I've been covering really very large patterns with you very deliberately, and many people ask me a question about being a comprehensivist, and then being competent. And what I've learned to do in disciplining myself, is that I can plunge in depth for various periods of time into something you really need to know about, and you really go after it and let nothing impede you. Having, however, started from as comprehensive a basis as possible, I never really lose the fundamental comprehensivity, and I can come back really quite rapidly from any subject.
Ever since I, there was a period when I really needed to get at the sanitation for dwellings, and I spent two years just developing a mass production bathroom, but I really did find a great deal just such simple matters as there was only one man in the United States designing all the toilet bowls of all the different companies, and I found there was nobody who really knew why a toilet bowl was the way it was except this one man. And he had inherited his art from some English craftsman, and he was in a little top room of a building in Toledo Ohio Standard Sanitary and Kohler and all of them were getting their information from him. And I found that in making the toilet bowls, the tolerances that can be maintained between forming the original, regular ceramics, and you're getting your clays, and then before you bake it and one thing or another a lot of weight. Things go out of round, so you could not have any of the machinists kind of tolerance at all. If you could hold to a quarter of an inch in the diameter of an opening, you are doing very, very well. And all this became really very impressive to me, so I decided to really go in pretty deeply, and I found that no scientist had really ever really looked at the plumbing. Just think what we're really saying here. This is, in our day, scientists are not looking at plumbing. They find fault with the plumbing, and they call the plumber but nobody is asking scientists to look at plumbing and say that, you know, you're a pure scientists and you shouldn't be looking at this kind of nonsense.
And here are all these extraordinary chemistries that are going into the toilet, and then very valuable chemistries are getting all pushed together nature has taken a lot of trouble to separate them out and then we deliberately push them together. And when Nature does separating out, if you ever get into mining, or refining, you'll find it's quite a job to separate out. You spend a lot of energy separating out, so to deliberately let things go back together could not be more unintelligent.
So I found that right in the very life of the people who are being educated to do logical things, right under their nose, in their everyday life they were missing things. So this came up for a whole lot of attention, and I did, I say, get deeply into I found that I could produce stainless steel toilet bowls in two halves of stampings, and get absolutely fine tolerances and I was able to really find out how many gallons of water we need to flush out the toilet. We find that people are getting rid of a pint of water and using seven gallons to flush it away. All this beautiful, valuable water coming down the hills that we need so badly. It just, it all began to hurt all of it, as I began to get into it deeply, to see how much advantage could be found for humanity if you got into so it was very easy to get these fine tolerances in beautiful stainless steel and so forth.
And, anyway, the big point is that once you understand, you can be a generalist and plunge. And, you really, really dare to pay no attention to anything else on the side. Because I've been into so many different fields plunged very deeply into many things, whether it was cartography. When I do, it's maybe six months two months maybe two years that you're really off there, and then you come back into the big swing again. But, there are a number of subjects, which, and I have very good records of all these things.
And so there are a lot of slides. First, I've just been handling thoughts themselves, and I wanted to come back now, really, to artifacts and slides and particularly to artifacts because I want to review with you for a moment my own grand strategy of how I carry on.
When I made up my mind to peel off and commit myself precessionally to what had been called the side-effects, but that is to really how to make ecology you get on with ecology and play the game that Life is trying to play of making the big show work and not just looking out for yourself. When I did that, risking realizing that there was nobody to mark your paper from there on there was nobody to pay you that only Nature would support you if you really were doing what my theory was that if I was doing what Nature wanted you to do, I would find myself supported, but it would be absolutely so indirect that you would never be able to say "This was for that." And, that you must not get scared because you didn't seem to be supported right now, or whatever it is, you must really keep on. And, in doing that, I realized I must not waste anytime. You're going to have to be terribly sensitive you're going to have to use everything you were born with as a child with intuition and sensitivity, and realizing, "Am I really doing the right thing?" This is the way a child can really get into the forest, and when he gets to some critical point he'll get to doing things pretty carefully. So you can really be sensitive.
So, this meant then, that if I had just my one lifetime to try to get somewhere, then I'd really have to get a whole lot out of my time. And I said, "My experience tells me then that I have listened to a great many people talking to a great many people, and one trying to persuade the other this is the way things are." And I made up my mind that people that I listened to were really not listening too much, they kind of waited for their turn to speak and sound off a little. I decided that what I would do, that I would never I would discipline myself not to talk to people unless they asked me to talk to them. I have, in effect, really asked to talk here, and so I am talking to you because you are here to be talked to. Because I am sure that this is the only time people really listen, when they want to hear what you have to say and have really said so.
