So I say, "Mr. Mathematician, now I've given you a tool that is much more reliable for you, but never kid yourself again, never call it a straight line. You are simply dealing in wave phenomena, and now we can go on and do all the geometry we ever did. This is the first time the mathematician really began to be a little friendly with me, because he did not like really being excluded from the experiential club of the physicist. And his salary is very much smaller than the physicist.

Now, this brings me then to I've given you some agglomeration of spheres, and I, there are many things that I would do if I had more models around me; but I'm fairly limited in my choices of the things I see because I see one frequency and so forth and not agglomeration. I've already done octahedron and tetrahedron with you, and you've felt those and you've felt those grow.

And the you'll find if we take tetrahedra, I do get four tetrahedra together but no nucleus right? If I, however, then, make another layer I can have a tetrahedron where you see three balls on an edge where there is actually frequency two. And it's number of balls is four on the top and six on the bottom. There are ten balls. Six-ten Then if I have another layer here's another ten it gets to twenty. In this layer, however, the twenty layer no, when there are twenty altogether, this is the ten. And you have one, two, three, four, five, six, seven, eight, nine, ten that's where the nucleus begins to show up again, on the "ten" layer. So, on this surface of the tetrahedron on each of the four surfaces, you see a new nucleus A nucleus beginning to show for the first time, because the tetrahedron did not have a nucleus of its own. I'll then have to put one more layer, and the next layer will have 15 in it so it goes to 35. One, two, three, four, five, six seven, eight, nine, ten eleven, twelve, thirteen, fourteen, fifteen there are fifteen balls. When you do, then, for the first time we have a nucleated tetrahedron. So there is a nucleated tetrahedron, the same way we can get then to how do you get an octahedron with a nucleus.

Whereas this same formula then for the number of balls in the outer layer of the nucleated tetrahedron in contradistinction to the vector equilibrium where it was ten times frequency to the second power plus 2, it comes out four times frequency to the second power plus two. But the four as the ten was two times five, the four is two times two times frequency to the second power plus two. That's the number of balls in the outer layer when it is tetrahedronal.

When you do it octahedronal, the number comes out four. We have there is a multiplicative two here, and I take that out, so there is a prime "oneness." We find tetrahedron coming out the prime number "one." The octahedron comes out the prime number "two." And the cube is a prime number "three." And the vector equilibrium and icosahedron are the prime number "five." These are the first four prime numbers one, two, three, five of all numbers. And we find as we're going to go on here, some very interesting things, the number really goes up only to four. So it's like the four of the vertexes of the basic structural system of Universe. You get four positive and four negative, we get to the number "eight" and I'm going to try to show you that. I'm sorry we don't have the good pages and models and everything all printed out. We will come back in our video WE HAVE ALL THESE PAGES IN SYNERGETICS, SO WE'LL BE ABLE TO TAKE PAGES FROM SYNERGETICS AND REINTRODUCE THEM INTO THE VIDEO.

May I have your chart then. I wonder if I could sit on here would that? in that very, very bright light there and everybody can see it a little better. You'll find, this is the SYNERGETIC HIERARCHY OF TOPOLOGICAL CHARACTERISTICS OF OMNITRIANGULATED POLYHEDRAL SYSTEMS (See pages 46 and 47 of SYNERGETICS I). And you must remember when you are talking about the cube, in order to have a cube you must put a diagonal in its face. It always must be triangulated. These are structural systems. In other words, they are absolutely stable in doing what they are doing. And, there are a great many other items on here, but this is where we begin with the vector-edged tetrahedron, with a volume of one. The octahedron has a quality of always doubling on itself. Which, you may remember as I pump this down here, octahedron seems to occur in double bond always. You see two octahedra congruent one with the other. The more you get familiar with synergetic geometry, you'll realize that this is fundamental for the octahedron so it occurs twice, keeps showing the number "four" when it really represents the prime number "two." This doubles itself, and we find then that this is this hierarchy, and I'll go through then the vector-edged tetrahedron and the vector-edged octahedron, and the vector-diagonaled cube and so forth, and vector equilibrium. We find then that the vector-edged icosahedron, combined volumetrically with the vector-edged cube, where the cube likes to be edged this way, it's number comes out to the two come out together altogether they come out the number twenty-seven. And we find all the vector-edged octahedra and so forth, these are all beautiful, rational numbers.

