And then it goes the negative five, the negative four, the negative three. It's just the same transferring between the northern and southern hemisphere that kind of idea.

Next picture please. Now here I've if we can sharpen this up all you can yes. I have here an array, and want to look into something on what would be the left hand column as you are looking at it. Up at the top there is one ball, and then below it there are two balls, and then there are three balls and four balls. What I am studying here are relationships between numbers of unique events. And up at the top one ball doesn't have any communication to anybody. It takes two balls to have communication or relationship. Like I said, no "otherness" no me. My awareness really begins with "otherness" because there must be some relationship. So, if I have two balls, I want to have the way we're going to say it is "How many private telephones do we have to have to talk between this "A" and "B"? You need only one private telephone because there is just "A" and "B". So two people have just one telephone. Now I'm going to have three people, so that there is "A", "B" and "C". So I'm going to have to have a telephone "AB" "AC" and "BC". I'm going to have to have three telephones in our telephone system. Any two people must have an absolute private wire. So now I'm going to have four people "A" "B" "C" and "D". I'm going to have to have and what you're looking at in the left hand column there are the telephone wires between little points. So between "A", "B" , "C", "D", I'm going to have to have AB, AC, AD that's three of them, and then I have to have BC and BD that 's five, and then I have to have "CD". It takes six. Four people have to have six private wires. This is our friend tetrahedron, it has to have six edges. The four requires six connectors. So, in fact you'll find, this gets to be a sort of fundamental model that way.

You find, then, the next there are two columns I have up there. One is the actual number of telephone wires you have, and then we put up the number, because, the first case we find we needed none , then we start with one between two. Three requires three. Four requires six. And five requires now I tell you, it is N to the second power minus N over 2. So you finally, you actually learn the equation. If I have 5, N is 5. N to the second power is twenty-five, minus five is twenty, so divide it by two is ten. You're going to find you have ten. So the numbers are going like one, three, six, ten.

Now the next one would be six. So six to the second power would be thirty six minus six is thirty divided by two is fifteen. (From the audience someone said "Do you want to draw that on here? Bucky "It would be nice if we could have this chart on here" From the audience "Because we can't get it much clearer than it is, and you could probably see it better if it were drawn but do it, do it your way." Bucky "Are you going to introduce this later then? or what?" From the audience "If you wanted to just draw it, the connections on the board you could see it a lot better" Bucky, "oh, oh, oh, I see. I think you're getting it, on your own, perfectly clearly." So, if I could go back to the drawing itself, the first one is all the people and their telephone wires, then the next was a summary of how many telephones are needed. It's a vertical line of columns, and I find that those numbers, l, 3, 6, 10, 15 are actually the total number of balls in a triangular collection. First you have one ball, then three balls, then you get six balls in the next triangular collection. Remember one, two-three, four-five-six-seven-eight-nine-ten. The next one is fifteen. So we find that the numbers of relationships are triangular numbers, as I call them. That gets to be pretty interesting.

Then we find that those triangular numbers are fascinating because, if I take three balls for instance, and I sit them on the what, the next is six. Three plus six equals nine. Or if I took the six and had it sit on the ten I get sixteen. Or if I took the ten and put it on the fifteen I get 25. What do we get? 9, 16, 25. Now these are second powers. In other words, any two sets make what we call the second power. I don't use the word "square" any more you notice. I always say the words "second power", "third power", "fourth power", I never get caught with saying "squaring" and "cubing". So we find that, here is a triangle sitting on top a triangle always one ball less, sitting on top of one, and it makes this second power number. You'll find as you take that top one and hinge it over, it lays over and makes a diamond. And when you look at that it's the diamond. "May we have the picture itself back here, because we have the, these pictures are " The diamond then, you'll see they're stacked up there. And then we get into the diamonds where, a diamond simply is a square, but remember these nest at 60 degree angles instead of 90. So it's a diamond. You can count up the you can understand it's the second power to see how the second half completes how the six completes the ten.

