Now these, there are twelve points here and when spheres are closest packed around spheres, these points are where they touch the next sphere, so if energy were traveling through space through atoms that were in closest packing, you find that energy follows a convex surface not the concave. It's very easy to understand. Just take a piece of paper and just bend it. The exterior this goes into a little more tension on the outside doesn't it it tries to resist you. So tension, high tension, and energy follows the higher tension. We have a great copper sphere hollow sphere, maybe 20 feet in diameter Van De Graaff generator where you simply keep loading electrical charges to it, and they always stay on the outside. You can get up to a couple of million volts and they are used for making artificial lightning. But people can walk around on the inside with absolutely no trouble at all they will never be short-circuited because energy always stays on the convex side. For this reason, when you're trying to plate, silver plate, any plate metals, the convex is very easy to plate. The concave is almost impossible. You have to get your anode almost in practically touching, in order to get it to flow it on there at all.

So we find energy is always following the convex. So that energy, going from here to there in Universe, following the convex would follow the outside of the sphere, where we came to the point of it could only get to the next sphere through a point of tangency. And it could get on such great circles as these, and so these begin to be the beginnings of railroad tracks. All the great circles that go through these twelve points. Just sort of fundamental symmetry are going to the way in which energy can get from here to there in Universe thru closest packed vector equilibrium (VA).

Now, it gets terribly interesting in this particular device when we begin to pay a little more attention to it. This is interesting, these are the same four planes, and I want you to see remember this is our friend vector equilibrium how you could pump that around. It is an extraordinary thing. You can flatten it down to get all the planes congruent or it opens four completely different ways, and you can flatten it any one of those ways. It comes out a different looking pattern altogether. These are typical of the intertransformability, starting from our wonderful vector equilibrium . And, I have mentioned, the other day, the vector equilibrium really was the limit domain of the nucleus, and everything that goes on within the vector equilibrium is unique to nuclei and to atoms, what goes on outside of them when they join up with others, is unique to chemical compounding and molecules and so forth, where things join up. Joining is outside, this is the domain of the non-joining, inside. It's very fundamental it's the basic patterns.

Now, I want to talk about other great circles. And this one is very easy to make because they are all the same and you can do your own improvising really quite easily. But I have slide pictures of other great circles. And I want you to think about what they might be. As, for instance, we have the tetrahedron, and I'd like to find symmetries in it. For instance, I could, you might say, it doesn't seem to have a pole there. But I take a mid-edge and a mid-edge, and suddenly it does have symmetry. Tetrahedron, supposing I were to take a knife it's made out of cheese, and I cut parallel to this plane here, but up here. I could truncate this little corner couldn't I? And leave a little triangular unit, can you see that? I could cut off this corner, I could truncate each of the four corners and get little additional triangles on here. If I did that, having cut here, you can see where I've cut into here you find you have a hexagon. So I get four hexagonal faces plus four triangular corners. You also see that figure showing up. Then, suppose I wanted to take a knife, or a plane, and I slice parallel to this edge itself, in other words I cut off, truncate the edge, can you see how I do that? My lines would look like that parallel to the edge. So I cut the cheese off so I've got a little flat plane on each of the six edges of the tetrahedron. And so, that will leave me still four flat faces out here and I could then truncate these corners. Sum totally, I could get facets on here I could get up to the four faces already there, plus six facets if I truncated the edges makes ten plus four facets at the corners that's fourteen, and these always they are opposites they must be in pairs, so there are actually seven axes of symmetry in the "fourteenness" of the four, plus four, plus six. And that fourteenness shows up as seven sets of the great circles.

And these seven sets of great circles have very interesting properties. We're going to look at those, and they are really all the axes of symmetry of all crystallography. There are seven fundamental symmetries. And the let's come for instance to the, may I have the first picture now next picture. We're looking at the vector equilibrium again remember the four great circles. Now you're getting familiar with it all of a sudden, and we're looking at it made in colors. Next picture, next picture again. This is one where I get what I call a concave and a convex one and you're going to find those very interesting as I said Vector Equilibrium was the limit case. And if I take the vectors edge of the VE I could bend them and make them into arcs. This means that all the vertexes go inwardly a little . Or if I bent the exterior edges concave, it would give you a shortening of the lines, therefore the vertexes would have to come in. In this seemingly straight condition it takes the most room in Universe. And those concave and convex qualities you see in that picture, relate then to the first like knocking out the central ball and it becomes an icosahedron. These are the first degrees of contraction where you have to follow the hierarchy of forms that begins to generate.

Next picture. This is a little difficult to see. That is a transparent four great circle.

Next picture. What we're looking at here now is I've tried to make just take two great circles and cross them. And they really become unstable, they just look like this. They have a common axis but they flap, and I try to make, then, the

Next picture please. There we tried to make the central angles of the tetrahedron what we call one hundred and ninety degrees and twenty-eight minutes, where , that doesn't work, you'll find that the one hundred and ninety and twenty-eight is what each one of these arcs are and they don't come out in whole great circles.

Next picture. Here is the octahedron and you'll say, well those are 90 degrees, if you try to make those in supposing I try to make a bowtie the way I have here-90,90,90,90 what do I get there, four times 90 that's alright, that's 360. But then you find that you can't make a, you have to take two whole great circles.

Next picture please No, you take six of them! You take six great circles folded to make the three great circles. I've told you this before, octahedra always appear double, they always appear congruent, so to make the octahedron in great circles, folded great circles, it has to be double., again. So it's really six great circles that look like three.

Next picture. There you are looking at the octahedron. No, that's the attempted central angles of the tetrahedron and they do not work.

Next picture. Now we're looking at the six great circles. And the six great circles you will like to know where they come from. Let me then take the vector equilibrium itself, just let's see what it's got. It's got those six square faces, eight triangular faces. It has twenty-four edges, it has twelve vertexes. So if you take twelve vertexes they will then have six equators they are opposite each other. The twelve vertexes are in pairs north and south. There are twelve vertexes that are opposite from each other and you have six great circle planes as I revolve it it goes perpendicular the perpendicular bisector triangle goes square, triangle, triangle square, triangle, triangle and that gives you, that is the axis of symmetry that gives you six great circles which I have been showing you.