Next picture. This is looking at the same six great circles.

Next picture oh, incidentally, you get the six great circles if you want to by, take a cube and put both sets of diagonals you have the two tetrahedra crossing one another inside the cube and that gives you the six great circles. You find that, six great circles have four times six twenty-four triangular faces. They are not equilateral, but they are isosceles.

Next picture please. Now we're looking at the twelve great circles. Say, where did those come from? Well, remember, there are twenty-four edges here in the icosahedron. So, if I take the mid-edges of the twenty-four edges it gives me twelve axes. See that. That would give me then twelve axes of spin, so this is really quite a complicated one. You see it goes through mid-edge, corner, mid-edge, mid-edge corner, mid-edge, mid-edge, corner, mid-edge, mid-edge, corner. So there is a symmetry about it, but it makes it quite a complicated one. Look behind me there and you'll see it's quite a complicated form.

Next picture. There is another of the...

Next picture. Another of the twelve great circles.

Next picture. Now, I come back again to the Vector Equilibrium. I have here, how many ? There are six faces square faces aren't there? If I take opposite the mid of each square face; the six of them would give me three axes, and this will revolve, go vertex, vertex, vertex. It has a square section in there, can you see that? I can't put my finger here, but, I have to hold onto it to do it, but this is how it revolves. This is, then, what they call the three great circles.

How do we get the four great circles? I go to the, there are eight triangular faces, so I take the eight triangles, and take their mid their centers of gravity, and there would be four axes between the eight faces and so I revolve it on those, and there you see the four great circles. See that great circle? Here is a triangular one again. As always that's what gave us this beautiful form here. Those come out of the triangular faces, so the three great circles of the square faces; the four great circles of the triangular faces; there are what other features do we have here? Then we have the twelve great circles of the mid edges. There are three four six, we have there were twelve of these vertexes so there are six of these three, four six, twelve. So three and four make seven and six make thirteen and twelve makes 25. There are twenty five great circles on the vector equilibrium. They are 3, 4, 6 and l2. They are four of the seven of the basic symmetries of crystallography. And you can see why how absolutely simple and fundamental ...

Now, I want you to watch each one of the ones that I have just done with you. Go back to the three great circles which were square faces. It goes vertex, vertex, vertex, vertex. There are four vertexes involved, right? And then next we go to the triangular faces, and I have six vertexes all the time. So they go through many more vertexes the four great circles go through many, many more stations of tangency more spheres it can this is a railroad track, it could get you into more stations than the three great circles. And then we take the six great circles where the, where we get vertex, no vertex, no vertex, vertex you get two vertexes on the six great circles, but none the less they do transfer at the main grand central station of tangency to other spheres so that the energy can travel over the convex surface of spheres the most the shortest distance, because all great circles are the shortest distance they are going to travel, so they can travel on the six great circles. Then look at the twelve great circles, mid-edges, remember? Sure enough it goes thru a vertex, mid-edge, mid-edge; vertex so it goes only thru two again on the twelve, these are really very fascinating characteristics, but this is part of the main switching of energy in Universe, and every one of the ones I have just given you the three, four, six and twelve, are all foldable out of whole great circles. You have to do your spherical trigonometry to know what the central angles are, but once you have you can fold this up very neatly and you will literally take twelve great circles, fold them up, and make the twelve great circles, and come out the continuous great circles out of these bow tie forms which come together. So it tells you that everyone of them has a holding action where you can go around locally, or go on and travel, but the fact that some of them have two stations, four stations, six stations means that they really are quite a different set of options for travel on those different sets of great circles. But and one of them has more of them than the other. The one that has twelve has only two stations, so that is really twenty four cases there; the one that has six great circles we had two again so you see the twelve opportunities there. So the things are not coming out the same number I want you to realize.

Now, the next thing we come to is the icosahedron. May I have the next picture. Here is our friend the icosahedron. You see some pentagons and right away you say, this is due to that "fiveness" something to do with the icosahedron. So, what do we have here? We have the same twelve vertexes, so it has six great circles. It was interesting, there was one the six great circles of the twelve vertexes, but also don't forget, where you get the three great circles was the octahedron and it took six of them to do it. So it really is a six there is six appearing in here twice in the vector equilibrium. There are also six on the icosahedron. Now look what it does. I spin it and it doesn't go through any of the stations! So, suddenly, there is a cut off. Then, let me see, what other features do you have here? The other one has squares and triangles, this one has only triangles, so I have twenty triangular faces I have twenty faces, therefore I have ten axes of spin. This is the ten axes of spin here, and you'll find that it is a very amazing thing on the icosahedron it keeps missing the vertexes. And then I have the what else do I have? I have thirty edges which gives me fifteen and this is the only one where they transfer. It goes yes, yes, no; yes, yes, no; yes, yes, no something like that other kind of pumping you ran into. The yes, yes, no shows up quite often in basic series here, and makes it possible to do yes, yes, no; and a yes, no, yes, no so that you don't have any interferences. So, icosahedron has only two chances.

Now we find the icosahedra, they do not carry on and fill all space, therefore they are not what you get in closest packing closest packing you only get with the vector-equilibrium. Then it contracted in order to be the icosahedron, so it doesn't have the contacts. Time and again I want you to feel this kind of neutral condition, like a neutron without vitality; and the one that does have the nucleus you go into the proton. Same number system, but just a little bit contracted. And the difference in the contraction is just the difference of an electron. So, we find that this thing cannot have many layers. In fact, it tends to act only as an electron. It really has to be a free space actor.

And it does have only one way in which it can actually ever make contact with things and get something out of the system. And that was this one, of which there are fifteen of those, great circles, thirty edges.

So, let's look at the slides again, and everyone of those are foldable, out of whole great circles. That is the six great circles.