So that became a basic discipline, and I made up my mind, then, if I was not going to use words which so many people do use as my prime approach, what else did I have. And I said, I see that Nature is transforming continually, and it would be possible if we could comprehend the principles that she really is using structurally and mechanically, associative and disassociative really feel your chemistry, feel your technology, feel your hydraulics and pneumatics, electromagnetics, interattractions or repulsions if you really could FEEL those things, it would be possible then for you to take Nature is continually transforming the environment, that you could really participate in the environmental transforming, and the only reason that you really are doing what you're doing is because you feel it, you've discovered that this is why we are here. We are here for one another, and in one sense you have already discovered that older people have very powerfully conditioned reflexes it is not easy for them to adjust to the new to take advantage of the new. They tend to hold onto the old. Therefore, I said, "My focus is going to be on, not just looking out but primarily on the youngest life that has no conditioned reflexes and to try to give that youngest life, provide environments for that youngest life so that, within which environment that child would prosper."
That would be one reason why you would find me lying down, remembering how I acted when I was a little kid on a bed trying to get off of a bed or whatever it is, and I paid a great deal of attention to saying, "inasmuch as there really are only a certain number of transformations, motions such as I have given you I find that there are categories of hierarchies of tools; and no matter how fancy the tools look, there is pounding, there is pushing, there is scraping off, the horizontal the pushings and pullings. Things get down to really relatively few things that can be done. I say, I think that child all the things that child is doing is very experimental. It's finding out how it can stand up. It is a superb research operation. And when that little child later on begins to tear paper, it isn't because he's being mischievous, the child needs to know what coheres and what doesn't cohere. It needs to keep testing things, because having been so informed, so insistently informed by gravity about falling he must know what he can hold onto that won't come apart when falling. Now that is logical, isn't it. So he grabs at the bed and so forth he instinctively does that. So he has to keep testing what holds together. So you find that they are not tearing newspaper, they are looking for things that look tougher. They tend to take your best papers, they tend to look at the things, try pulling apart the things that you that people consider very valuable around the house, linens and so forth, they want to find if they can pull that apart. Finally they find that things do hold. So I said, "If you realize what it is they are trying to find out, it would be very easy to really arrange things in their environment so they'll find the things that they need to experiment with, and they don't have to use a lamp cord to find out about tension, they can have another kind of a cord and it will be much easier for them to do it."
So, that, the idea was then, how do you develop environments that are favorable for the new life, within which the new life can get all the information it really needs in the most logical way, and it doesn't tend to engender fear, or that when they are experimenting doing whatever they are doing they don't suddenly get hit in the head, or that society is going to say "Stop That!" and they find themselves obnoxious to society.
I felt that all of this could really be done. So my first focus, then, becomes developing artifacts instead of words and by artifacts or tools I would mean a building is a tool; a great ship is a tool. So the artifact may be I'm not talking about spoons and rulers and a lot of the small devices, but any of the participation in using the principles of nature to take apart and reassociate and so forth. I'm really generalizing this for you very much, but the point is, I find that Nature already has things in certain associations. She herself disassociates them gradually. She takes her own rock apart, and we could learn, then, how you take things apart. And you find then that they are very valuable chemistries that are temporarily associated this way that can be disassociated and reassociated in preferred ways. So that's what we do in mining going around the world finding there are certain resources, and then deciding where you are going to begin to separate that out. Are you going to get ore? You have local energies available that makes it possible to do next steps, grinding or whatever you want to do. Or do you have to forward it in ships to get to another place to where there are energies, or so forth. There are many critical decisions to be made of that kind, but, by and large there is a very, big, big pattern here.
The metals are deposited very unevenly around the whole earth. In effect man goes half way around the world and takes ore out of the ground, and starts separating it. And as he goes along sometimes leaves things behind, he separates it and has the residues, and keeps forwarding. And finally he gets highly concentrated metals at various centers around the world. And as there is finally a maximum degree to which he separates these metals out from everything else, the chemical elements. Then he starts reassociating them in preferred manner. As alloys. And, after that we get to the point where things get to be made into special parts of special engines and so forth. And we begin to then, we start assembling again in preferred ways.
