I made a diagram last summer of the trigonometric functions, and I thought I would complete it for you, and in trying to get it run to your head, because I am a little slower at looking at my paper and putting it on the board. But when I finish it, I think it is going to be useful to you. Remember then that we are always starting with Universe and then we're subdividing, so we start with a sphere our geometry begins with a sphere, or at least with a system, and a sphere is simply a high frequency omni-structural system. The sphere is a high frequency structural system all triangulated. And so it would be approximately unit radius, high frequency structural system. And I'm going to go through some of the arguments of the geometries of the early days, about how you play the game of geometry of Egypt and Greece which is really worth our feeling here, because they took for granted a flat earth, so they started on a plane. But if I'm going to play the game, in starting with totality, I've got to say, "What do I have to prove things, and so forth," and I think you're going to find that it comes out really very satisfactorily.
So, I'll complete my picture here, this is then a section through a sphere. Radius and we just make a working assumption radius is one. It is sub-frequency, it is just unity. Radius is unity. And I have the angle that is considered is called theta, that's from here to here, and this is half theta, "A" is half theta. And it gives, when it's you can have a right to have a perpendicular bisector of the chord, so we can assume it is a 90 degree angle so it is a right angle out of which we then come into trigonometric functioning.
(I have spoken publicly for years and years and years. I have never drank water at speaking till I came here, and always, I was a long-distance runner, and I found that nature makes her own juices, and has her second winds and things, and I know that you don't drink water when you're running you let nature do that, so that I'm a little puzzled by this I think it's a tale end of a cold, I had a very bad cold in late December, and came back from Europe I'd been speaking in Europe and I came back from Europe to speak here in Philadelphia, and I literally couldn't speak it was the first time it ever happened to me. And, so this is some kind of tale end of something going on there, I don't know what it is.. I'm sorry about it.)
Now, coming back to our diagram. I want to complete it, and it will just take me a little bit to do so. Now I want to make a green line here, and I want to have a (there are so many things to keep in your fingers!) and an orange line here, vertically, and I'm going to have a red line this line DB, E up here, and I'm going to make a diagram of that here. I'm just sort of making a side picture, where it's in the diagram on the side that seemed to be useful...(Bucky is drawing this whole time). Now, what's missing here, there's a M missing here. I hope that's clear, this goes there. Now, I think everything is on here, and we are talking about a trigonometric angle and this is angle theta. So she is always isosceles, and then to get the advantage of a right triangle, and all the laws of right triangles, we then have this perpendicular to the chord, and we get another half theta, which is A. And that is the angle we always do our checking at this is what we're checking up on all the time.
And, so, I'm now going to mark what these are. This orange one over here is FD F and goes down to D here. So FD is cotangent. It actually says all these things right on the diagram, but I thought I'd separate them out too. F and O always the center of the sphere. This is F and O and it is your cosecant. And the red one here has the E and O for me here, and this is your secant. And then, E to B is tangent, and M to K is the famous sine s-i-n-e. And this is your cosine. Those two sine sides of the right triangle, the sine and cosine. And so they are of this famous angle half theta, A, the one we're looking at. O.K.?
I think this makes things look fairly simple for you, in looking up you know what you are really finding out. The data you get will be either expressed in degrees and minutes, it could be in sometimes they do it in hundredths of a degree and it can even be done in radians and so forth. But it doesn't make any difference which language you use, you are multiplying so your formulas will be multiplying and dividing one of these things times another all the way through to find out various trigonometric phenomena. I think this is quite, quite a simple diagram, and it was not given to me when I was young, so I really worked on something to have it be seemed to me pretty easy.
Now, I'd like you to think about the game of geometry as played by the Greeks and the Egyptians. They played the game with straightedge dividers. So with your dividers you can strike circle, or you can measure distances straight edge and dividers. You had to demonstrate an unfamiliar geometrical form consists of demonstrated forms and terms of your original constructs. But where as they played the game of making using a straight edge and then taking any point on the line, put the dividers on there and they can strike a circle. And it will cross the straight line at two points, and those will be the radii and they know they are equal to one another, and they are half one diameter. This is really a very simple game.
They could then, from the where the circle crossed the straight line they could go out there with their dividers to the end, put their dividers on the crossing point and strike break into the circle from either end, and they never could prove that the space between was the radius. Therefore, they never assumed the equilateral triangle. And starting from a plane you get into that trouble. And I don't get into that trouble. I'm sure that's one reason why the 60 degreeness has not gotten into the popular game of geometry. A great deal seems to be self-evident, but that's all. And you knew that the dividers seemed to span exactly, but you couldn't prove it.
