EIK - Session 6 Part 6
Then came a day, about six months after that map business, he wrote me a note and he said "Bucky, you're right." he said "You didn't know this, but ", excuse me, this occurred before the map, and he had heard me on these things before, and I had made my prediction about the English in NINE CHAINS TO THE MOON published in 1938. And Henry, while I was on FORTUNE between 1938 and 1948 this note came from Henry saying, "Bucky you will be interested to know that the British have just sent their secret archives to Ottawa, and he said you really are right, but you didn't know the war was coming." And I said, " Of course, I knew the war was coming." But the point is I, how it occurs, and why, is irrelevant. All I say, I'll say it to you what is going to occur, but what are the actual critical factors that make it do it they are very happen stance. They are utterly unpredictable. The big things are predictable.
I think it is important for you to have the feeling of some of the personal experiences in my life that are related to conducting myself the way I am, to give you a sense of confidence yourself.
Now, this is not a bad time to tell you about how the method of projection works. Just thinking about the well-known kinds the Mercator is a very simple thing. You have a cylinder of paper around a globe. And this cylinder of paper is tangent to the globe at the equator. And you have a globe, let's say it is transparent, and there are the outlines of the continents on it, and there are, on the globe there the latitude and longitude lines, and we have a light at the exact center of the globe, and so the lines whether it is the continent or the great circles, make lines on the paper. They are shadow lines. So we see that the light coming from the center, that where the cylinder is tangent to the globe the things are very accurate absolutely correct. So the Mercator is exactly correct along the equator.
Then, as you get further and further north, the angles get wider and wider, and finally the cylinder paper gets where the light at the center, it will never get to the edge of the cylinder, because the axis of the earth and the cylinder are parallel, so there is a hole in the top where you never will be able to get any shadow projected. So the further north you get with a Mercator, the more and more distorted things are. So, all the Mercators are actually "fudged" there at the top. They deliberately pull down the data that they know about and just changed the pace.
Then we have something called polyconic, and the polyconic is what most of the important charting will be done with. You have, instead of a cylinder, you have a cone. And the cone could just touch the earth at one point, so where it touched would be accurate, and again, as you go away from it you get inaccurate. But the polyconic, we have a cone of paper that goes into the earth and comes out of the earth again. Can you see? A sphere where there is a cone coming like this and it penetrates here, making a lesser circle, and then it comes out into another circle, can you see that? So, where the two circles occur, they both are accurate, so the intervening space is less distorted, so the polyconic has been by far the least distorted of all the methods of projection in the past.
And this way you have the light at the center, and this is hitting on a piece of paper you have what you call a polyazimuthal. And so, you may get a circular mapping very accurate at the point of contact, but it gets more and more distorted as you go further and further away. The polyazimuthal, very much so.
There are other modifications of what I am saying, but those are the main classes. Now, common to all of those, wherever the sphere touches the sheet of paper, everything was accurate. And as you went away from the point of contact or line of contact it increases in error, quite rapidly.
What I did was to take the globe with a spherical icosahedron. I used an icosahedron because it has again, then, the largest number of identical triangles. You remember that as you get a smaller and smaller triangle, the sums of the angles get less and less. If I'm going to want to have something out in the flat, then I'd like to get as near to the flat condition as possible, so that there are 20 identical equilateral triangles in the icosahedron, and that is the largest number of absolutely similar forms. You remember we saw yesterday that you can divided that into 120 small right triangles. The amount of the sums of the angles of a spherical triangle add up to over 360 degrees. It's called spherical excess. When the surveyors survey, they use, always watch out for this "spherical excess." So I would like to have the minimum spherical excess.
For instance, if I were to use the spherical tetrahedron, it's corner angles would be 120 degrees each, so I would have 120 degrees spherical excess in my triangle. Alright? So, if I use the spherical octahedron, the next system, I would have 90, 90, 90 270 degrees above 180 degrees, so I have 90 degrees spherical excess. And getting down to the spherical icosa, each of the corners is 73 degrees and 26 minutes. So I have the, so when you have the five come together around one, you divide 360 by 5 and you come out 720 is an equilateral triangle, so 720 goes to 60 each corner. 72 at each corner, a spherical triangle icosahedron has 72 degrees at each corner. So that when you flatten it into an equilateral triangle each corner is 60 so that there is only 12 degrees spherical excess at each corner, and a total of 36 degrees for the whole total triangle. In other words, it is the one which has the least spherical excess to start off with. So I said, that is an optimum condition, I'm going to have to have it beautiful, because the excess is divided three ways which is symmetry. So what will happen, is because I have twenty of those triangles then, with three corners each, so there are sixty packages of l2 degrees each around the map that will subside to 60 degrees, and that is really an invisible amount, because the total triangle subsides. It means that everything shrinks symmetrically, so it is just the interior of the triangle shrinks a little faster than the edge the edge, absolutely no change at all. Because where the edge is is true contact, so I have by far the most with the thirty edges of it, 63 degrees and 26 minutes each you're going to get thirty times 63 degrees thirty times sixty all this is absolutely true. As I have by far the most great circling, it is absolutely true to start of with, and I am going to have symmetrical subsidence locally, and the beautiful thing about it would be that where it is contacting this triangle itself that is all true. So all the change is internal rather than external. Now, when you make two circles, one of a circle radius of one and another of circle radius two, you find that you dismiss your error outwardly in the circle radius two, the area of circle radius two is three times the area of circle radius one, so if I dismiss my error outwardly, I have three times as much error as if I dismiss it inwardly. So, there is no way that you can get a better condition on the spherical excess than on the icosahedron.
Just thinking about conditioned reflexes of human beings, I spoke at Harvard University two years before we published the world map, and I had all the pieces with me. And the mathematics department asked me to show my method of projection. And, I did, and he asked me if I'd come home to tea at his house with him and bring the pieces. And he had some children, and he wanted his children to play with my map and see what they would do with it. There were the pieces on the floor, and they began to realize how to put the edges together so it would work. And he said "Darling, you have the world upside down." And, of course, there is no up and down in Universe, and the mathematics department man was showing how ignorant he was and the children felt absolutely comfortable, because there isn't any up and down in the world. It's just how you want to look at it. And the kids had this freedom. I just want to stress this. It wasn't a matter of the children being ignorant, but how quickly the reflexes can be mis-conditioned if people he was a Ph.D., beautiful, extraordinary man
Now I'd like to go into, watching our time, I think it is time to have a break and then I'm going to get a little more into the mathematics. That will end the map for the moment.
Are we still on? If we are I would like to carry on for just a second. There are a couple things more to tell you about that method of projection. I'd like you to feel this with me, and construct a little model with me in your mind's eye. I'm going to make a steel band, just nice and evenly. You can call it a steel ruler, and you can have inches or whatever you want.