I then found, as I began to have the balls coming around, rolling around it in all directions, if I had the twelve on here, then this next layer I get it filling in, every time it comes out the same shape the vector equilibrium. Always has the eight triangles and the six squares, every time I keep enclosing it it comes out that way. I don't think we have any models of that here in the room. Now, I found then, that the first we have a ball at the center, it is "0." There are no layers so if I call these layers, this is the "zero" layer, and then we come to the next one, we have twelve balls in the first layer, the next layer we get there are 42 balls, and the third layer there are 92 balls. Now the fourth layer, it turns out to be 162 balls. By the fifth layer it has 252 balls. I found that no matter how many layers I had, it always ends in the number "two." When I then recognized that this is decimal system that I am counting in, congruence to modular ten it is called, I have a constant suffix of the "ten" so I take the two away, and that leaves me instead of the it gives me ten, forty, if I just take out 2 and say +2 to all of them and then this would be 90 and 160 and 250. Now each one of those are divisible by ten, so I do that, and I get the numbers 1,4,9,16, and 25 and you suddenly recognize that as first 1 to the second power, 2 to the second power, 3 to the second power, 4 to the second power so what's going on here is, I call this then FREQUENCY OF MODULAR SUBDIVISION. F is my frequency beautifully done! And we find that the frequency, I gave you also something the physicists have two kinds of acceleration when I talked about "precession," remember? There is angular acceleration when you're holding onto the ball that is going around in a circle, and linear when it is going away. The radials are going away, they are the linears, and this is the angular, going around like that.

Now, whereas in a square system, on an x,y,z system in order to identify any point in geometry, they always have to go follow the line, you can't take a diagonal you can't take a diagonal like that. In other words there is no short way, you can't go from point "a" here to "b" on the diagonal. You have to go thru "c" in fundamental analysis algebraic analysis of any positioning of any points. But in the 60 degree coordination, because you see then the hypotenuse and the legs are never the same so the angular acceleration would not be the same language as the radial linear. But, in here the linear and the angular are the same. Exactly the same size vectors, same energy vectors remember what a vector means a vector is a represents an energy event and it represents a mass times a velocity in a given direction in respect to observer there is an axis of observation, and we have a special angle to observe its moving. And the mass then times velocity gives you discreet length of line. Vectors are not lines that go to infinity they are inherently limited, so that when I talk about a vector these vectors in a vector equilibrium represent forces of the Universe in balance the tendency to explode I showed you the other day being exactly countered by the contractive forces. So that the hexagon has six radials trying to explode and six chords. The six chords are more favorably arranged because it is a chain, and there is mass interattraction, so they get into critical proximity end to end and they hold together, where as the six others try exploding, disintegrating, not helping one another. And the other six help one another. So that we have then, in the Synergetics accounting, a space between two balls in closest packing, is then a wave length, and you don't have a frequency until you have at least two wave lengths. So frequency doesn't really begin until you get to this layer out here. In other words, frequency doesn't occur until that turbining, the disturbing quality enters is trying to do something, is trying to go somewhere. So this is "frequency," this is, when you get out here is frequency 2 second power. So I'm going to then find a point that we have down here the numbers of balls that we have in any layer in the closest packing of spheres will always be frequency to the second power times ten plus the number two. That became a really fascinating kind of a matter. There was every layer had two balls assigned to the function of being a neutral axis. There were two extra balls for every layer to take care of the neutral axis of spin, so Nature provided for that. If any of you have ever thought about a Victrola record this part is going this way, and this part is going that way two opposite sides, but you get to the center where there isn't anything going anywhere. This is literally a neutral axis theoretically there, but you've never been able to demonstrate it in three dimensions. In the four dimensions you can. I'm going to show you that immediately now.

