I've really had very distinguished mathematicians and they never catch it. My grandson and I, he has accompanied me for a whole year around we went to fifty different Universities last year. And we were at Rhode Island, and there was a little girl, the professor's daughter. And she was getting quite good at school, she was about 10. And I gave her a model of this. We had to leave, and she couldn't get it, so I showed her. My grandson said, "Grandpa, you cheated that little child." That's just what you've got to find out for yourself! If she were given the chance, she would have found out, he said. You cheated that child of the right to find out. And if you did find out then you would feel the 60 degreeness you have to explain it to yourself if you found out for yourself. But just telling it like that you lose the whole beauty of it. So, this is one of my most beautiful lessons I've ever had. It's still part of that how to get on with that child. I was amazed how my nonsense of wanting to feel gratification really, you show off to the kid. So the big thing is here, the 60 degree convergence. Everything is converging and diverging. That's the way nuclear things are they converge and diverge. They don't parallel in. And all of humanity keeps working on parallel lines and cubes and squares. That's not the way Universe is. So when you begin to think "convergence" "divergence" then things really fit.

Now, next picture please. Now, there's the one I just put together. Next picture. Now you see a really very big one. This is a very exciting one. There are, count the edges, 1, 2, 3, 4, 5, 6, 7, 8. That would be seven frequency.

Next picture. Take these two apart. There they are. And the two parts are hollow. They really are very strange. This is even more provoking. When I make this one up it's so big I don't want to try to stop here to do it, but there are in each half there 50 balls. So the two balls coming together have exactly l00 balls. Because they are hollow. The do really surprise people. Now what they have inside is exactly room for a four ball edge which is a three frequency just goes inside. And it has, we may remember, exactly 20 balls when I put this together. Four times five. What fits inside the one that's up on the wall is "twentiness" and the enclosure is 100. So I could get, and we know there is no ball in the center of this group. Remember, there is a tetrahedron there. It's center is absolutely space. So twenty ball is absolute space in the center, so I can get 100 on the outside, I can get 120 balls around a common vacant nucleus. This is the largest number of balls I really can do in a Then, next thing, if I want to have a nucleus I'd have to put one ball layer on the 120 and then you would suddenly have a nucleus.

Next picture. Now, you see something else rather interesting. You remember that I showed you the nestability of the tetrahedron. I can make a three ball edged triangle, alright. Now, that is nestable. So fasten that one in the center. I'd like to make another one of the same to make another one just like that. That is a second case nestability remember? I told you it is a one-eighth octahedron. And if you're making a one eighth octahedron, you'll find, sure enough, the angle is exactly right. Then I'm going to ask you to do something logical with these two pieces. Something that feels good to everybody. We have been talking about precession. How could you precess? What could possibly match there? You match the little faces of the tetrahedra, somebody else want to try? What else could you match. The triangular faces. I have something to tell you. When you finish you'll know you're right. You got so close to it, it makes me I bet she sees it. Precess. There it is, put it down on the floor. It's a cube. You got it. These pins are bad. Come near me, I'll fix the pins. That's a cube a beautiful cube. Just put it down on the floor if you can. (The young woman who had been experimenting with Bucky says "I didn't even recognize it." That's because those angles, you remember were the right angles. It's a quarter-eighth and one ninety degrees in those corners. It's one-eighth octahedron. So two one-eighth octahedron give you the cube. They will not do it just by themselves in planar, so they've got to do it with only in the balls, with vertexes.

Next picture. That's going to be up in here in the show. Now you see a big cube, so there are big cubes, and you'll find that on each corner of the big cube is one of those one-eighth octahedra. And in the big cube those corners have been put onto a vector equilibrium.

Next picture. There is the vector equilibrium. You are taking a one-eighth octahedron off of the corners of the cube and there is your vector equilibrium, and it is a, count the number of balls at the edges, it is 1, 2, 3, 4, 5, it is a four frequency vector equilibrium, and that really is the limit case of the nucleus. I'm quite certain as we get into the post-uraniums, because you get outside of the 92, but this arrangement still is the nucleus.

Next picture. Now you are looking at an icosahedron made out of balls, but you have cut the balls so that they are down to the planar side, so that you can see what the balls look like together. That is the icosahedron where there is no ball at the center.

Next picture. You'll see when you do the counting whether it is the vector equilibrium, or it is the icosahedron. You'll find if you cut the balls away like that, the poles, the plus "twoness" is quite clearly, is a different color than the others. And the other faces get together where I gave you that three come together with two, or with one, or whatever it may be but you'll see all the triangles coming together and giving you the second power area, where the numbers of balls in the outer layer will be frequency to the second power, and then the poles, plus two. And you'll always see the balls are right there. The count is there every time. It is really a very beautiful thing.

Next picture. These particular models are supposed to show it to you but some how or other, they have faded away. You see the icosahedron there. This is where you get the multi-frequencies, and I want you to understand how well they work for the single just the plane tetrahedron, or where you see then twenty face icosahedron, they break into ten diamonds. You remember how the diamonds, then, compliment. You are looking at the ten diamonds grasping each other here. The blacks and whites. But it's always one extra north pole and one extra south pole. They are lovely things to make the count.

Next picture. There it is, that's vector equilibrium. And, the poles have been identified there. This coloring is really quite badly faded. That is the same one you saw as vector equilibrium becoming the icosahedron. But it cannot have any layers inside or it will not be able to collapse to do it. No matter what the frequency is, the icosahedron closest packed surface it can only be one layer. And I'm quite certain this has to do with it's "electronness", but the icosahedron's electronness cannot have the nucleus, but it has the same count as the vector equilibrium which is the nucleus.

Next picture. Over, could you put that over, I guess that's alright. You're looking at pictures made by Linus Pauling, or rather models that he made from his Nobel Laureate book paper. There you see, one of the things he does, is take the vector equilibrium, and he takes the top three balls and rotates them, because remember there are six nests on top here and we only use three. Three alternate ones are what you do. The minute you rotate, the top then becomes polarized. It's absolutely omni-directionally equilibrius. Until you take the three top and rotate them it becomes polarized. And when you take the three top, so you've got 12 balls , three balls at the bottom and three balls like that, then you get around the equator you get pairs of squares, pairs of triangles, pairs of squares, and then a triangle and triangle on the top. So that it is completely polarized, and when you make a section through the polarized what used to be the vector equilibrium , it's no longer equilibrius because it is polarized, then a cross section of it, is the chemical hex. So I want to bring you into proximity with other phenomena that you All of hexes have to do with polarizations where things, I said you never will catch nature in that vector equilibrium, she is always going to be in the polarized, she is always going to be offset one way or the other.

Next picture. These are more of Linus Pauling's pictures. Next picture of polarized sets, how he could bring together "threeness" in various positions all polarized. Now the top one is a vector equilibrium and it has a red ball inside, and you knock the red ball out of the center and it becomes the icosahedron. You can make a rubber model like this, and have it fastened together with rubber bands pull the middle one out and it would immediately snap back into the vector equilibrium the icosahedron.