So , I'm now going to start and, I am going to say I am working on a plane, I am going to make a sphere where I have my dividers and I have a point in Universe and I have a radius going in all directions. O.K. So I now have a surface to work on and I know all of that surface is equidistant from a given point. And I'm going to take any point on the surface of the sphere, and I'm going to strike a circle on the surface of the sphere. You and I call it a lesser circle, but we know that every point on that circle is equidistant from that point on the outside of the sphere. We also know that every point on that circle is equidistant from the center of the sphere. That's quite nice, so now we have quite a little of two kinds of things we have a radius of the sphere that is known and we have the radius to the surface circle which is known. They are both the same divider.
I can now, then, take any point on the surface circle dividers, and strike another point on that same circle can you see that? Now, I'm going to go to more or less the opposite side of the circle. It doesn't make any difference where I come on there some other part of the lesser circle and I find quite quickly with my dividers that the lesser circle really seems to consist of quite a number of increments of this, so I am going any opposite side where it doesn't make any difference, I just take another point on the lesser circle, on the surface, put my dividers on that and strike against the lesser circle again, I get two increments there just look at the globe of the world and I've got a circle up here, but this one here, we'll say, which is about the same radius of it, and I come to an arbitrary point out here, and I strike on this circle here. I now know what that distance is. Quite clearly if I try stepping my dividers around, it takes quite a few divisions, so all I have to do is to go to the opposite side to pick any point I want, somewhere in here, divide it and strike another. I now get two arc sections of that lesser circle, that I know are going to have a unit radius distance now, which is the original radius of a sphere, has the radius of the circle around the surface, and I've got two increments here which are exactly the same. I now am going to run a line back to the center surface point here and two lines from there. This now gives me two triangles on the surface that I know the radii of the edges of those triangles are all equilateral. I know I've got two equilateral triangles with a common apex at the center of the surface circle, which goes down to the center of the sphere. And we know they are all equidistant. So then I was able to take those, and you may remember the way I made the four great circles.
It looks kind of familiar, because I want you to look for instance, here is the sphere line. I got on the opposite point here, and I struck a circle, now I've got these two units here, and all I've really got to do with them is remember how I made them what I have is this and these are all radii, all radii, everything is radii in here that's the reason I constructed it. So there is nothing to stop me rotating around this radii and coming down like that. Now, I have a, I have opened up my I know all those edges are identical in length. And they are all freely rotatable about this, so I now have a hexagon. These are all chorded. I am able to extract the hexagon, then from the sphere, and I really know what it is. Is that comfortable to all of you here? Remember how I did, I first used my dividers starting at this point in developing a sphere, omnidirectional from a point. And then from that I go to the surface of it. And I know every point on the surface is equidistant so I take a point on the surface and I struck this circle out here a surface circle. Then I took any point on that surface circle, and struck over here and got that length. So I got a chord, this is a chord, this we know, we struck that I know all those three points are equidistant from one another now, because they are all constructed that way, by my dividers, the same thing over here. And they don't have to be, you know, diametrically opposite, please understand; because you've proven with your dividers going around that this circle broke up into much more than two, it broke up apparently into six. So I just take, really quite randomly, because this is a hinge, and these are able to hinge exactly like that, so I now have constructed two tetrahedra, edge to edge. I'm able to open the two tetrahedra out in the flat, and now I find them actually consisting then I know I have a hexagon, which every part has been constructed by the dividers, which is a privilege of starting with the whole and working to the particular that you couldn't get to this is synergy the behavior of that whole. It did not permit it if you tried to start with the part. Is there anyone here who can find any geometrical fault with what I am saying? Janet? You're thinking good and hard. Does it seem alright? Good. I haven't found anybody who has ever found any real fault with it. But I'm now really privileged to think tetrahedra. And you can really understand how from there on I can re-establish these four great circles making these four great circles really a part of my game of geometry. It is a very beautiful thing, that you know that you had the circle, so everything is absolutely superbly proven all the way through. Now I have a very nice kind of a geometrical situation where I can then use the diagonals of these squares and so forth, and I can get my x,y,z coordinates. Everything all everything that you are familiar with can come out of here from now on in those great circlings that I gave you. So all the important coordinates that man has ever used are manifest here and constructed so that if you want to take things out in the flat, you are extremely comfortable about what the increments are.
