Next picture. I'm reviewing this quite rapidly because I want to get to some other things that I have added in, over and above what you had yesterday.

Next picture. There are the two halves of the three frequency.

Next picture. They come together as the tetrahedron precessionally again.

Next picture. And we have, then, the eight frequency, and they come together again as the tetrahedron, again.

Next picture. And, we then have the two one-eighth octahedra. And I'm sorry to say the top one is not clear, and they precess together to give us the cube. This is the first cube to appear in spheres. In other words, you can't have a sphere with eight cubes, just the corners that's all. There are in this, then, the base is six and the seven, so there are fourteen. So this first cube is apparently fourteen, and it's very possibly something to do with carbon. It seems very logically so.

Next picture. There we see the completion of the corners of the vector equilibrium by putting on the one-eighth octahedra in each corner.

Next picture. Then we take it off and we get the vector equilibrium itself.

Next picture. And then we take this whole thing apart in these various slices.

Next picture please. I also, then, want to remind you of something I gave you the very first day. The vector equilibrium, remember those shells, remember. First shell twelve, next shell forty-two, next shell ninety two counting up to 146 plus 92 gives you 238. I'd like to show you something about that that I didn't mention that day. So we have the formula for the number of balls in the outer layer was always, remember, frequency to the second power times ten plus two. Remember that? And I made it very clear why that was so. So we found then that we had this layer of twelve and forty-two, ninety-two and one hundred and sixty two and so forth two fifty two. Then, however, this was the first layer, but there was a ball inside, so it was zero layer. In other words, it can't be a layer, unless, I say, unless the ball itself really there is really a ball there, and it has an outside and an inside. And these layers have been enclosing, so I'll have to remember then this is a, being frequency to the second power times ten plus two; we then found that was ten and then we took so this was twelve was one was the number one, the first layer, this is number two, this is number three and number four. Those are the numbers that became second powered; so the center ball is zero, so I find out, remember what it's formula is. So it is frequency to the second power times ten plus two. So zero to the second power is zero, times ten is zero, plus 2 is 2. The value of the inner ball there turns out always to be 2. In other words it's own concave and its own convex. And I have given you, unity is two and you'll find this showing up time and again when we come to this extreme. I find this a very, very exciting point and I hadn't given it to you the first day, and it was very important that it come in today.

Next picture please. Here we have the knocking the central ball out you remember, from the vector equilibrium, and it immediately became the icosahedron. Same twelve balls simply the squares disappeared and they became triangulated as the pumping model shows very clearly.

Let's make a quick review of it, then we went from open vector equilibrium, down to here, and then we add these in here and here's the icosahedron. So like everything else it's the degree of contraction everything happens in the vector equilibrium, where the realities begin to occur there are always some degree of contraction. So this is a much further degree of contraction, and even further so when it came down to the tetrahedron.

Incidentally, there is another tetrahedron in the vector in the jitterbug which I haven't tried to make for you before this is a polarized one. See the precessional edges in my left hand and my right hand I have the double edges, and all the rest are single. All these things occur without ever breaking the edge. In other words, the integrity of the system is always there.

Next picture please. On the left you will see the these are pictures from Linus Pauling's book. And you will see a column there, he's got a polarized column, and he has the same picture I showed you, taking off one whole large corner of that cube. I'm sorry there's something else over on the other side that is very interesting. I made, in the late 30's and early 40's, I was able to get a hold of beautiful little clear crystal balls, and I found that they were transparent, and gluing them together I was able to make very fine models. I think we still have them, but they began to tell me, this upper row here, I began to really get faithfulness in respect to various atoms, and this is all a part of what kept me going.

You can imagine when I first began to discover all of these rational relationships, and I was able to really talk about them in the 30's and I began to confront scientists with what I was finding, and they found no identity with what they were thinking. . And they were not thinking models, and they felt any attempt to bring the models in was really tending to really roll backwards into the Platonic era, and this was all nonsense. So that, what I would have to say to myself. I would ask the scientists if they could find anything wrong with my arithmetic or my geometry. And they would say no. I had found this beautiful hierarchy of rational values, and I'd say, "Do you think I ought to go on?" And he would say "Yeah, I think you seem to have logic in it alright, you might as well pursue it, but it doesn't have any significance, you know, in physics, it's just sort of a mathematical pastime. And, so I finally had to ask myself a good question. I said "Am I so important that Universe would secrete a great cul-de-sac of incredible beauty and elegance, just to fool me?" I said "I'm just not that important." So I have to assume that it really is very important and that the other people are wrong. This had to be my argument, and I've carried that on since it's been really a very long time.

Next picture. On the left we then see our friend the tetrahedron with the octahedron in the center. I didn't have a nice model like that to show you the first day, so I thought I'd show you that again.