Now, what I found, I spoke to you about, that Euler didn't think to do, was to identify that the "plus 'twoness'" of his equation with poles. So I find that every system always every system is inherently as he himself knew, is rotatable in other words there is a neutral axis of spin of the system. So that you have to have two vertexes have to have the function of being poles. So when I take the Euler formulas, as nobody had done, and automatically subtract two take out let's go through some of these (he's still looking at the above-mentioned chart on SYNERGETICS HIERARCHY). The tetrahedron has four vertexes plus four faces, equals six edges plus the number two. Four plus four equals eight, and six plus two. In the octahedron we have six vertexes plus eight faces equals fourteen which is twelve edges plus the number two fourteen. Or we get to the cube, and it is now triangulated so it has eight vertexes, plus, instead of six, I have twelve faces that's eighteen, equals then, it's eighteen edges plus two (= 20). We keep coming out all right.

Now, what I did was to take all of the formulas as given by Euler, and no topologist looking at this recognized some further order in it because they didn't take out the two vertexes for spin. I now take out the two vertexes for spin, and that leaves me for the tetrahedron two plus four equals six. Remember, I've got two taken out for poles. This leaves me on the octahedron is four plus eight equals twelve; the cube is six plus twelve equals eighteen; the vector equilibrium is ten plus twenty equals thirty; the icosahedron is ten plus twenty equals thirty. Now each one of these, then, is coming out in even numbers.

I find then, because they are all even numbers, I can divide them all by two. So I try that. So I get, instead of, for the tetrahedron 2 + 4 = 6, I get 1 + 2 = 3. That's a very simple kind of relationship: 1 + 2 = 3. Then, the next was the octahedron, and that had been 4 + 8 = 12, so I divided it by 2 and I get 2 + 4 = 6. Let me write those down. The first one I got 1 + 2 = 3: now I'm getting 2 + 4 = 6. Then I get to the cube and I've got 6 + 12 = 18. So I've said, I divide those by two and that gives me 3 + 6 = 9. I wish I had done it 1 + 2 = 3; 2 + 4 = 6 and 3 + 6 = 9 so what's the next one, the vector equilibrium or the icosahedron which was 10 + 20 = 30 and I divide that by two and I get 5 + l0 = 15. Now these are very interesting numbers because you find 1 + 2 = 3, you couldn't have something simpler. But the next one 2 + 4 = 6 is 1 + 2 = 3 x 2! And the next one 3 + 6 = 9 divide that by 3 and it's 1 + 2 = 3! And the next one is 5 + 10 = 15. Divide that by 5 and you get 1 + 2 = 3! So we have then, we have in every case here 1 + 2 = 3 times tetrahedron is by 1, octahedron is by 2, multiplied by 2, and cube by 3, and icosa or vector equilibrium by 5 those first four prime numbers.

We have, then, I found there is what you call a multiplicative "two" and an additive "two." There was an additive two of the poles for EVERY system in Universe. There was also a multiplicative two because there is a concave and a convex there is inherent duality of this congruence of an inside system because concave and convex are not the same. You just have to realize that you have a fundamental congruence of the macrocosm and microcosm. There is negative and positive simply congruent there, but you can't separate them. But the concave radiation impinging on concave, converts concentrates the radiation, convex diffuses it. So, and energy-wise you find that they are absolutely not they are just not the same, yet they are congruent, you can't separate them, so this is what I call then the "duality twoness." So you find every system has a multiplicative a duality twoness and it has a plus twoness of poles for axial rotation. When I take that out, then the constant there is a constant relative abundance for every vertex, two faces and three edges. And the only difference there is a prime number, that a tetrahedron is a "one," and octahedron is a "two," a cube is a "three," and a vector equilibrium (VE) or an icosahedron are the number "five." Now this gets to be very, very exciting.

Then I gave you frequency the other day, and then I showed you a series of triangles, the edge reads two, then you have four triangles the edge reads three edge is frequency. So I have there frequency to the second power and you remember it came out then alright as triangulation. So as we get into any of these, we find that they all are triangulated, so simply increase the frequency, so then in addition to the duality twoness of every system, a polarity twoness of every system (that is the plus twoness) (the duality twoness is a multiplicative twoness) a multiplicative twoness, an additive twoness then there are the four prime numbers, and everything else is just frequency to the second power times that frequency, whatever it is. This tells you all about all the structural systems in Universe. Which is very, very exciting, because then you find, because there is a duality, you do have to have the multiplicative twoness therefore you find that for every positive one vertex, you're going to have a negative one in the system, or the opposite. So I said, 1 + 2 = 3, but instead of that I've got to say 2 + 4 = 6.

That is, quite clearly, all the numbers or points in Universe will be divisible by two, and for every point in Universe there are always going to be three vectors, because there are always going to be pairs of points, then you are always going to have six I said the other day, then, there are six basic because there are six vectors always with every event in Universe you have six vectors. And those are the six each one is a positive and negative, so there are my twelve degrees of freedom I gave you the other day. You want to see how beautifully these things begin to prove themselves up and there is a very swift simplification of a great comprehensive accounting as we get into SYNERGETICS HIERARCHY. Everything coming out rational and whole.