This steel band is a very nice and flexible one rather thin, and clearly marked off in these basic increments. And then, I'm going to take three such steel bands, and putting a hole in the end of the steel band. And I'm going to put a line or a rivet in there that has a stovepipe through it so it makes sort of a journal. I'm going to put a rod two rods one through this end of the steel ruler and a rod through the other. These are very powerful strong rods, they don't bend. And I'm going to take then, these rods, exactly perpendicular to the ends of the steel ruler. I going to have them protrude clear through down deeply and I'm going to take hold of the ends of those rods, and bring them together, and you're going to find that it makes the steel bend very beautifully, and the rods come to a common center here. So it makes kind of a circular triangle with arc radius. Now, I'm going to take three such bands, and I'm going to where the corner make a triangle with them, and bring their corners together, and again have a swiveling rivet go through the hole of the corners, and these powerful rods can come through. We have three rods and three corners, and the thing is standing up with a flat triangle like that with these three legs down. I take hold of the bottom of these legs, which are stiff, and bring them together. As I do so we found that any two of them coming together made the single band bend, didn't we, so all three bands have to bend, because there are three rods, each one any pair of them with ends at any one steel band. So as I pull them together, all three steel bands bend. And as they do so the farthest any one side of the triangle with the bend rotates away from the opposite angle, being due to the rods, also, being brought to a common center because they come to the center of gravity. And so we can see that it makes each one of these arcs bend outwardly and makes them able to do a rotating away from each other. We, then, really get at that spherical triangle that I had I simply took those rods, those powerful rods, and brought them to a common universal joint, and we're able then the bands in this case were a delicate, very powerful aluminum high alloy 73 ST Aluminum, so you could go, that really was a sphere showing itself here, and as the rods went northerly and southerly, it kept just embracing the earth with the spherical triangles, opening a little further and closing a little further.

I want you to understand then, how, then you are able to take a band of even module absolute module and make it into the spherical. This is just to get your own confidence that the edges of the triangles which are done that way are exactly that way. It would be possible to take a light at the center of the sphere and project through a spherical icosahedron, but if you did to a planar paper on the outside, you would find as it went through the arc to the plane, the angle would begin to open up so it is not uniform module scale uniform boundary scale. This is absolutely uniform boundary scale and it is the only projection in history where you don't break open your package. In the Mercator you are breaking open the top. You always have something open-ended. You have a line of true reference and a triangle that ends, but in this one the line of true reference is continuous to the triangle it never breaks open always contains and brings about the symmetry.

I want you to really feel quite confident about what goes on here. So this really amounts to, it's really topological transformation and not a shadow graph. So the word projection would have to be a mathematical projection but not a shadowgram. It's a true projection all right, but it is a mathematical transformation and not a shadowgram. Now I'm ready to break.

(Break)

As we get into the techniques of the medium that we're dealing in the videoscope. And we have a great supply of slides, as you know. One of the things we learn is that the video frames the picture frames are just a little smaller than the 35 mm slides. So that, what I usually like to show vertically may have to be shown horizontally. And I'm going to review really quite quickly those great circles that I had, you remember the pumping of the great circles. Instead of them being vertical like that, coming down, it's going to be horizontal. So, may I have the first slide?

And in this, I think you'll enjoy it a little more. It's a good idea to get this feeling about spherical trigonometry. You see the triangle up at the top? Mounted horizontally this time. We're dealing in about a 72 degree angle up there.

Next picture. And here we are down to 120 degree angles.

Next picture. Now we're at 180 degree angles.

Next picture. Now we're going into the negative spherical trigonometry down in the southern hemisphere, down to about 90 degrees.

You really feel that transformation of how a triangle can rotate this way, and change it's we've had absolutely uniform boundary scale the whole time and just that we changed the angle the angle is variable.

Remember when we were talking about the necklace structure when I was getting into structures? We found that the struts didn't change, only the angles changed, and I look for, what are the varying things. And you begin to get feeling very strongly about angles. In fact, I've discovered that you can describe all designing can be done with just two phenomena. One is called angle and the other frequency. I'm going to have an axis of reference. So in relation to the axis of reference as I said a vector I'm going to go off deliberately at the start of this angle. I'm going to go off like that. And now, I say it's angle, so I may not go off in a plane, I can say my angle goes this way the point is I am going to go for so many frequencies for so many frequencies this way, and then I change my angle so many frequencies change the angle, so many frequencies and so forth. All you do is change the angle and frequency, and finally you can outline the shape of anything you want. I think that might be said another way. You just said I understand angle alright, but you say "measurement", you go so far, but I use the word "frequency" for "measurement".

Now, next picture, please. I'm going to go thru more and more of my slides. Here again we are showing something horizontally. Up at the top you see three possible structures. The tetrahedron three triangles around each corner. Octahedron four triangles around each corner. Icosahedron five triangles around each corner. And at the bottom of that array, you will see a tetrahedron it's about by my shoulder here. And then, it's not really very well done, we have a series of the corners; the corners with three triangles coming together; then it goes four triangles coming together; then five and then it goes six and it's a plane. You can see it's a plane. That's why it can't be part of a system, because it doesn't come back on itself.