In this lesson we're going to be describing what we mean by the concept of symmetry. Symmetry is an intuitive concept. We see it everyday and many times we take it for granted. We define symmetry in terms of elements and operations. In element for example, can be an axis, a plane or a point. And an operation is symmetric if the object is returned to an identical position from which it started. And when we have a group of these Symmetry elements that operate on a structure. We refer to this then as a group. For example, when we talk about symmetry, we can have symmetry in phrases, words, and number sequences. For example, there is a quotation that's due to Napoleon, who said when he came to his exile point in Elba, Able was I ere I saw Elba. But spelled backwards was Able was I ere I saw Elba. No x in Nixon is when spelled backwards no x in Nixon. An example of a number sequence would be 1881 and read backwards is 1881. These are all referred to as Palindromes. And these are symmetries in phrases, words and number sequences. Now when we talk about symmetry in crystalline materials, we talk about a variety of different symmetry elements. One is rotation, and we can have n-fold axes of rotation, when n can be an integer, one, two, three, four, five, et cetera. We can have something that's referred to as an improper rotation where we rotate and we go through an operation that involves a mirror plane. We have planes of symmetry which are mirror planes, and we can have reflections across these mirror planes. We have an inversion axis, or a point that acts as a center of symmetry inside of the crystal. And then what we have for a complete list is something that's referred to as the identity operation. So, the structure comes back to it's original position. In order for us to understand and see how symmetry that occurs in everyday life can be useful to describe what's happening in crystalline materials. We'll use a simple stop sign. When we look at the stop sign, of course it's missing the word stop. But what it has is a structure that has eight-fold rotational symmetry. That is, you can rotate by 360 degrees divided by 8. And the sign will wind up reproducing itself as it goes through that eight fold axis of rotational symmetry. Now, if we wind up changing the sign slightly, that is what we have done is we maintain a hole at the top. We have a hole at the bottom so we can wind up affixing the sign to a pole. We now have reduced the symmetry from eight fold to two fold or C2 axis of rotation. When we add the word stop, we completely reduce the symmetry to the point that the only symmetry we have is the identity operation. That is the stop sign has none of the symmetry elements that we described previously. Another example would be a tennis ball. And when you look at various objects that you come in contact on a daily business, I would encourage you to take a look at those and identify the particular symmetry elements that you see. Now, in addition to having rotational symmetry, we can have a mirror plane. That is, a reflection across the plane. And what it does is, it takes a right-handed system and converts it to a left. So the left hand appears as the right. The right appears as a left. This is referred to as a Mirror plane operation or a reflection through a plane. Now, if we talk about the basic concepts of rotational symmetry, we can combine them together. So for example, let's begin by talking about the symmetry element of a pentagon. So, notice the red point that is fixed to one of the corners of the Pentagon. And now what I'm going to do is to go through a rotation that is associated with a five fold axis. And as I go through and I rotate, I can see how that point moves around, and I then create that image again as I rotate completely through those fivefold rotations. If I look at a triangle and I fix the triangle, and I indicate one point on that triangle, I can go through a rotation of 120 degrees, and I can go through another rotation, and then I wind up being at my original position. I can do the same thing with a rectangle, and here we have 180 or two-fold axis of rotation, and I wind up repeating my position. So those are my rotational symmetries. Now what I can do is I can take those rotational symmetry´s, and I can add a mirror to it for example. So I can see this very complicated structure I have which has five fold symmetry. And I have included on it a number of aspects that provide the opportunity to see how mirror planes can couple with rotational axis of symmetry. And so in order to see the mirror images, I wind up using the letters B and D. Because when they've reflect across the plane what you'll see is the B reflecting into a D and the D reflecting in to a B. When we look at something as complex for example as a soccer ball, there are lots of symmetry elements that are associated with the soccer ball. There are a variety of faces that are referred to as regular faces. And we have a regular pentagon. And it turns out when we look at a complete soccer ball, 360 degrees, we see that there are 12 of those pentagonal faces. We have a total of 20 hexagonal faces. And when we continue to describe the structure, we wind up having a total of 60 vertices and edges. So the structure then is composed of faces, edges and vertices. And we then have all of the axes of rotation that are shown by the geometry of five-fold, three-fold, and again, the five-fold symmetry axes and the two-fold symmetry axes. We can take that soccer ball and we can cut it apart and lay it out on the plane so we can see these axis of rotations, the five, the three, and the two. So we've just completed some discussion of basic symmetry operations that can occur in nature and we'll see more about this when we get into crystalline materials. Thank you.