There was, doing spherical trigonometry, which is dealing in the laws of that right triangle it is a beautiful thing right triangle because you by doing so you have one thing that is known. And that is really great. Then, you found that there was sort of constant variation between the whole, it added up to 180, so that if I make one angle smaller of the non right angle, the other one is going to be larger, and the sum total is always going to be 90 degrees. So you've got two very important things known. You've got a known ninety degrees, and you've got a known sum of the other two add up to 90. And they are going to be varying they are going to co-vary with absolute arithmetical accuracy. So this brought about the development of the trigonometric tables, and as I gave you yesterday, the maximum variation you really can get to is then where you have an isosceles 90 degreeness, and it is an isosceles and the two corners are 45 degrees each. So forty five is the inasmuch as they co-vary, 45, the largest gets to be 45 and the smallest gets to be 45. And inasmuch as you have learned that that is so, all you have to know from now on is up to 45. That's why I gave you yesterday all of the prime numbers. There are some people who spoke to me amongst you about the prime numbers, not quite certain why I carried two to the twelfth power, and three to the eighth power and so forth. I simply find that there is, then, trisecting that goes on. You need threeness and twoness, and as you get into multiplication, you need a whole lot of twoness of the prime number two. To give you great flexibility in complex computations. But you don't have, very often that the prime number 43 comes in. One of that is enough to be sure to accommodate, and if it is in the dividend it will accommodate all but there is quite a number of occurrences of the 2's and the 3's and the 5's and you can understand that quite readily. That's why I have many powers of those numbers, and I did not really know, and I haven't got to the control of it yet, how many you would need very certainly; but I what I did was to keep going through and developing, multiplying powers times themselves and sometimes doubling and tripling to see if I came to any interesting numbers. And if you'll look at your book SYNERGETICS you'll find that ever so often the numbers are very regular, and it suddenly comes out absolutely 9 million or whatever it is. There is something incredibly beautiful simple number. And you say, Nature has come in she's clicked here. She's deliberately made it very simple sublimely simple. So I began to give these names that I call sublimely rememberable comprehensive dividends. I kept exploring and multiplying, I have been doing this for years and years, and every so often she came into phase and I've got a listing of those. I used to do all of these things long hand. There wasn't any computer to do it with.
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.
It had been learned during W.W.II in the riveting of airplanes, when there was an enormous production of airplanes, rivet holes were of the greatest importance. So parts would be made here, and parts would be made there could you brings parts with holes punched in them and bring them in great complex numbers, and still have them all register and have rivets come in. Some of the production tricks the Germans and the English had developed, the United States inherited a lot of these techniques. With extraordinary kinds of controls of light focusing on metals and so forth and using light sensitive things that punched and so forth. At any rate, they discovered, thinking about what kind of strength differences do you get in the airplane where you have two holes and they are a little off like that and the rivet has to be then small enough so it can go between the two because there is a little slack in it. Got into an enormous amount of testing laboratory testing of the relative strengths arrived at where as you got better registering of the two holes as far as the center holes go and the hole sizing, and they found that the strength went up very, very rapidly as you get to greater and greater accuracy both of positioning and the sizing of the hole and the sizing of the rivet that slid in.
For a very simple reason. If rivets if there were some slop in it, it could be, everything was fastened down good and tight, but a big stress on the thing, if there were a little more slop opening in this particularly direction, suddenly she starts to all the thousands, you get a little slop in that side and it would yield in that direction. This starts a shearing action and things begin to go. So if you don't shearing kind of starts it's very much like having a pole that is balanced, you have a pole balanced in your fingers and there is nothing really to keep that no effort to keep that can you keep on that dead center. And if you don't as you get off that, it takes much more effort to withstand, and keep where you like it. So, when I did the Ford Motor Company dome, I also found the best people in the Ford Motor Company, by far, were the tool men. The men who made the mass production tools they are the brownies of Detroit. No matter how the management they are the ones who get things tooled up and get things ready for three years from today when they start selling things. But they are the people who really are keen. And the tooling men liked what I did. Anyway, I designed the rivet holes for the there were some thousands and thousands of these struts many, many thousands, some fifty thousands or so, and they I designed the rivet hole positionings, and the rivets themselves. We kept the tolerance to l0,000ths of an inch! Now bringing a l0,000th of an inch into a big building where the quarter of an inch had been fine, you can imagine the general contractors when I faced them, they said you can't have any nonsense like this, so I said "I have designed all of the tooling, I have designed all of your logistics, as general contractors, simply, I will handle the whole thing for you. You really have to do nothing but just write out the checks and pay the bills here." So I had to put on a show for the Ford Motor Company in my office I took an office in Detroit, where I gave them the complete logistics, the complete design of every part, how every part was to be manufactured, how they would be organized at the job. How it would be assembled the roof, what workman would do each job and how many there would be. I had set out the absolutely whole thing, and the general contractor said he had never had anything done like that before. This was the kind of work his estimators and engineers had to go to work and do, and he found it all done for him. So he was really very agreeably surprised, and that helped me a lot.
It was an operation that was very difficult because the Ford Motor Company had already been spending a lot of money getting things ready for the 50th. They actually had spent, had invested over 25 million dollars in TV shows and the big show they were putting on. And so they didn't want the work they were already doing being messed up, with something going on over head. So I had to build a bridge across this whole thing and I built I used a high-voltage cross-continent mast below to and at the top of it with this bridge going across, I introduced this hydraulic arm that went upwardly, and revolved around. We mounted the dome they were very scared about the danger of the men working there. So all of the it was an enormous steel arm that went around with the dome on it, and people could reach it from the bridge. They didn't have to go up the scaffolding anywhere, and we were able then to continually revolve the dome going up and they kept adding onto just the bottom of it from the bridge, so nobody went aloft, until it was all up and then we let it down onto the actual parapet of the roof, so all of that had to be calculated very accurately. There would be no stresses in that. And the roof was not very beautifully done, so I had to allow for all kinds of come and go, shimming of the thing.