In other words, Nature has "come-apart-ability," she has reassociability, and you are simply participating in that in a big way. So I find that the total operation going half way around the world to get the right metals, to bring them to certain concentrated places while concentrating them themselves, separating them out into a lot of assorted, very valuable materials, and then start reassociating them in preferred ways whether it is going to be some kind of an engine, or it's going to be some kind of a building, or some other kind of tool. And, then having gone into this very great complex undertaking, which is going to take months, it's going to take years to get those mines operating, or ships. It's a very big complex thing, taking a lot of time; and I got to then to demonstrate all the time that has been invested by man and all the energies invested by man are going to be worthwhile having invested that way, and so what I have to do then, is arrange to get what you produced available to the most people around the world.
So, in effect, I now have produced something that has been assembled from all over the world. And now I must arrange to get what I have to all the world again. To make it most available to the most people. This gets to be the optimum big pattern that you are concerned with. I find, then, if I do get it available to the most people, then we find out very quickly, "Was this really worthwhile doing, and how do you improve on it if it wasn't quite if it was pretty good?" In the end humanity ought to be gaining advantage humanity ought to be gaining advantage of greater health, and more time, a little greater longevity more freedom, to undertake more things. We're continually trying to free up humanity from being locally preoccupied as a local machine. And to get freer to use more and more of it's head , to look at more and more of the patterns, to be able then to be more and more effective with its mind understanding principles and realizing how much more it could do with the energies that are available in Universe on the ends of levers to do the physical work that we're really here to do the mental work.
So, my total pattern is, then, half way around the world inbound, half way around the world outbound, which sum totally is once around the world. These are a whole lot of energies that are involved. I find then that this makes it possible for me to get into very discreet patterns.
Next thing, I can really understand my totality and have a way of judging whether what you're doing is worth the doing. Such patterns as I have just spoken to you about really are, then, highly documentable. What the data is involved, and make some very good calculations in advance whether this is going to turn out to be worthwhile to humanity. I do not look on these projects in the terms about whether the people like the looks of what I am doing for the moment. I am always concerned with how it works, and it usually looks alright, because I am concerned with not only how do you get it there, but how do you maintain it? And how do you in the end recover it and take it away when that becomes obsolete cause you've got something better. So you get that into recirculation.
So I see a total responsibility in design, getting things to people, not trying to sell it to people, but trying to make it available to people. I'm very glad to be doing this program under the Bell Service because I, one of the examples I use in industrialization I consider by far the best example operating in the whole history of industrialization, is the telephone service. That is, you don't try to sell people telephones! What you do, you're selling service. You're making it possible for people to communicate, and the easier you make it to communicate and the more accurate they can communicate, the more people are going to use the service. So, it is a very interesting matter. People used to think that you've got to sell things to people, because in order to get it improved you've got to people are going to demand a very good product.
The telephone company learned that they didn't wait until the people said "I don't know whether I can afford a new telephone." They didn't sell the telephone to people, if they'd have sold the people telephones, then they would have had telephone architects and they would have had to develop "Napoleonic" and "Voodoo" and "Georgian" telephones and nobody could possibly sell the telephones to anyone else anyway. So the telephone company simply sold the service, and they found that every time they found a little better way, that they could really afford to scrap enormous systems because they could give so much more service, that the number of people who used the service would go up that fast. And, so the telephone has continually been improved.
So when I'm thinking about big systems, and world systems I find this a very good field to work in to take working examples. I also then spoke about recirculating and using materials. The telephone company when I was young we lived just outside of Boston, and our telephone number was number l0, so it was early in the experience of telephones. And, it was not long a few years before they began learning to get more messages over the same cross section of copper wire. First it was just one, and then they began to get more we got up to 28. And then there was an increase, I think we went up to over 200. The next increase if I remember it, went up to over ll00. Then it went up into some thousands. By l930 the chief engineer of the Bell Labs said that, at that time the telephone service was being employed by about 10% of humanity. He said they'd be able to increase the telephone service so that the whole world would be furnished with telephone service, and that during that time the telephone companies would not have to mine or buy another pound of copper! That during that whole time they would be copper sellers, because they found the rate at which they were learning how to get more messages per cross section of copper was so, so vast that there actually was a gain. There would be the amount of copper they already had was a copper mine, and it would be adequate. That has turned out to be the fact!