Come back to our model, the vector equilibrium here, and I would have then, I could get, there are two balls I say in every layer that can account for being a neutral axis, and I'm going to take, in the vector equilibrium like this, and I am going to I said "lower the top triangle towards the triangle on the floor. I'm not allowed to twist this is the axis, here, I'm not allowed to twist the axis. It simply contracts in length, it does not twist! I do so, but the equator goes around! Here you see then the axis absolutely neutralized and yet it is able to introduce the motions, the equatorial motions. So we are able to also make this model as you will see later on where we make these with wheels that are going, so it doesn't have to stop they can keep on and on and on accommodating. But the center axis is absolutely immobile. When you get into these four-dimensional systems, one system then, like this, can latch onto another on the neutral axis, without in any way frustrating the motions in which they are participating. It becomes an extraordinary kind of accommodation that we experience in our actual life, but we have never been able to accommodate in any three-dimensional model. But with a four-dimensional model it is right there!

So we find that "twoness" is a fascinating matter.

Euler, I told you, developing his incredible realization that all visual experiences were reducible to three main aspects: lines, the crossings of the lines and the areas bound by the lines never to be confused one for the other and that in a picture on a polyhedral face or a polyhedron itself, the numbers of vertexes plus the numbers of the areas will equal the numbers of the lines plus the number two absolutely infallibly. So if you make a donut I said put a cord thru it really where we've got that axis there, then the numbers he said are the vertexes plus the numbers of areas equal the number of the lines. The two had disappeared. I do not know why Euler did not identify that with the axis because Euler also made one of the what we call structural engineering analysis engineering analysis structural analysis goes back to Euler. He was the first to develop then the concept of a neutral axis of spin of all systems. And so it's a structural member and for us to find out what its neutral is for its dynamic and he knows exactly how you're going to get your bendings and so forth. We find then, why he didn't think of the "twoness" of his own formula as representing the poles of the neutral axis, I don't know but he didn't. But when I found he hadn't, therefore it became very exciting to me, and I said, "I am going to now always assign two of every layer of my balls so he didn't get into this kind of a ball-kind of pattern. He didn't get into these layers of closest packed spheres. And closest packed spheres. And closest packed spheres is the way the atoms are all packed, so it is a very extraordinary kind of pattern to be considering. And as we're dealing in atoms and we're dealing in nucleus, and it has an inherent nucleus and no other geometry that I know of starts with an inherent nucleus. It's the only one. Closest packed and has nucleus!

And this in every way conforms to all of our experience with the atoms. So, I found then, by taking two of every layer, they would always, then, take care of the neutral axis of the system. Therefore it would be able to latch on to any other system, and we can keep on accommodating all of the kinds of things we do. Now, this was a very exciting discovery.

I've spoken about an inherent nucleus. It is possible to get a nucleated cube, but it has to have many, many layers before a sphere tends to come into the center of the aggregation in pure symmetry. You can get a nucleated tetrahedron, and you can get a nucleated octahedron, and they occur very much more early, and very much less aggregation than does the cube. There is a hierarchy of behavior going on here.

Now, in this particular formula, we are then dealing with the vector equilibrium which has a volume of twenty do you remember? Also, I showed you that going from this closest packed condition where there is a nucleus, I took this thing, and I made it I made it contract symmetrically, remember? All the twelve vertexes worked towards common center at the same rate. Remember it finally gets to be octahedron they're all doubled up. As all the vectors came towards common center at a constant rate. The, as I dropped this then, lowered this towards the other side, notice the six squares begin to change triangles cannot change they are structural. Squares are unstable and they do change. We are now at a point where this thing has contracted slightly, but it is at a point where the short diagonal of those diamonds is exactly the same length of this chord, so you put this in there are six squares, become six diamonds, and you put in the six cross members, and you have the icosahedron so the icosahedron is a contracted form of the vector equilibrium. It still has the same twelve vertices, same balls, but because it is contracted, there's no room for the nucleus anymore. It becomes exactly the same phenomena except for one thing, it does not have a nucleus. Or, you have compressed the nucleus, but you say "You really can't compress that nucleus," so I have to really consider that it does not have it. So I find then, the vector equilibrium is in a sense a vitiated or an empty an inoperative one and I have an operative one which has the nucleus. This gets into very much the relationships, then, between our proton and our neutron. So, I find then, the icosahedron has a volume of 18.51 where vector equilibrium is 20.

