Next picture please these are revolving the icosahedron on the ten and the fifteen I just want to and the six.

Next picture. There is the icosahedron showing all of it's what does it have it has six, ten and fifteen thirty one great circles. O.K.? But the first one's where you use the same twelve vertexes that you had in the icosahedron in the vector equilibrium. And those twelve vertexes gave me this very nice great circle where you did have two vertexes you went thru, so it was contact, but the sixth great circle on the icosahedron does not, it is absolutely pure equator a great equidistant from all things. It would not conduct at all. So we take the, remember, I had twenty-five great circles on the vector equilibrium. There are twenty-five that really match them that are taut or twisted on the icosahedron, and then there is a sixth additional that goes around, but does not touch anything, so each one has one has thirty-one and the other twenty-five, but twenty five plus six is the thirty-one, the extra six which does not go thru any of the grand central stations.

Next picture, please. Now I am going to, see if we can make this bright enough for you to see it, I spoke to you a little while ago, when I had the vector equilibrium, remember, pumping up and down, and the equator was rotating, but the axis was not rotating. That is the big thing, right. Now, I can make this same kind of a model I have eight triangles, you can see them alright. Then I have four axes to the eight faces those would be the same as the perpendiculars to the faces of the tetrahedron the four axes. You can find those four axes if you want just go to a cube, and there are eight corners, and they are symmetrical to one another. And take the diagonal from this corner of the cube down to that one there, and there are the four diagonals between the eight corners, and they are the same lines and the same central angles as the perpendiculars to these square faces here. Then I could take this vector equilibrium and put a one-eighth octahedron on here, and the whole thing it becomes a cube. So it's just coming from this center. Now, because that's so, between vector equilibrium there is something I call each one of these triangular faces has a one-eighth octahedron, so if eight of them come together, they make one octahedron. So it's what I call an exterior octahedron, and inside, when I bring vector equilibrium to vector equilibrium this square face touches it there is an interior octahedron and there is an interior. Two types of octahedron that keep showing up interior and exterior to the nucleus. And they have to do with the loanings and the joinings of molecules, of the chemistry of atoms coming together. How you can loan so many charges one to the other. And this is what is done in here.

Now the I'm going to, instead of I'm going to put eight and four rods coming thru a common center here, and weld them together nice shiny rods, we'll say a quarter inch in diameter. And now that they are welded together I'm going to take, instead of eight triangles, I'm going to take eight little automobile tires. I'm going to have this rod, then, it's diameter will be the size of the hole thru, get little toy automobile tires with the little metal wheel in the center, and then it has a little hole for a journal going through so we can slide it onto a rod. And I'm going to slide the eight automobile tires, toy automobile tires onto these rods, so that the plane of the tire this is the wheel, it's over like this sliding in thru its hub at the center of gravity where the triangle would be here. So each one of those wheels will be touching another wheel at three points. Can you see that? There's one here, there's another one here, and so the automobile tires slide in on the rods until they keep meeting each other because they are converging so they begin to push very hard the rubber` on one another. So I bring them all in a equal distance and in tight contact with each others surface, and then I put a little journal on the outside of the rod so that they can't slide outwardly we'll use a little metal washer, and then some tape on there to hold them where they are, so they are held in tight friction with all the other automobile tires. That is a model you see up there behind my head. Once you have it on your minds maybe it will be more clear. There are then these eight wheels, and I find then they are absolutely independently journaled, free on here, yet they are touching one another. So if I take and put this if I hold onto this as a system, these rods then stick out and I can hold on to these rods independently, if I rotate one of these wheels here, then this one has to move they all move. If I rotate one all eight rotate reciprocally very beautifully. I can try anyone of them and I found all eight of them absolutely beautiful to go round and round, so this motion that you saw, I want you to suddenly realize this could be the same motion I say they're rotating on each other, but this top one here is staying put and the ones around the equator are rolling along, can you see them? This one could go like that and then keep on going. Can you really feel them going around the equator? Well, alright. Now, for the first time, then, this has a limit until you come to the end of the hinges, but the one model I give you now there are no hinges, so they keep rotating one way or another and the whole thing is reciprocal. Then you'll find , going thru these four pictures, I have up in the top left hand side a little white marker, and what I do now is to take a hold of one of the wheels with my fingers like this, so I immobilize that one wheel. And I take a hold of the wires that are sticking out and pull the system around the one that I am holding onto, because you'll find the three touching they just roll nicely around they roll around, they're ball bearings. And these ones are rolling this way. The three are rolling the other way and there is one at the top.

As I hold this one fixed and I roll them around so there are out of eight of them three of them in the northern hemisphere, three of them in the southern hemisphere are rolling beautifully. But the top one is absolutely immobile. If I immobilize the bottom one, the top one is absolutely immobilized. And that is what you can see in this picture as I go around.

Next picture. You see the marker will stay up there at the same position all the time.

Next picture, I'm sorry, I seem to be so much in the way of this thing.

Next picture. The hands had to really stay fixed at any rate.

Now, what I have shown you is the I've given you an independence of the axes that you can fasten onto another system, yet the rest of the system can be carrying on. So, I said, every system I find always has axes, it always has an isolatable axes this is four dimensionality.

Now, four axes of the basic symmetry. So the next thing I wanted to point out to you is that those rubber tires, I could have made them a distorted donut, to be a little triangular can you see? So they just look like a cam. Here's the circle and I begin to make it go like that, so this is a little shorter radius, and this is a little bigger radius on the side, I could make each one of those a triangle. And if there were springs holding them in towards each other as they rotated, they would go into the position of the octahedron when they simply get down into this closest position of the sides of the triangles versus being on the corners of the triangles. So as I rotate the system everyone of those triangles is going to be pumping like that by just their own friction, and around and around they go.

Now, the next thing about it that I am going to say remember I had an involuting and evoluting donut? Rubber donut? So I'm going to make each one of the triangular cam rubber tires into also involuting and evoluting so, when I hold onto one triangle at the bottom end, I'm holding onto it, which makes the one at the top do something, you'll find this whole things goes through now I'll take a hold of one edge and start to move it around force doing that, and the whole thing pumps like this and continues involuting and evoluting . And when you see something called turbulence this is what you are looking at. It's a very, very beautiful thing. When we begin to really study what is turbulence, this is the big show!

I find it fascinating that with just a relatively few models, begin again to be able to do this in your imagination.