I was really so terribly impressed when I was a kid by the fact that whereas that chemistry was always associating in whole, rational low-order numbers, associating and disassociating in beautiful, whole rational numbers physics was always coming out with irrational numbers. And I felt that what was really causing it was that we were really using yardsticks that were not the logical yardsticks that we came in the attic window and were trying to measure all the rest of the windows by the attic window or something. So it just was an unreasonable unreasonable story, so I feel that man, then, being fairly monological, thinking of it as a flat earth I can understand his making cubes and cubes were nice, and they seemed to fill all space they were building blocks. Tetrahedron wouldn't, all by itself, so you had to cast that out. But it was a flat earth anyway so you might as well plan on cubes, and that's the way to divide the Universe. The minute you get into the spherical you're going to realize that they are not going to work very nice, but you could have a triangle on the surface and then it went to the center of the sphere and you get a beautiful tetrahedron right there all the time.

Now, this chart goes on to get into really these complex forms that we get into here, they are all superbly accounted you'll never get in trouble, because all of them are some combination of those first four prime numbers. That's all you have to have, and the minute you get a three you know you're dealing in cubes that's all there is to it, it's always going to come out that way.

I'm going to run a few slides now that confirm some of the things I've talked about earlier, but I must ask you to imagine. The ones I'm going to use now I'd like to have first, Bob that, the half octahedra. You see two one-half octahedra and a whole octahedron. And you remember, the octahedron does have a volume of four so that each half octahedra has a volume of two. And each one of those you remember nests very neatly into the square faces of the vector equilibrium.

May I have the next slide? Now you see a one-half octahedron cut into four one-eighth octahedra there on the left hand side. The gray ones each one of those are one-eighth octahedra, and they have an equilateral triangular face on the outside, but at the center they have the 90° angle and subtended by two 45 degrees on the outside.

Next picture. Now you can see that one-eighth octahedron extracted from the octahedron coming out from the center of gravity.

Next picture. Now I'm going to take, there is a round tetrahedron and four one-eighth octahedra. I wonder if that picture couldn't be elevated? At any rate, addressing the four one-eighth octahedra the equilateral triangular faces of them which would be their outside faces when they are an octahedron to the equilateral triangles of four tetrahedra's equilateral triangular faces, and together they make the cube. Next picture. Can you see this coming together to make the cube?

Next picture. Now this time, I'm going to cut the picture out, just hold onto that for a minute. You see a great circle. I'm going to remember how I like to be sure you have a limit case, you come to the end of things I like to deal in where there is no question about our dealing in unity. And here is a circle, and it is very interesting, that a circle, you can take any two points on that circle doesn't make any difference, any two points, and it will always, if you make the edge there, it always goes congruent no trouble at all. And then you fold it and you have to half circles alright. This is a very simple kind of a folding.

I now want to do something I'm going to try to divide these in thirds, so can you see how I am taking this part and making it match as two halves, alright? Then I fold back on the other side in just the same way. I've now divided my circle into approximately six, sixty degree equal parts. Now I'm going to do that for several more great circles. Here's a half, and again, I'm going to try to make that just as even as I can between the two halves, and this fold back,, the other corner. Now this way. And do that four times all together. I'm taking four great circles. I'm taking four great circles because of the interest we really have in that "fourness" and four great circles of a plane... I want you to remember what a great circle is. A great circle is a line formed on a sphere by a plane going through the center of the sphere. I think I had mentioned to you before that the great circle is the shortest distance between two points on the sphere. Remember how I took the latitude of eighty degrees North latitude and superimposed it on the equator, crossing the equator do you remember that, and it was a shorter distance between "a" and "b" where the little circle crossed the bigger circle, much shorter distance to stay on the equator than to go off on the detour of the little circle. This is typical of the great circle being a shorter distance.

The word geodesic in mathematics, SYNERGETICS, means "the most economical relationship between events." One event would be a bird flying in the sky, and the other event might be you, and I don't know why you would want to do it, but suppose you wanted to fire a gun at the bird, which I am sorry to say many people do, if they want to hit the bird they don't fire the gun at where the bird is, because the bird is in flight. They fire where they figure it is going to be. And they find, while there is not much gravity effect, there is always a gravity effect. So that the firing is pulled a little like this, towards the earth. It may be infinitesimal to your eye nevertheless there is such a measurement, and in due course it is going to go right towards the earth.

So we have, then, the bird is in flight and there is always some wind. There is also a little inequity of the surface of this bullet and so one side has a little more drag than the other. If you take, which they often do, during World War II there were a great many photographs taken at night of two airplanes in a dogfight, where they were using tracer bullets and the picture is usually taken from another plane, of the two. And it doesn't make any difference if it is taken from one of the planes, or another plane. What you saw was absolute corkscrew fire. That is the shortest distance, most economical distance between these two was a geodesic line. And they are not straight they are always curves, waves whatever.

There would be for instance the earth revolving before the sun, very rapidly. We have a vine growing on top of the earth. And this top of the vine, growing each day. And it is very flexible, and it wants the sun. So in the morning the little stem will come out and the leaf opens toward the east to get the sun. And then as the day goes on the earth is revolving the earth is revolving but the leaf keeps growing apparently towards the sun and so in the afternoon it seems to be reaching towards the sun. And then tomorrow morning it's over here again. That's why they are spiraling the reaching this way but this leaf was always much near to the sun than was it's roots. And if you really take a total picture of it go around the total sun, revolving, it describes a line very much nearer to the sun than the rest of the earth. So these are geodesics they are interesting things.

So, the great circles are the shortest, most economical distance between the points on a sphere. Therefore, great circles are called geodesics. Now I'm taking I made four of these great circles, and folded them up, you saw me, into thirds. And I'm going to put them together using bobby pins. I'll put one to they get two tetrahedra here. And, another one. There. We now have our eight tetrahedra of the vector equilibrium, in pairs. I'm going to take just two of these you see when they sit like this they tend, really to come together in in sort of a natural way. A bobby pin there. And another bobby pin here. It's quite a neat form it gets to be. Then, put two more of these together. Then take those two and sit them on the top of here. Get some more pins. Another pair. Now these are absolutely perfect they are whole great circles and there is nothing extra in them, and so there begins to be a little tension as you begin to pull them together. There's quite a little gap there. So another pin, and sure enough the slack is in there. Now suddenly I took four great circles, and you see four great circle planes all over again. Here's one, here's one there they are. The four great circles have been, then, folded locally, so in local energy holding patterns, and we have a very extraordinary thing here where we can either go completely around, or we can go around locally with the same amount of energy. You remember those six moves that you can make; a very local holding pattern that can go on and on.