Now, then we find, if I stack these layers, two layers together, then I get tetrahedron vertically. So I find that whereas you and I need a private telephone, or any two of us want that private telephone, I have that in there. I could also call those the relationships between our experiences. You and a child have an experience, and have another experience. So it's a relationship with. If you begin to understand, you begin to understand the relationships between experiences.

So then, I find, if I stack the relationship of all my experiences together, in the end it comes out to be a tetrahedron of such and such a frequency. In other words the numbers of the telephone were triangular numbers, and the sums of all of which these numbers of telephones, then, were all the experiences I have had in my life. These were all the relationships between all the experiences I have had in my life. These are all of the understandings I have had of all of the experiences in my life. These are understanding relationships between those points. That's what those triangular numbers are the understandings of our relationships. And then those are individual experiences. Now, I keep integrating this set of understandings with a new one. I've just had a new experience so the final tetrahedronal stack up there is the sum of all the relationships between all of the experiences you have had. To really understand being a comprehensivist and the way I carry on with you, is very much relating then to all those relationships. Well, I'm really carrying on in a very "tetrahedronal" manner as far as the that it comes out in this beautiful, elegant, tetrahedron is just one more whatever it is always comes out a whole, rational tetrahedronal form a whole triangular form. I find this very exciting. I call this then the underlying orderliness in superficial disorder. Where the experiences seem to you and I to be very disorderly and random, but suddenly find underlying the whole thing absolute order. You cannot become disorderly. It becomes really a very exciting matter.

May I have the next picture then. I told you earlier about the two General Dynamic Scientists who were making experiments with titanium sheet. They were making experiments with titanium relative to re-entry problems in the rocketry. Do you remember that? And they had two hemispheres, one a half an inch less in radius than the other, and that they were concentrically arranged, and then they were actually sealed up, welded up at the bottom, and so there was a space between each one of them, a half inch. And then they clamped it into a frame, and this is what you are looking at in the picture. And the atmosphere is able to come in underneath the frame, into the inside of the dome. So the atmospheric pressure pushes the inner dome outwardly like that very normally.

But then they exhaust with a vacuum pump the air in between the two so the atmosphere operating in the outside one pushes it in towards the other one and it dimples in, I spoke about "dimpling in", in the same exact icosahedronal this happens to be, it turns into a four frequency geodesic tensegrity structure. In other words, I had also been giving you the way the molecules of gases operate and so forth. And so they always want to get the most economical, which means always A great circle. So they insist, not on lesser circles, they get beautifully into whole great circles. And when two great circles, remember, crossed the disk you've got three great circles triangle, and now with all this triangulation in there, there is an enormous amount of action in there, so they average out an equilateral they keep trying to get absolutely equilateral. So the whole thing just makes itself get orderly. Time and again I get so excited to find how beautifully life is really carrying on, how the Universe under all these things going on around us, and there's this lovely order.

Next picture. It's fun to see what's going to come up here! Now, we see three lines crossing one another. Now one of the great differences between myself and the mathematicians is that all the mathematicians assume you can have a plurality of lines going thru the same point at the same time. As you get into geometry where they get into the Non-Euclidean Geometry, get into the hyperbolic or they assume, still, a plurality of lines going through the same point at the same time. This is exactly what physics finds can't happen! And I say the line HAS to be an action. They have their lines going thru the same point at the same time is the point. And, I simply say to you, then that the lines, if one could go through it and then another if you had a machine gun and then another machine gun and synchronize them so that one went thru, then the other, and it might look like it was a couple of lines crossing but they're not.

Hold the picture a minute. I want you to notice then how those three lines are really crossing. And what they do then, is that one has to be superimposed on the other. So they, just automatically, do what is going on here.

May I have the next picture please. And in this next picture here, then, we see what the physicist is saying. The difference between the mathematician and the physicist is that we find that when we have two events because a line is an event there is just no question about it. You cannot have just an imaginary I proved that to you yesterday. We have an event and there is already an action taking place. Therefore, it's what we call an interference. And with an interference it could be a glancing blow, and be what you call then a refraction just a little ticking here, and it changes it's angle a little pssssss that brings about refraction of life, incidentally. Just exactly that.