We have now one communications satellite weighing a quarter of a ton, outperforming the transoceanic communications capability and fidelity of l75,000 tons of copper cable one quarter of a ton! This is the rate which is suddenly an enormous step up, doing more with less. And my whole hope of how we are going to get all of humanity at a higher standard of living starts in looking out for the young life you don't have to quarrel with. Because you give them something that really works, and that child, that's what he's going to use. And his reflexes are going to be conditioned to that which works, and what is intelligent, and he won't have anything else from there on. And I said, "It's a long pull job we're doing here, we have to start with the children, and we have to get a whole lot understood before we get anywhere."
Because I found that at the time I started what I was doing that people were thinking about architecture and buildings in such a powerfully conditioned reflex that it was just incredible. In 1920, when I in a first presenting the Dymaxion House to people, where I developed a machine for living. It really is a machine and using the most advanced technology we had, and I was able to devise a three ton house, that I was out performing a two hundred ton conventional residence. People, you know, they were so taken up with Georgian and so forth, it was amazing, the conditioned reflexes. And how much is really in there, almost on a fear basis, because we get, it goes back to the power structure, and the castles, and what the strong man has. The man down the street with his high dazzle that's what he has, what do you have? The people their eyes were just powerfully conditioned this way.
So, in all the things that I talk to you about, I want you to realize that I never allow myself to say what they're going to look like. I'm perfectly confident that if you're doing it the right way it's going to look strange, but in due course people discover that that is the way things really work and they begin to like it, because they can understand it and feel it.
As we begin to get into the space programs, the devices you see going off into space look very strange to people. And they didn't mind about that going off into space, but if it is something you're going to have around the house, that you're going to have to live with, they've been very, very sticky about it.
Now, I'm saying these things to you. I'm going to be getting into these strategies and a number of projects I've gone into. But I want you to have a little feeling of the overall controls. I am looking then as: Always a world project. I'm looking at it as total history. I'm seeing the total inventory of all the metals of our earth, being separated out and progressively more easy to get out. And, so I began to see that there may come a time when you wouldn't have to mine anything. In fact, this copper I also spent two years in depth in the Phelps Dodge Corporation as Assistant Director of Research to Phelps Dodge, which is the third largest copper company in the world. And it was there then that I did, also then, plunged into doing some bathrooms and several of the things that I got into. New types of automotive breaks using the steel we use, the steel is a very poor conductor of heat or electricity. I got into the conducting metals like the aluminums and coppers for the breaking that carried off the heat very much more rapidly. They just carried away break fade, and more effectiveness, and then finally they even had metal to metal. The Japanese found this you have carbon brushes and copper together, the metal didn't get worn away, and they've been using this with an electrical trolley going along to pick up its current, and getting no wear, and finally I could get that into a break.
I developed at Phelps Dodge various things I got into in depth there. One was, Phelps Dodge was primarily copper, but the gold and silver co-occur with the copper and they also became very much involved with tin. And America and all the world was going to need a lot of tin in World War I. Tin ores coming from Bolivia, and when you get into low-grade tin ores, and there is lots and lots of low-grade tin ore. And there is it is something very difficult to separate out. And I found that taking the ground, powdered ore I was able to develop a centrifuge. And I had a centrifuge that had to be water cooled, so the metals didn't get to some critical heat where they would break up. Because when you get to spinning great weights at great velocities, there is an enormous tendency to come apart. So I wanted to introduce a very powerful flame in the blow torch flame into the powder, and be able to centrifuge. So I had to design a completely water-cooled apparatus which this went on, so that all the heat was just at the contact of this flame with the powder. And, I figured that you might like a cream separator really be able to separate the tin out, because the weights are very much more. And sure enough it worked. I didn't they didn't get that into production. They were really very scared of the centrifuge, and that it might really kill a lot of people. What I did find is that it is really possible, then, to take low grade ores and centrifuge them just like cream. And this tin was just running out, it was really beautiful.
I've been able to get in in quite depths in a great many directions, and I've become very deeply involved with the metals, and know a lot about their histories, and I could I will go into talk to you about that in due course. But, I had made up my mind that this morning I would go through some various slides, and I'm introducing to you a number of projects. Some of them rather short where I have very good slides of them I want to take advantage if I could, we do have something for you to visualize. And, for instance, you'll find me getting into my map, and I was able, then, to develop a better method of projection than any known, where there was no visible distortion.
