[MUSIC] We have already looked in detail at plane nets. We've looked at triangular nets, and square nets as simple examples. We've also seen how nets can be modified to create very intricate patterns. As we see, for example, in Islamic tiling. What we want to do now is move from two-dimensional or plane symmetry into three-dimensional symmetry and three-dimensional structures. And in front of me here are two examples of polyhedron. Which make use of plane nets which are then folded to create three-dimensional shapes. If we take, as our first example, this red polyhedron here, it's known as a tetrahedron and as you probably know tetra means four. The tetra is referring not to the number of corners on this shape, but the number of faces. So there are four faces with this tetrahedron. What's important is that when you look at any particular face, you can see it's a triangle. So intuitively you know that if you were to fold up a triangular net, you could create a tetrahedron. The other solid figure here is what's known as an octahedron. In this case the octa, which is for eight in the Greek lexicon, the octa is referring to the eight faces. So you've got four here at the top, you have four underneath. Actually in the case there are only six corners. But again, if you have a look at this shape, you can see each face is a triangle. So, by folding the triangular net in a different way, you can create an octahedron rather than a tetrahedron. The purpose, then, of this lecture is to consider in detail how do we develop these three-dimensional shapes from the two-dimensional networks? In particular, we want to define the so-called Schafli notation. The Schafli notation provides us with a quantitative way in which we can define the solid figure by counting out the number of faces that surround each vertex. Historically, there are five very important solid figures that are known as the Platonic solids. And these Platonic solids come up time and time again in symmetry. Whether it be in architecture, in nature or in crystallography. So it's important that we understand the way in which these figures are constructed. There are, of course, many other solid shapes beyond these five Platonic solids and we can't look at all of these in this course. But we will look at one which is known as an Archimedean solid. It's known as a cuboctahedron. To create a tetrahedron from a triangular network, we select four adjacent triangles. The central triangle in the animation is shown in blue, and the surrounding triangles in green and red and yellow make up the four faces of the tetrahedron. When these are folded together to create the solid figure, we also create four vertices. It's also possible to count the number of edges, and for the tetrahedron, there are six of these. The formal description of the tetrahedron is derived from counting the number of triangles around each vertex to derive the Schafli notation symbol. Which is 3.3.3 or 3 cubed. The tetrahedron can be viewed down its principal axis. If we look down the threefold axis, it's represented by a triangle indicating threefold rotation and the associated mirror planes. We can also view the tetrahedron in such a way that we view a single mirror. Or we can rotate so that we can view a fourfold inversion axis. Four fold inversion is something which we will cover in more detail in coming lectures, but it involves rotation of 90 degrees and then inversion through an inversion point. To construct an octahedron, we are going to start with a triangular network. But in this case, it requires eight triangles to build the solid figure. These eight triangles of course create the eight faces that are observed. Once these are folded together, we find that we have six vertices. In addition, we can count the number of edges. And for the octahedron, there are 12 of these. Each vertex of the octahedron is surrounded by four triangles. So the Schafli symbol for the octahedron is 3.3.3.3, or in short hand, 3 to the fourth. To view the octahedron we select three principal directions. The direction which is easiest to visualize is looking down a four fold rotation axis. As we know every four fold rotation axis is associated with mirror planes and these mirror planes will be of two types. Those that passed through the edges of the octahedron, and those that passed through the corner. We can also rotate the octahedron to see two fold rotation and in addition, three fold inversion. To construct the cube, we begin with a square net, rather than the triangular networks that we used for the tetrahedron and the octahedron. In this case, we select six squares that will end up forming the six faces when we fold it up into the three-dimensional body. After folding up the square faces, we find that we end up with a solid that shows eight vertices. And we can also count up the number of edges and there are 12 of these. The Schafli symbol for the cube is 4.4.4 or 4 cubed, which is telling us that we have 3 squares around every vertex. If we look at the cube down its fourfold rotation axis, we see the two types of mirror planes associated with that rotation. We can then rotate the cube to look down the body diagonal which is shown as a threefold inversion axis. Finally, we can look in across an edge of a cube. And in this case, we end up with a twofold rotation axis, again associated with a mirror plane. Now, we've looked, in some detail, at three of the platonic solids. The tetrahedron, the octahedron, and the cube. There are two other platonic solids. These are the icosahedron and the dodecahedron. The dodecahedron is a little bit different to those we've considered so far because it's based on a pentagonal network. And in this case, you join together these 5-sided pentagons to create a figure with 20 vertices, 30 edges, and 12 faces. This is a fairly complicated figure, and we won't spend a lot of time on it in this course, but I think you'll agree, it does look like quite a beautiful shape. The final platonic solid is the icosahedron. In this case, it is a gang made up of triangles, but there are total of 20 triangular faces. So it's much more complicated than, say the octahedron, which only used six triangles. So to summarize, all the points that we've covered to date, and there are quite a number. The tetrahedron, the octahedron, the cube, the dodecahedron and the icosahedron. We define these formally using the Schlafli symbols. And remember, the Schlafli symbols are derived by counting the number of faces around each vertex. All the Platonic solids are composed of a combination of triangles only, squares only, or pentagons only. Because you use faces which are always the same shape, these are known as regular solids. You will notice that the way in which the names of the solids are derived is by counting the number of faces. So in the simplest of example, the tetrahedron, you count up four of these faces. If you look in detail at the diagram at the top of this slide you will see that all the vertices of the platonic solids lie on the surface of spheres. And that is also a property of these types of solids. Finally, we will look at one example of an Archimedean solid. The Archimedean solids are known as semi-regular solids. They're semi-regular because they're composed of faces of two types. And in this case, it's a square face and a triangular face. The same principles apply, however, we can take the plane net and fold it to create the three-dimensional solid. But in the case of the cuboctahedron, where you have surrounding each vertex 2 squares and 2 triangles, so the Schläfli symbol is 3.4.3.4., we create a solid figure with 12 vertices, 24 edges and 14 faces. In general, the Platonic and the Archimedean solids obey the Euler rule. Which states the sum of the edges and faces subtracted from the number of vertices will always equal two. So to conclude this lecture, let's recap everything that we've learned. Firstly, there are 5 regular polyhedra, which are known as the Platonic solids. These platonic solids are the tetrahedron, octahedron, cube, icosahedron and dodecahedron. For the regular solids, every face has the same shape. Either a triangle or a square or a pentagon. There are also semi-regular shapes. In this case, you have two types of polygons making up the solid figure. An example that we looked at was an Archimedean solid known as a cuboctahedron.
