Next picture and the six great circles again in transparency. I've done them in quite a number of different ways different opaques.

Next picture please. And this is showing the mathematics with...

Next picture. Same thing again, still six great circles.

Now we are into the ten great circles.

Next picture. That's the ten great circles.

Next picture. And there are the fifteen great circles. This is a very beautiful one. Fifteen great circles are as large as we get, and the Babylonians discovered this long ago, that, I gave you you remember structural systems where I had tetrahedron, omni-triangulated, inside and outsideness a system. Octahedron, inside and out. Icosahedron. But tetrahedron has three triangles around each corner. Octahedron four. Icosahedron five, and you couldn't have six because they would add up to 360 degrees and would not come back to themselves. They could not be a system. So there was absolutely a limit of three possible cases you remember that.

Now, in the, I've lost track in coming back to my picture. Can somebody give me a help? (From the audience: "fifteen great circles" ) Right! So the most equilateral triangles the tetrahedron has only four, the octahedron has eight, but twenty is the largest number of equilateral triangles you could possibly have in the system quite clearly.. Because otherwise they would add up to more than 360 degrees this is a limit case. Now, each one of those equilateral triangles, quite obviously, you can divided an equilateral triangle by a perpendicular bisector very nice symmetry, so each triangle has three perpendicular bisectors, which will then divided it into six right triangles. There are three positive and three negative. Yes. six of them. We have six then times twenty faces l20. May I have the l5 great circles back again?

The Babylonians, mathematicians, discovered then these l5 great circles. And I want you to realize the difference between a spherical great circle triangle than a chordal. Because, look at the right and lefts in those if they were flat edged they would be hinges, but they are arc edge, so they will not hinge to the side. So you find that the concave and the convex cannot rotate the one cannot take the place of the other, on the flat they can. The positive and negative right triangles. One is red on the inside and white on the inside. You have a red and white seemingly congruent this way but in the concave-convex they can't do it, so there are 60 positive and 60 negative right spherical triangles into which you divide unity. This is a limit case of similarity of subdivision of unity. IT'S A BIG ONE! So the Greeks the Babylonians discovered that, and therefore, this is where they came to trying to coordinate time and circles. The two kind of unity. So they came to the sixty second, sixty minute. This is where the sixtiness comes from. This became, then, to them, really the top necessary number, and they included the prime numbers l, 2, 3 and 5. So I just wanted you to know that the Babylonians show this figure in the old things and it is very exciting to see it.

Next picture please. This is the fifteen great circles folded, and you find that they are folded, there are fifteen of them but you will find that they make a total of l20 triangles. Each one gets folded into divide l5 into l20, what do you get? Eight. Yes. Well, each one of these has to be in a special fold. You'll find that they are not these are each very nice and symmetrical they are bow ties, just as neat as can be. This is the model I have used different colors, I've used yellows and blues and blacks and so forth, and the model these are strange kite-tales, where the one tetrahedron is edge to edge with the next tetrahedron, and they come together to make tetrahedron spaces outside of themselves and inside of themselves. So each one of them has four, and each one of them has two no it makes up four on the outside four inside and four outside of these strange things, and they do not come together in a symmetrical manner. It is absolutely impossible to make them symmetrical.

The icosahedron has these very interesting, very independent properties where it seems to peel off. And Vector Equilibrium is where everything really is passing through Unity and from thereon everything that goes on is some kind of an aberration a folding up, or a skewing, or whatever it may be.

Next picture please. Looking at the fifteen, same

Next picture 120 triangles.