So at any rate the work that I laid out, there were a certain number then of hole pattern. There were six prime variations, and the Ford tooling men made up Class A steel dies to stamp out the parts and to punch the holes. They were able to pre-punch the holes in the flat, before they got into the bending. But everything was under such tight controls, that I kept this down to a ten thousandth of an inch. Now, the interesting thing is that the dome weighted exactly one half of what it would have weighed if it had been laid out by the best sheet metal workers in the world at the tolerance a sheet metal worker can lay out. By I had absolutely under these invisible tolerances. And the dome was you could have exactly two domes for one the same amount of metal, simply by that tolerance difference. That there could be that when you get that kind of economy inherent in competence you can really understand the reason why I would feel that I'm not putting it upon you to take some of your time to begin to get into the subject of the geometry and how we do good calculations and so forth.
Now I am going to come back to man, then as a navigator going around the world and trusting this great ship and all the people in it, and the enormous commitment to this navigator to get you from here to there, going in unknown seas, and you don't know where the rocks are, and you're sailing at night you can't stop the wind blowing in this direction. It's a very hazardous undertaking. And so Napier in England developed some very beautiful rules in trigonometry to simplify the navigator's problems so that he couldn't make what we call a 180 degree error. Because of these quadrants, these x,y,z quadrants you're going around, it is really very easy to make a plus or minus error, and find yourself going exactly the opposite direction from where you ought to be going.
So he simplified this out, and then developed a game of calculation trigonometrically which you could find in Bowditch's Practical Navigator and which was the Bible of navigation for the United States Navy, for the Naval Academy. And you'll still find it there.
But I'm going to give you these rules, because I found it very useful
All through the years immediately, just before the Ford Motor Company Dome a few years after, I found myself being invited all around the world to architectural schools, whether it would be in Ghana or it would be in India and so forth, I would have a class of students, and they would vary from 18 to possible 30 in number and I would teach them, then, all of this mathematics, and I would teach them, then how we calculate geodesic domes and everything; and we would literally make organize production. I would organize the students into aeronautical production. How you go how you develop prototypes and get into production in aeronautics, using their kind of techniques. And there was purchasing to be done and so forth, so really getting the students immediately hooked up with real life. Not only do you get them hooked up with real life, but they found, time and again we had, at Cornell, the President of Dupont happened to be a friend, he had a student, son, up there at Cornell. He was interested. One reason and another there would be somebody you could get in touch with, and you said we are going we are making an experiment, and we really really there is no use in going through this without using the most advanced materials, and we understand you have a little better clear Plexiglass, or whatever it is, and these men would send you a special airplane of materials. So they found enormous cooperation on the part of industry, and began to get enormous insights.
The students I have organized into teams were purchasing agents, and mathematics had a mathematics department, design engineering, and production engineering, installation engineering all the logistics were worked out everything we did; this is what I call COMPREHENSIVE PARTICIPATORY DESIGN SCIENCE. YOU ARE RESPONSIBLE FOR EVERYTHING. You are also responsible for how it's going to be removed and so forth.
Just one sort of last work reflectively on that Ford Motor Company Dome. When we finished lowering that dome onto the and it really worked, and people climbed over it and it was finished. The head of the Ford Motor Company's Engineering Department said "I not only congratulate you" they had been very quiet with me, but he said "I'm going to dismay you very much to tell you what I'm going to tell you, but he said we were so sure you couldn't do it, that we had paid in advance, we paid the contractors, it was going to have to be done so fast, he required a premium, we paid him in advance, we paid a contractor to remove all your unfinished no-good work so we could get so we could go on with our show. We assumed it wouldn't be done. So we're paying him much more than we are paying you for a very beautiful and successful job." Well, they really felt very conscience stricken about it, and as a consequence the Ford Motor Company went way out of their way to tell the Air Force who were going on my radomes and didn't know, they couldn't understand why they seemed to work, and the engineers couldn't calculate that they would work. The Ford Motor Company gave me a really terrific send off and that's one of the reasons why the geodesic domes really did proliferate so rapidly as a consequence of that project.
Coming back now to the Napier developing a trick for the navigator. He got down to very, very simple devices. Assuming you still had held on somehow to your tables, and could look things up if you couldn't the navigator might know his mathematics well enough to know actually how you arrive at each of these calculations, that is a fairly simple kind of formula, you can get yourself within good, sort of practical range, so you know within a hundred miles possibly in some case down to 10 miles within where you are.