This 18.51 is a very interesting number because the if you'll take the relative mass relative weight of the electron in respect to the proton, and it is 1/18.51. It's really kind of a familiar kind of a number in here. So this has something, when we get into the icosahedron level, with nothing at its center it has something to do with the electron. But, you cannot take the icosahedron and pack them with other icosahedron and fill all space. They will join to one another, and will finally produce a geometry they will come back to the octahedron but they make a very wide-open octahedron. But they will not make themselves. We find then, they cannot be multi-layered, because not only can they not have a ball at the center, but as you go from the outer layer in towards this they collapse and there is not room for another layer so it can only be a single layer. Icosahedron is always single layer.

But, it has other qualities very close to the same as the vector equilibrium. And those the vector equilibrium has a characteristic of "twentiness," and the prime number five is in it. Whereas in the octahedron it has the prime number six there are six vertexes, and there are four faces and so forth the prime number two is in there, the prime number three is in there. But in vector equilibrium we first come to the prime number five is in there. We find the icosahedron the same prime number five, see? Pending five around each corner and so forth. It is a very fundamentally "fiveness." Now then, that "fiveness" is in here as a basic characteristic of the either the vector equilibrium and by the way the vector equilibrium I now write this way "VE" because I have to keep saying it all the time, so I use that symbol for it. And so this here is really two times five times frequency to the second power plus two. There is a multiplicative two showing here and there is an additive two, with the prime number five and frequency to the second power. Frequency to the second power is a very intriguing matter, because we have now layers something growing around, absolutely symmetrically, like waves. It's an omnidirectional wave phenomena and every way characterized by the great electromagnetic fundamental wave phenomena omni-directional wavings. Propagation.

In respect to it we have, remember Einstein's equation for energy, how much energy is locked up in a given mass, and I went into the knots and so forth here it's self interferences. But, Universe is, the physical energy is, the physical Universe is, the physical is energy and energy is either energy as radiation, unfettered, or mass brought together. So we have energy = M, that's the brought together side, times, is modified, how much energy is in there by it's relationship to the speed of radiation to the second power. See, the speed of radiation to the second power, as we said that is the rate that a surface wave grows this is the second power. Then we come down to the gravitational constant, and we come back again to our friend the second power which I spoke to you about, the exponential two that shows up, which is something apparently then, to do with surfaces, and we find out here is a system growing, rationally, beautiful rational number, absolutely in relation to this frequency to the second power! Which characterizes both gravity gravitational constant and the radiation constant. It gets to be very, very intriguing. As we go on in these kids of numbers the 12-42-92, I'd like you to add up those numbers 12-42-92. Why am I interested in that? Because, incidentally, I am going to stop for just a minute and double back on myself for just a little (turns the page of his drawing board).

I'm going to get into a little more discussion about nuclear phenomena. I have one ball is not a nucleus by itself. And I'm going to take start triangulation of balls, and here is one ball, and then I'm going to have two balls tangent like that. There's no ball at the center of the group. Then I have another ball, and another ball. No ball at the center of the group, is there? Now I'm going to have another layer of balls. For the first time there is a ball at the center of the group. It's going to be red nucleus. Now, let me have another layer. This is the center, there is no ball at the center of the layer. Now we'll have another layer. There's no ball at the center of the layer. Now I need to have one more layer. I don't know if I can really work this or not we'll try. And, suddenly there's another ball at the center of the layer. So it went, No, No, Yes No, No, Yes No, No Yes. It's not Yes, No, Yes, No at all. It's a very interesting kind of periodicity.