So I began to get used to the rate at which you accumulate where you have to put something over in the next column. And I began then to see that where things looked a little messy, it was simply because there was a spilling only to the left. If you had some way to spill right as well, you might keep up the symmetry. But up to the time you get up to more than ten, so the number 7 you saw the numbers, 1,3,3,1 then the next one goes 1,2,3, with a 5 in the middle, it's a very interesting compound, it suddenly goes beyond 10 and then it begins to spill over, and then the symmetry tends to go. I'm so familiar with that, that it makes the number I recited to you yesterday is comfortable to me, and I could show you exactly, it's incredible beautiful cross symmetry of the two sides of it but some spillings over, and I'm used to how they do, so that's why I can remember it so readily.

This is the largest one that I have come to. It is interesting having done those long hand, I can't tell you how long it used to take to do it and check yourself. Now that we do have a computer, I've got some of the big university computers to working, and all my figures have been checked, and this is an absolutely checked checked figures, and I feel very, very comfortable about it now.

In the early days of navigation, then, because you could solve problems with triangles, and there was yourself and two stars or there was your ship and the horizon and the star. There were three time and again to give you some kind of a fix, and obviously they gave you some kind of angular control and you could get converging lines of angles. One of the great significance of the models I have given you to make, and the significance of people not getting things right was that they didn't seem to have them approach like this because you were thinking about perpendiculars of two quadrangles, see, they fit this way. You were trying to match surfaces by parallel motion rather than by convergence. So they were on where one side reads more than another, and you find of course, you're going this way, of course it does more. But we do not, the navigator does think in convergent angles, and I find the landsmen thinking in parallel lines they think parallels all the time way over balanced on thinking about the rectilinearity of x,y,z and just the word three dimensional. I have never seen a scientist going to the board doing a problem when he didn't say "superscript three always says cube, or superscript 2 he just says squared. If it's superscript 4, well then he says fourth power. But he does not say second power, third power he's got absolute identity of that squaring and that cubing.

So, part of my Synergetics has really been, I've had a hard time getting on with people because they still come back talking three dimensions. The word is in them even though they concede some of the points I make, they go back to three-dimensional thinking. So, when we are having this lovely convergence, and convergence does bring you to nuclei. There was nothing in the geometry of the Greeks of the nucleus. There was no inherent center it was always boundaries. They start with the game of boundaries. And so I said they accredited only the area that was bound by and the rest of the Universe didn't count, so they had an automatic bias and really a myopic bias of looking at things in a small way. I hope you like doing with me what I am doing, I was trying to understand the significance of the fixations of the conditioned reflexes that are heightened very greatly heightened by the education system to make you look at things in a myopic way, rather in the beautiful complementaries that are always there. So you can always think of the complementaries, and they're equally rational, and it gives you a chance then to be very comprehensive and to be synergetic.

Now, with navigation I'm sure many a navigator lost tools that he had, a great sea going on, and all the things washed overboard. So the navigator tries to contrive to do things in a very simple way so that those Maori and the Pacific Naga sailormen, so I said, were naked they could have things around their wrists and arms and their neck, that wouldn't come off, and that was pretty useful. They would even have things in their ears. So, these were the only pockets they had. I, I the more I learn, the more certain I am that probably those rings on their neck, and people thought of them being such simple people, that they were just children and sort of decorated themselves in some superstitious way. I think those rings were literally like the abacus. These are things that slide up and down your neck counting devices. And it could be that the person who is wearing it doesn't necessarily know that that is what the navigator uses, because it is very useful to have different people on board having different equipment if you were the navigator, so you have several pockets around different people's necks and arms and fingers.

Now, we come to days of fancier ships, and big rib ships, big bellied ships and getting into great circumnavigation such as Magellan and Drake, and the there are storms, and there are battles, and the things get lost. What was the minimum number of things that the navigator had to have with him to make calculations in a hurry, in relation to his observation. And part of the trick of helping that Navigator was to try to simplify how few of the number of things he would need. Certainly when calculations had been made, it was very important to have tables of calculations that had been made, and those were made by monks. Really up to the time of the computer coming in up to, yes even in the time of W.W.II with the great depression of 1929 and of the 30's, the big government projects in America, England, Germany of what are you going to do with the spare time in hiring people one of the things is what do you do with artists and what do you do with scientists so there were very large projects in America, mathematical projects checking checking the trigonometric function tables. That was a big undertaking, see if you could carry them out into finer degree. It had been done by monks for centuries and centuries.