Napier made this diagram, and incidentally, in trigonometry if we are dealing with sphericals, we draw a triangle like this we put a little curvature in it, and you can see quite easily whether you are going to if it really is a 90 degree I meant to draw that for a 90 degree corner, so this is a 90 degree. And this corner is big C, this is big A, this is big B. And the side opposite C is little c, and this is little b and this is little a. That's the standard convention in trigonometry. Then, Napier divided developed a diagram where he had and he had big A here, big B, and so this would be little 'a' here, and this would be little 'b' here, and this little 'c' here. And then he put a complementary, little 'c' modifying those three. This is his basic diagram. If you can remember that diagram, then that's all you really need so that you'd never get into trouble. So then he said, I'm going to make a rather poor poem, but you can probably remember it. He said, the sine of any part will always equal the product of the tangents of the adjacent, or the he used the word cosine, he made a rhyme out of and tangents and adjacents he made a rhyme out of at any rate, = the sine of the product the sine of any part will always equal then the product of the cosines of the opposites or the tangents of the adjacents. If you can remember that statement it is not too difficult, the sine of any part will always equal the products (means multiplying) the cosines of the opposites or the product of the tangents of the adjacents. And then I'll show you what he means by opposites and adjacents.
So, this A you now, this is 90 degrees and you've learned that the one thing you know is that this is 30 degrees. So, you can look up the sine of 30 degrees. Oh, excuse me, I'm going to say what I have known in my maybe I know that that is 30 degrees, o.k.? And I know that little b is 22 degrees and some minutes, whatever it is 30 degrees and 20 minutes I know two parts. So what I know is little b, and I know big A. O.K. I've got to find then something called "opps". Well, they are beside each other. Therefore I have to get something opposite to it, so I can say that the sine of little a, and every time I write little a I'm going to put this c above it, the sine of the little a equals these are opps, the cosine of big A times the cosine of little b. Also putting a little like that. Now I've got my formula written out, and I've got to substitute, where the c's occur, I've got to get that out of the thing. These are put in here so you can't make the 180 degree error. So then you convert the c you revert to, so this instead of being sine has to read cosine; cosine of little a equals cosine of big A times the sine of little b. Just the opposite of cosine. Now I've got a formula that I can look up in my tables what the sine and so forth are, and I've got to multiply them, so if you also then have your in those days we used to use logs, so you had the log of any of those numbers and all you had to do was add them, instead of multiplying them. So trigonometric tables were usually also given with the logs in those days, so you looked up your logs and simply added or subtracted, whatever you had to do. Which was fine. So this gave you then, by this time, so now you have learned what little a is, and you go on to find out what the other parts really very rapidly. This is a fantastic, simple thing I've given you here. That's all there is to it.
Now there are many other formulas in trigonometry I assure you, and there are special kinds of tasks to be done, and there are some really quite fancy formulas for doing. But this is the essence. You can get today, Hewlett Packard has a little computer and that's now out at $350, or you can get a $750 one, but it has all the trigonometric functions and everything on it, and you can do all these problems with this little computer. The $750 one they now have programs, strips that you can put in, and so if it were something you did quite regularly geodesic domes, just put in the five frequency, four frequency alternate, and you'd get your answers just like that wham, wham, whatever it's length of every part is of whatever you are building.
Once you've discovered then what the trigonometry is for the radius is one then I showed you here before you want to build a geodesic dome. And what I need to know is the length of a structural member. So what I really want to know is from here to here, I want to know the chords instead of the arcs. So what I do when I get what my angle is, I know, then, that the chord is two times the sine of one-half theta. So, this is quite different. You can't just automatically multiply let's look up to a large angle you can't do that. Because, I want you to realize the difference between the because these things, this would be the chord of that, and that's not at all that number. So you can't just take it's two times the sine of half theta, and just never kid yourself about that. So, just when I have then I began developing what you call chord factors the phrase had never been used before, geodesic domes I could give you the chord factors for any radius, so all you had to know was your radius, and you could do it in meters, or centimeters or anything you want, but, so, as people began to catch on to what I was doing, then they found they could publish all the geodesic domes of various frequencies and the chord factors and that's all you had to know and you start putting together and there's your dome.
So, I can tell you that when I did the first calculation of a geodesic the, I say there were no electric computers whatsoever. There were no kinds of everything had to be done longhand. You did have your log tables and you did have rather poor tables, I didn't have the Edwards ones. So, it was just very clear to me, coming at things the way I do, you're used to my kind of argument, that an omni-triangulated sphere, and particularly if it were tensegrity that is operative, was simply going to make since tension has no limit. Therefore, a tensegrity dome would have no limits, but all the arched domes had very limited clear spans and St. Peter's was the largest in the world at the time of __Mem -few __days(ms?) 150 feet in diameter, and I looked at we really get into some very large sizes, they no longer were building the domes just with bricks and so forth, and having chains around them where you did Santa Sophia or St. Peter's. But they were doing what they called radial arch. They had great enormous steel beams running from the perimeter all the way up, which makes a very long beam. The slenderness ratio made it very heavy, and then they had a centering steel ring and they were all brought to that centering steel ring. And the weights involved were enormous.
