to rotate to D, B is going to go to A as you can see illustrated already on the screen. We get that C is going to go to B when we rotate it as illustrated right here and then D goes to C. Also, we see this in the illustration. So we get this permutation right here, we'll call that row one. The next one we'll call it row two. Row two is actually constructed by rotating the square counterclockwise by 90 degrees twice. That is, if row one is a 90 degree rotation, row two is 180 degree rotation. And so in this situation, A is going to go to C and then C is going to go back to A. So A goes to C, C goes to A just like illustrated right there. Also, D is going to go to B and B is going to go to D. So B goes to D and D goes to B in this rotation. So row two is just 180 degrees counterclockwise. Row three is going to be the matrix where we actually are going to rotate three times by 100, but we're going to rotate three times by 90 degrees, which when you compose those all together, that's a counterclockwise rotation of 270 degrees. So notice what happens here. A goes to B, D goes to A, C goes to D, and then this is getting super messy here, and B goes to C as illustrated in my permutation tableau. A goes to B, B goes to C, C goes to D, and D goes to A. Notice if I were to rotate another time by 90 degrees, that would just be a 360 degree rotation, which is none other than just, of course, the identity we had before. And so these right here, these four permutations, the identity, row one, row two, row three, are just the rotational symmetries. If you rotate by 90 degrees any number of times, you're going to get back the original square. Something I do want to mention here is that row two right here, this is just rotating, whoops, that's a P, if you rotate row one twice, that will give you 180 degrees. If you rotate row one three times, that's the same thing as row three. That's the numbering scheme we're using right here. I also want to mention that when we rotated 200 or 270 degrees counterclockwise, that really is just the same thing as rotating 90 degrees clockwise, which is the inverse operation of row one. So I want to point out here that row three is actually the same thing as row one inverse, and also rotating 90 degrees counterclockwise is the same thing as rotating 270 degrees clockwise. So row one is the inverse of row three. Let me write that correctly. And then also rotating 180 degrees counterclockwise is actually the same thing as rotating 180 degrees clockwise. So row two is its own inverse. So when we look at, and of course the identity is its own inverse as well. So when we look at the four rotational symmetries of the square, we can see that they're inverses of each other, and any product of rotations will also be a rotation, like composing two 90 degrees and rotations is 180. If we do a 90 degree and a 180 that's a 270, if any other combinations you can see, those are going to be rotations again. So those symmetries seem to be taken care of okay. The square also has four reflective symmetries, and suddenly I explain what's going on here. If we take the diagonal line that goes from A to C, that is it's the northwest to the southeast corner of the square. If we reflect across that diagonal, well the number, the points A and C won't get rotated, or they won't get reflective along the line since they're on the line themselves, but B and D will swap locations. So A goes to A, C goes to C because they're on the reflective line, but B and D will swap locations. This is an example of a reflective symmetry which we'll call this one sigma one. Another reflective symmetry is if you take the horizontal line, you can think of like if our square was centered in the origin on the plane, then this is just the x-axis right here. If we reflect across the x-axis, A and D will swap locations and C and B will swap. So A goes to D, D goes to A, B goes to C, C goes to B. This is another example of a rotation, not a rotation reflection excuse me, switching over to the next slide. We also get a diagonal reflection if we go from D to B. This is the southwest to northeast corner. In this situation B and D will stay fixed because they're on the reflective line, but then A and C will swap locations. So A goes to C, C goes to A, B is fixed and D is fixed. And then the last reflection we can see here is if you take the y-axis that is a vertical axis of reflection. In this case A and B will swap locations and then C and D will swap locations as well. So A goes to B, B goes to A, C goes to D and D goes to C. We're going to call these ones sigma 3 and sigma 4. And so I claim that all four or all eight of these symmetries of the square, the four rotations and the four reflections, this encompasses every possible symmetry of the square. And I should mention that the composition of two symmetries is itself a symmetry. We would call this symmetry multiplication or these are just permutations. This is permutation multiplication. So let me show you some examples of this, right? If you were to take row 1 times sigma 2, as these are permutations, remember that we actually work right to left. So if you take sigma 2, which sends A to D, B to C, C to B, D to A, and then row 1, because remember sigma 2 right here, this is just reflection across the x-axis, the horizontal axis. Row 1 is a counterclockwise rotation by 90. A goes to D, D goes to C, C goes to B, and B goes to A again, right? When you multiply together permutations, we're just going to take sigma 1 and we're going to put it right here, right? And then using row, scrambling things up a little bit, row 1 sends D to C, so we put that right here. C goes to B, so we write that right here. B goes to A, we write that here, and then A goes to D, we go right here. So we've now composed the two permutations together. If we nix out the middleman right here, we see that A goes to C, B goes to B, C goes to A, and D goes to D. And if we interpret this geometrically, this right here is none other than just sigma 3, right? Because B and D were fixed and then C and A swapped locations. So the product of row 1 sigma 2 is actually equal to the reflection sigma 3. And likewise, if you take sigma 4 times sigma 1, remember what happens, we're going to take sigma 1 right here. This is the reflection across the diagonal associated to AC, so A and C are fixed, but then B and D swap locations. So we're just going to record that right here, just verbatim what we see right here. And then comparing the bottom row of sigma 1 with the row right here in sigma 4, right? So sigma 4 will send A, it sends A to B, it sends, so we have ABCD right here, we have ADCB, since then I might have a typo right here, so I'll have to double check to make sure this is accurate. So A goes to A and then A goes to B, so this should say B right here, like so. B goes to D and then D goes to C, so that should be a C right here. C goes to C and then C goes to D right here, and so the last one should be A, right? A goes to B, sorry, D goes to B, and then B goes to A right there. So the bottom row should look like the following. And so we get BCDA, and let's make sure that's the right map. Is that, that doesn't sound like row 1, that sounds actually like row 3 from our previous list, row 3, right? So A goes to B, B goes to C, C goes to D, D goes to A. So although I had a typo on this slide right here, the principle is still the same. If you take a product of two symmetries, you actually get a symmetry back again. The identity with the identity we saw before, we already mentioned the inverses of the rows, row 2 is its own inverse, and it turns out that all these reflections are also their own inverses. Sigma 1, sigma 2, sigma 3, sigma 4, if you do the symmetry twice, you actually get back the identity. And so this gives us an example of a group with eight elements. It actually is a non-Abelian group. If you swap the order, you're going to get something different. It's not the same thing anymore. I'll let you kind of verify that on your own. And in fact, you can use the following Cayley table to help you out here, that as we're looking at this Cayley table, we're going to get row 1 times row 1 is row 2. Row 1 times row 2 is row 3. Row 3 times row 1, I should say row 1 times row 3 is the identity, it's non-Abelian, so make sure you get the right order, right? Row 1 times sigma 1 is going to be sigma 2. Row 1 times sigma 2 is going to be sigma 3. And you can use this table to help you out with these calculations with the dihedral group of degree 4 right here. Now in a previous video, we introduced a different non-Abelian group of order 8, the Couturian group there. I want to mention that these two groups are actually different from each other. We have two different non-Abelian groups with eight elements to it. It turns out there potentially could be more, we'll talk about this more in the future, but this presents us a family of groups, like we did the dihedral group for the square and you could also talk about the symmetries of an equilateral triangle, symmetries of a regular pentagon. These will give you the dihedral groups d3 and d5 respectively.