Next picture please. Something has gone wrong on my picture. That was going back. This is a picture of great importance because we are looking at the skeletonized tetrahedron with the octahedron inside it, and on the left hand side lower left hand side you'll see a blackened tetrahedron. It's base is on the table, and immediately to the right of its base, is then the base of an asymmetrical tetrahedron. It has the same triangular base both equilateral, but you'll see a dotted line going from the lower mid-right side of the big tetrahedron. There is a dotted line going to the top of the small tetrahedron in the left-hand corner, and that is, then, one of the x,y,z coordinates running between because the octahedron then has three corners, and this is one of the and that edge then makes an asymmetrical tetrahedron which is leaning leftwards a leaning tower of Pisa, but it has exactly the same sized base as the regular tetrahedron on our side. So the bases are congruent, so we know they are the same, and we know their top vertexes apexes are congruent, and the base is the same area, so their volumes must be the same. And that is a one quarter octahedron. You can see that it is just one quadrant of the octahedron if you study it carefully. So you can really feel very comfortable when I give you the octahedron as a volume of four when the tetrahedron is volume one.

Next picture. I didn't have the opportunity to show that to you the other day. I talked about it, but no model. I'm just confirming to you from the other day the count of the tetrahedroning instead of cubing. When instead of superscript 3, we call this then tetrahedroning instead of cubing.

And there is the count for each of those layers. The one, and then the two get up to eight, twenty-seven, sixty-four as they combine. So now you feel quite content to because we have also found that structure is triangle and if it isn't triangulated it's not a structure unstable. And that tetrahedron then was the simplest structure, prime structural system of Universe. And so when you count in tetrahedra, and cube take three, we are being more economical, and if you use a cube you use up three times as much space as Nature is using, because she is always most economical. So you want to catch on to what she is doing the most economical, and you've got to use the tetrahedron for your accounting.

Next picture. Again we really can't see. There is a vector equilibrium to the right and it shows the eight, little one eighth octahedra no the eight tetrahedra go in and the six one half.

Next picture. And that's showing a completion of one of the corners of the vector equilibrium to make it the cube, by the one-eighth octahedra.

Next picture. This, now is very, very important. We haven't come to this yet. Yes, we did, in the terms of spheres in packing, and let me remind you again, when we are talking about spheres in packing we are talking about vertices. And when we are talking about edges we are talking about when we see lines in structures, we are talking about the edges, and the counts are very different. There is one vertex, for two areas, and three edges, so you can see the difference. We are now looking at a skeletonized tetrahedron and the center of gravity of the tetrahedron. And we pull out from the center of gravity of the tetrahedron, a one-quarter tetrahedron, and that one-quarter tetrahedron, you may remember, we formed by we had a triangle of closest packed spheres, and it's edge read four and there was one ball on top. It was a three frequency. There was then nestability. Do you remember that? We found that where we had three balls there was no space for it, not until we had a three frequency, or four balls did we have then no, that gave us a ball at the center. It had to be five balls or four frequency before we get the one-quarter tetrahedron.

And, this, then was in a hierarchy of nestabilities. In other words it wasn't in just being arbitrary and saying we're going to have one-quarter tetrahedrons, and one-eigth octahedra, we found that those occurred where a ball could nest, and they were the sequence of the first time, and the second time that a ball could nest on top. Your first nest on top would make a tetrahedron. The next time it would make a one-eighth octahedron, and the next time it would make a one-quarter tetrahedron.

Next picture. Now those one-quarter tetrahedra up at the top. I've taken an octahedron, taken the octahedron and, that's alright, on each of its eight faces we put a one quarter tetrahedron. Again it is a regular equilateral triangle, it is all the same vector edge, so that it fits perfectly well. I've got one here, then another one here. And when you do, you'll find that the apex of those one-quarter tetrahedra are on the same plane as that line between the two. And what it does is to form the rhombic dodecahedron. It's a fascinating thing. As they come up here, this makes then for each of the twelve, there are twelve edges on the octahedron, and these apexes come up here and it becomes a flat it makes a diamond it makes a diamond on each edge of the octahedron. There are twelve edges so you get twelve diamond faces, and there is your rhombic dodecahedron. Now the rhombic dodecahedron is a very exciting kind of a form. It's volume is exactly six when the tetrahedron is one, and the cube is three, and the octahedron is four and so forth. Six. And its "sixness", what is this rhombic dodecahedron. Remember the vector equilibrium has twelve vertexes. And those twelve vertexes of the vector equilibrium are where every sphere in closest packing is in contact to the next sphere. Now, they are also the spaces and the spheres in closest packing, and what the rhombic dodecahedron does each one of those diamond faces occurs at the point of tangency. There are twelve of those diamond faces and each one, the center of it is where each sphere touches the other spheres. So what the rhombic dodecahedron the rhombic dodecahedron, like the cube, fills all space. So what it is, is both the sphere and the space. It goes in exactly, and there's an octahedronal space, and a vector equilibrium space, and it exactly goes to the center of gravity of that space. So it represents the volume of the sphere and the volume of the space that belongs to that sphere. And there it is, the volume is six. It's this beautiful rational number.

Now, we're going to see a lot more about this rhombic dodecahedron because it is then, the epitome of the behavior of the spheres in closest packing, and it is, I simply call it, "the domain of a sphere," and sometimes I call it a spheric. Because it is the domain of the sphere. It's a sphere and the sphere's own share of the space that is not a 20 like the vector equilibrium, nor 4 like the octahedron. It's a very important number. 6 is a spheric space a domain. And when we are dealing in spheres, in a way, we are used to thinking about spheres and so much of the Pi business, and somehow there are some very nice numbers coming in here without so far getting into any calculations of that kind whatsoever.