Then they began to learn they could make it a little lighter. And then they began cross triangulating. Well this is what came out of my geodesic triangulation so you begin to find you can make those radials a little lighter by cross-triangulation. At any rate, the first my first calculations, I could see that the thing would probably work, and suddenly, supposing there were those who had thought that before, the whole thing was doing calculations so things would come out accurately. It was not a game you could do be just rough about, or things would really collapse. Errors would accumulate very, very rapidly as you went around the great circle.
So, my wife had a little money at this time, she decided to really help me buy time. We bought time, and I'm sure this is the reason why other people hadn't done it, because it really was going to take time. It took me two years to do the first calculations, really to know it was so. And today, anybody with a computer, I mean you really can run this out in less than half an hour you can knock out a dome. And nobody realizes the enormous advantage that has really moved forward to the man with the calculating capabilities. And the computer's carrying out tables to very, very many places and so forth. So that the accuracies are very, very great today. To me, one of the most interesting challenges here was in the calculating capability. If I hadn't done whole number long-hand work I would not realize the significance in numbers.
If you put it into the computer, you just miss it getting your simple answers. I realized I had really tended to make an exploration about the last moment in history when you would have the opportunity to really find something out, and that I'd really better pay fantastically strict attention all the way through here to the significance of everything as I went along.
This kind of talk I'm having with you tonight, then, is to do with a then what I call design science COMPREHENSIVE ANTICIPATORY DESIGN SCIENCE. You deal with things sum totally and in terms of total resources, everything you know about how the Universe is working, and how and why we have the energies available here. Why there is a biosphere. And how you really then employ the physical resources and the knowledge to the highest advantage for all humanity, and if possible to sustain all humanity for all generations to come. That is your challenge, and you must be responsible for how every way you participate in the transformations of nature, employing those principles, responsible for how the things gets where its going to go, responsible for how it goes while it's working there and how you take it away and get it into recirculation again. You must be responsible for the complete cycle. There is no point where you are not until whatever you produce is now melted up and is being used by somebody else. But as far as your using the original resources you make yourself responsible from beginning to end.
I gave you a way of realizing, yesterday, in big patterns the metals that are occurring around the earth, and therefore there is a half way around the world you go to find them all and they gradually converge until they get to maximum separation apart, and then they get into reassociation in preferred ways, and then when they are they finally have such an advantage for man to really justify such a big operation, you have to make them available to the most people around the world, which means you have to send them half way around the world again to make them available. So that is the size of the operation. And, often, really doing things the right way is much easier than doing them the wrong way.
At any rate, I never find myself shuttering at the size of the problem, and everyone of the projects that I have undertaken, and tomorrow we are going to go over a lot of the projects that I have undertaken, were always undertaken on this kind of a basis, and all the students who have ever worked with me have learned how I feel utterly responsible all the way through to humanity for having tampered at all with all this extraordinary phenomena we learn about.
I assure you this kind of carrying on is a very inspiring matter. It makes you tremendously conscious of everybody and why everybody is doing what they are doing at this particular moment I can understand why they are preoccupied. I can understand the fears of the father about his kids going to be able to eat, or whether he is going to lose his job. I understand all of those things. And so I feel very, very kin to everybody. Not at all annoyed at non-cooperation, but you have to find out,. then, how to get it done, and you keep at it until you suddenly begin to find, there are ways of getting things done.
Now, I've been giving you a few sort of clues I'm always looking for the simplest also. So it's nice to get this thing out of the way. That's the language of words, and they should not feel formidable at all, because it is really a very simple kind of a thing, because that is a tangent line, and you can understand what is really cotangent. You can see all of that. It's really, very, very self-explanatory. The only word sine you see this is the withoutness, the openings. This is really to do with angles, how much the angle is open. You can see that, that's a very nice measure because it's within the central angle, I think that is enough of talking about that diagram and talking about Napier, and
I am now going to switch over to tensegrity structures, because all of the geometries that I have been identifying for you in the terms of topological analysis and volumetric analysis and energy analysis quantum and so forth. All of those geometries can be made tensegrity. And so I point out to you also, that one of the things that inspired me very early in the game of structures was getting into the push-pulls and how they're accomplished, and finding that and I went over that with you. That compressions had limited length in relation to cross section and tension didn't. We went into all that. Discovering that the Universe is actually designed with islanded compression, usually spherical as the most effective use of the energy that way, and the whole thing cohered tensionally. And the tensions the compressions are inherently discontinuous and the tensions inherently continuous. We have a Universe with inherent continuous tension. And that was not the way man was building.