The kind of tables that I first had myself were all monks tables, and everybody knew there were some errors in them. And they were formulas that were carried out, but also with all these prime numbers washing around, making numbers really where you were very arbitrary about whether you would call it the next higher number or not which side of the fraction do you go? And how many places can you really carry out things with any degree of real accuracy. What did you really know? So five place tables and six place tables five place tables by W.W.I, that's about all you had really. Then the there was all of these, the WPA in America, mathematics project, put a great number of mathematicians and scientists to work, and they did get up to six places. Strangely enough, the English and the Germans, jointly, the English Navy and the German Navy, it was Goering's idea compounded their efforts in those countries developing a better trigonometric tables. And then what's called the Edward's that you and I can get today, called Edward's tables. But these were developed by those two, and Goering with his Lufthansa wanted much better calculating capability and much swifter calculations to be made in air observations and he really did get the English to cooperate, but when the war came, they broke company, but the Germans printed this work and after W.W.II the American Alien Popular Custodian, when the United States came into Berlin, one of the things the Americans got a hold of were the German trigonometric tables, and they were published by a firm in Ann Arbor, Michigan, and it's called the Edwards Table. And they are good for seven places of accuracy.

And when I was able to afford that, this book was quite expensive, I was absolutely broke, I got my Edwards Tables as soon as I could. And that was done in increments of seconds rather than where you had to interpolate between seconds and minutes and so forth up to the degrees, so interpolation was a very important part. But I have done so much trigonometry function in calculations that I am terribly sensitive to the errors I have found as I would get because in getting into geodesic domes where I saw that I really could get into comprehensive enclosure, and I could get into omni-triangulated, and I could get into tensegrity. When I saw that I am going to bring three struts together in space, I've really got to know very accurately, two of them might get there the other one overreaches, and then when I try to put five and six together they're going to be in very great redundance, and I really had to have very great accuracy.

In the building world, it will be interesting just to talk about the world that I had been in when I built those 240 buildings after W.W.II. In dimensioning of buildings, even today, as the workmen put together, a quarter inch is a perfectly good tolerance, but if you are building bearings for an automobile you can't have anything like that. So the automobile men get down to ten thousands of an inch. In building airplanes today and the space rocketry, where mild variations and enormous velocities are going to build-in errors, they are dealing in a millionth of an inch. But the building world is still a quarter of an inch kind of stuff. I couldn't have any such nonsense as that when I really was going to get into the geodesics, so really I was out to see how I really could reduce stress in forces.

To give you a little example of the significance of what I am talking about, I was asked to design the dome over the Ford Rotunda Building in Detroit for the fiftieth anniversary of the Ford Motor Company. Old Henry Ford liked his Rotunda Building very much, he had used it for the Chicago World's Fair, he had had it moved after the Chicago World's Fair to Detroit, and it was the reception building for the Ford Motor empire. But he wanted a dome over it, and so his grandson thought that for one of the items of the 50th Anniversary it would be very nice to have a dome put over it. The Ford engineers found that it was a world's fair structure very light steel work, that it could not take the load of the dead weight of the dome. It could theoretically take the snow loads, but the best known ways to build domes, they were called radial arch domes by this time, were steel. The weights went way over what the building could take and you had to really re-build the whole building. And young Henry was very disappointed, and his cousin, another Ford had heard about my geodesics and was familiar with it, and suggested that they ask me to come out and see what I could do. So I suddenly had a call from Ford Motor Company.

And, as far as I was concerned, I was very much of an unknown at this time, and they said could I come out? And I thought someone was kidding me, of course. They said come out to Detroit, and sure enough they had a very fancy automobile to rush me out to the rotunda, looking it over, and they said, could I put a dome over it? and I said yeah. And, they said, could you make some calculations of what it would weight? And so I did, and the calculations, they didn't tell me their dilemma, and my calculations came out well within the tolerance limits. So they decided to go ahead with me, and the engineers from the Ford operating management were tremendously skeptical of this character coming into their company, and doing something like that, so this was a very wonderful operation.

I showed you yesterday struts, where you could just take sheet metal and bend it. In the world of aluminum we had gotten up to very high tensile strengths with World War II. Aluminums, as we entered this war, were 20,000 pounds per square inch, was about it. During the war, a Japanese alloy came in, it was 71 ST, we got up to 71,000 pounds a square inch with it, with mild steel 60,000-50,000, so it was very strong. It was equivalent to the kind of strength you get in the first Brooklyn Bridge, and only one-third of the weight of the steel. So it was very, very high advantage metal. I could really only get it in sheet form, and there is nothing quite so that man produces in such quantity and at such speed as sheet. Whether it is sheet steel, or sheet aluminum, or paper. So we want to really take something that he has for membranes and control of the environment, sheet is very advantageous, so I found I could take my sheet and it won't break. You can bend it and make your angles, and we got into where they were quite fancy angles they weren't just angles like that a 60 degree angle. That's 70 degrees and 32 minutes business, and they were also they had little secondary ribbings along the edge, so that the edges of the metal would not curl and so forth. So they were actually it was a "V" like this this way this thing carried out is best strength. I was able to deal in a very light weight sheet of 032 032 where you got three square feet of material for a pound of metal, and so I designed the dome with that, and then where the parts overlap, they had to be riveted.