In this video, I want to look at duality and the arrangement of our platonic solids in the model we made. Now, this is a drawing by Kepler from 1619. And in it, he’s investigating the relationships between the platonic solids in the upper left corner is a cube. And he has joined the center of each of the faces of the cube. And this forms an octahedron. On the right, he starts with a dodecahedron. He does the same thing. And he comes up with an icosahedron. And lastly, there’s the tetrahedron. Now I wish to go through each of those examples. Here we have a cube, and the center of each of its faces is joined to form an octahedron. And this is possible, because the cube has six faces, and the octahedron has six vertices. So to do this, the faces of one must equal the vertices of the other. And this is where the magic happens. Here we start with an octahedron. It has eight faces, and the cube has eight vertices. So it works the other way as well. And this is called a dual pair. But there’s another jewel pair. If we start with an icosahedron and do the same thing, join the centers of each of the phases, we get a dodecahedron. The icosahedron has 20 faces, the dodecahedron has 20 vertices. And if we start with a dodecahedron and join the centers, we get an icosahedron. The dodecahedron has 12 faces and the icosahedron 12 vertices. Another jewel pair. Lastly, the tetrahedron has four faces, and four vertices. And if we join the center of each of those faces, it’s a jewel of itself. In summary, we have tetrahedron, a jewel of itself, the cube and the octahedron and the icosahedron and the dodecahedron. If platonic solids are jewel, they have the same number of symmetries. Secondly, they can fit endlessly inside one another, though, in a real model that need to be held in that position. In our model, the octahedron is held in that position by the tetrahedron and the same on the right. But if the octahedron was not held in that job position, a larger octahedron at an angle would fit inside the cube and obviously, the volume of that octahedral would be more than one sixth, which was what we covered in the last video. As for the arrangement of the solids in our model, I chose the one on the left, which has the icosahedron at the center. But other arrangements are possible. And the one on the right begins with the dodecahedron at the center. And then outside that is the icosahedron and so on. Now we know about your lt, there’s just one more section on the worksheet. And that’s about surface areas. And it would be helpful. If you have your model handy. They’ll help you to think clearly about what’s going On with the surface areas