But when I found all the geometries and all these interrelationships and all this coordination I was looking for nature's own coordinating system, I was sure it was rational due to the chemistry, and I found the rational coordinating system, and I found that all the geometries could be produced in tensegrity. I was looking for this and then I had the word.
At Black Mountain College I was a visiting professor the summer of 58 and 59 I was the summer Dean. And there was a student there in 58 and 59 both, Ken Snelson, he'd been in the Navy, and his father was a photographics camera store man in Oregon; and Ken had a great he was fantastically expert in cameras moving pictures and still. But he had real, real feeling about art, and he liked painting. And he'd come to Black Mountain on account of his painting, and to study with Albers there, and Ken Snelson fell in love with I talked to him and gave him my energetic geometry, and he was absolutely in love with my energetic geometry. And he was incredibly good at model making, so that for a whole year after Black Mountain he came to New York to wherever I was and worked. And then I went out to the Institute of Design in Chicago, and Ken Snelson moved out to Chicago.
Then in the second summer at Black Mountain, Ken showed me a sculpture that he had made, and, in an abstract world of sculpture, and what he had made was a-a tensegrity structure. And he had a structural member out here two structural members out here, that were not touching the base, and they were being held together held they were in tension. And I explained to Ken that this was a tensegrity. Man, I had found, had only developed tensegrity structure in wire wheels and in universal joints. Universal joints where he had a steel shaft, and the reason for needing a universal joint because you were changing the angle of the drive, and you had two shafts, and each one of them came to three arm points like this, tetrahedronally like this. They had forged members steel, and came to a pad with a hole in the end of the pad. So there were three pad holes, tripod like this, and they would bring those together at 60 degrees from one another with a flexible disk and they would try all kinds of fabrics and leathers, and so forth, to see how long they would last, and they would then rivet these onto that flexible disc, and as it drove this way, the disc would accommodate. In other words it was a tensional interconnection. So man had used tensegrity in this drive shaft of the universal joints. He had also made it with the wire wheel where he had an island of compression as the hub, and an atoll of compression at the rim, and the whole thing was tensionally cohered. So this is the only place I found that man actually had tensegrity. So when Ken Snelson showed me this little extension thing he did it was really just an arbitrary form, he saw that you could do it, but he was just, as I say, an artistic form or something startling to look at. And I said, "Ken, that really is the tensegrity and it's what I'm looking for because what you've done I can see relates to the octahedron and this gives me a clue of how this goes together in all the energetic geometry.
So Ken opened up my eyes to the way to go into the geometry. And, Ken, himself is an artist, and he's gone on to make all kinds of tensegrity sculptures that are getting to be very well known as one of the most, in the higher demand artists. But Ken himself kept himself alive by doing moving picture work, he has got a moving picture camera man's union card, and he turned out to be really one of the best, so anytime he wanted to work he'd go and make a whole lot of money on some big project and then go back and commit himself to his art work. At all times in my carrying on I have come to phases where I could see that something might be attractive from an art form you can suddenly get insights, I could suddenly see new patterns and then people would say and this could be extremely interesting, and I know personally I am deeply moved by it, that I could exploit it by stopping and just being an artist, but that was my commitment, I told you last night how my whole commitment was to be absolutely responsible to never exploit for self, and never just for self or fame or whatever, you must not exploit. And if people tried to make you fancy, then you must do everything you can to make sure that is deflated. So that I've never gone off with these forms, but it is interesting how many I have an enormous number of artist friends, and really deep friends, and they are very simpatico with what I do, and they do then go off, and they like what I can find out, also technically, which gives them a chance to do various things but at any rate. It couldn't be a more beautiful life than I have had with my artists friends.
But I cannot talk about tensegrity without talking about the fact that Ken Snelson really was a catalyst to my discovering how I really connect this up with all the geometry and all the coordination of everything I wanted to do, doing more with less. And, so I'm now going to go into the tensegrity with you and give you a little feeling. But I also did earlier in our time talk to you about pneumatics and tensegrity showing you how and why it did get into the regularities it did. So we don't have to get into that and you'll recognize that as we go.
May I have the first picture. Here I am also pointing out to you that where we have a balloon or a football, or somebody said, they think about it as impervious, but if you look at it with a very fine microscope, it is full of holes. The only thing is that the holes are smaller than the molecules of the gas that are inside, so they are really like a fish net, where the fish net is smaller than the fish and the fish simply hits the net. And I saw them really operating very much as fish. The molecules of gas hitting the bag and hitting it in so many places whatever it's stretchable shape is, it takes that shape. So you can make strange looking balloons of special shapes that are always getting pushed outwardly.
Next picture. Then this represents, look in the upper right hand side, or middle right hand side, what I gave you are the two swimmers coming together and shoving off from each other, and then hitting going careening off of the and there is no line that you can make inside there, in essence a radius that wouldn't be a chord. It could be a very deep chord, it could be a very light chord, and I find it really going around like that, hitting the skin a glancing blow. Now because there are two of them action and reaction shove off from each other, they each hit the skin a glancing blow. The fact is then that a chord, an arc, stays in the circle, but the center of the chord is nearer the center of the circle than an arc's ends, so it's ends are always emerging, hitting the skin at a very small angle, and there is a net of the two one going this way and the other both pushing outwardly, means that there is a single force going out like that, and the magnitude of that force is governed a great deal by that angularity of the but a frequent enough episode keeps it all moving out. So, you can get the bag harder by putting more gas in so increasing the frequency of the hitting very, very greatly.
Next picture. Then I showed you how what seemed to be randomness automatically worked itself into the circles and the omni triangulation and on the hexagons and the pentagons are simply incidental to the triangles here. The triangles are here. And these are just basic the triangles do all the stabilizing.
Next picture. Now, the simplest, could that picture be dropped, or turn it sideways? The simplest thing you can do for a tensegrity is two members like that, have their oneness one bowed like this, the other bowed like that our precessional effect of the two coming together. These are very much like taking a tetrahedron. I've got two balls and two balls coming like that at each other. Or let's take a tetrahedron, consists of four triangles, so pair them into two diamonds, so you take two diamonds and precess and come like this and grasp each other. And that is exactly the way that you make a baseball. A baseball skin, the two lobes in it, like in a tennis skin is really two of those balls. It is each lobe is a triangle of a tetrahedron, which you can then also draw as a circle, and then have, between the two complement that same radius. So you have two pairs of triangles, and two pairs of triangles precess them like that, and grasp them like that, and that's your baseball.
And, incidentally, your baseball, if you draw it, it is quite interesting because it is this. It is yin/yang. The yin/yang are these two complements, not in a plane, but really in the Universe. And the baseball form is exactly that, this uniform radius all the way through. It's a lovely thing. Baseball is telling you precession. Yin/yang to me tells me a great deal. I'm sure the Chinese thought in the terms of the whole too, and they came to a flat representation. I'm sure they were thinking this way. So they really felt the power of that yin/yang. So the complementaries do precess like this. O.K.
So in this first one here, you can't take two straight sticks like this you can if you want and they are two edges of a tetrahedron. Makes a very flat tetrahedron. All I have to do is having one tension member go right around, you call it a kite, you can make a diamond shaped kite. And you don't have to fasten these two to each other because it is simply a very flat tetrahedron, and by pushing, the tighter you make the perimeter the tighter this comes against it here. But it doesn't have to be fastened. It is the beginning of tensegrity, so it is two member tensegrity, so the two member tensegrity is really then a precessional affair, but it is a little set of arc a little like this the pull of those lines will make them do this to each other. That is what that model is that I have there. You can see how you can take a ship, now, a Naga ship, with great ends, and the ends come out of the water like that, and you could have a spar going like this in the sky, which is then supported from the ends of that spar to the end of the ship here, and out the other end of the spar, right to the other end of the ship which is sticking out of the water very far. And then you'd have to have a fore and aft tension to the top of this thing so it wouldn't fall over. It would be quite possible to make a boat that way, and I'm surprised that people have not built boats this way because it would be possible to drop sails from this thing just connect in tension. I think we probably will see just such a device one of these days, because we'll get into very, very light weight instead of having a mast that has to go vertical like that, this is a very much lower thing, yet can drop you a great square still, and give you an enormous amount of sail. At any rate, that is the simplest tensegrity.
Next picture please. The next one is one where you make the octahedron. It has x,y,z coordinates, and I suggest you try this someday and before you tie put tension from end to end, you take some little a box, say a cubical box, and you tape one of the tubes onto it. The box has six faces so we go on another face here and have one going that way, and have the x,y,z coordinates fastened onto the box, but a little away from each other. Do you understand that? Taped on. Then you take your tensions, omni-triangulated, you have eight triangles in high tension, then you remove the box from the center and you find they just don't touch each other, so here we have very clearly the non touchingness of the octahedron, and it makes it quite possible then with a little mild bowing by the tensioning, to make them quite fairly accurate. We find in nature all the crystals that are octahedronal or whatever they may be, always have, they skew one way or another. They are always turbining either one way or the other. Now this was turbined, there are two ways of turbining this.
May I have the next picture. No, I haven't come to it yet. This is a four member. I had three members, now I have three struts. This is the this is a tetrahedron made by Ken Snelson. It is really a very fascinating tetrahedron by the way. It's a four strut tetrahedron and it relates then to the vertexes and the opposite faces you have a suspension of the opposite face.