It was very important to be able to have a world map without any visible distortion. Because if you take the Mercator map and use the land as a background for say, percentage of resources, or how many people there are in that particular place if the background is distorted with Greenland three times as big as Australia on the map; but the actual fact is exactly the opposite, then the relative abundance within that particular area is very mis-informing. So I needed a world map that I could always, with absolutely no visible distortion against which I could show percentages of materials and people, whatever I wanted to see, so I needed to look at world all the time, so that's what brought about my world map. It is the only world map that is approximately distortion-free, both as to relative shape and relative size. So, that brought me into a great deal of experience with the world map. And many numbers of times I drafted the whole world plotted the whole world on paper that's been something that I've got a that's good for you to, to feel in depth. You get very familiar with your world.
I found that there is something worth defining. And what I found here was employed, in the beginning, in the space program. When you start going, suddenly, into rocketry, and humanity is awed by the prospect, and not at all sure how it's going to come out. And you start then, experimentally, sending enormous rockets into the sky to go great distances, people get to be quite apprehensive about where that's going to land and so forth, and so that I did make a discovery that there was a great circle around our world from America to America that didn't touch any other continent. Now, this would be a highly specialized kind of item, but you see North America there, and you see Florida where Cape Kennedy is, about here. Now that Cape Kennedy, I've got an axis where you leave Cape Kennedy and you just miss South America and you just miss Africa, and you just miss Australia, and you go over the neck of New Guinea, and you keep on going around and you come back to Cape Mendocino in California, and then right back to Cape Kennedy. In other words, it was possible to find a range at which you could fire, where you were not firing at the United States.
I'll go around that once more, and would it be alright for one of you to come up and do something with me? Would you come up dear? I want you to put your finger, if you sit down on the floor there, we'll use this camera here and put up your pencil somewhere out just about there. I'm going to put this Cape Kennedy on there you keep the pencil steady will you, and I'll do the turning. (O.K.) I just want to have it so that you can see where it is pointing. See, it is just missing South America. Just missing the tip of Africa, or if I did it right it would. I have to go back again. It just misses the tip of Africa. Just misses Australia, and does go over the neck of New Guinea and then comes right back to Cape Mendocino and back to Cape Kennedy.
Now, I have some slides of that and it would be sort of fun just to see it on the slide, because you can see the line itself. It's carefully drawn as a great circle. So it is the shortest way around the world. In relation to that line and we then have an axis of it because there is always an axis of a great circle. And it is interesting to see how many people are south of that line. For instance, South America is south of that line, and Australia is south of that line, and the Antarctic. So it's just Australia and South America. South America has four percent of humanity. Australia has not one percent, and Antarctica is not one percent, so only four percent of humanity are south of that line. Which says then that 96% are north of that line. Therefore, the pole, the North pole of that line, would be nearest to the most people. And the North Pole of that line is exactly it's just in Russia, and it is just South of Volgograd, and that is exactly where the Russians send all their rockets from. It is the nearest to everybody. I don't know whether they have been working on their geography about it, but it is interesting to me that that is so.
From the kind of work that I do, I often get extraordinary insights into what are the grand strategies of the big ideologies. And, I think that I'm now going to come to the next slide.
Next slide please. I'm changing my subject, now, going off of the map, and that world strategy. You're looking at three rods on a tower, and the three rods pierce a vertical circle. And that is made out of a very high alloy aluminum strip. And so there are three hinges in that strip. There is a rivet at the point where the three rods come out thru the vertical circular strip.
Next picture please. Now this same strip, now, has been depressed, and we see the rods still going thru the three corners.
Next picture please. Now we're seeing the same circle of rods. It is a spherical triangle, in which each of the angles are quite open about 120. We see the same strip up high, then down at the middle, at the equator. This is now a northerly great circle triangle, at the next one it was an equator, and then it was a southerly hemisphere spherical triangle the same three pieces, but transforming. And they went through a condition of l80 degrees at the middle.
Now, I want to talk about spherical trigonometry. And we can let that go. You all have been brought up on your geometry, with the sums of the angles of a triangle, always l80 degrees. And I find people just think that is absolutely fundamental. Now, I am going to have you go to the North Pole, and take a great circle, which is a meridian. When I use the words "great circle" now, I'm sure you're right along with me as to what they are. We take a meridian down to the equator, and at the equator we will then go one quarter of the way around the earth, and then we'll take a meridian back to the North Pole. Now meridians impinge on the equator at 90 degrees inherently. So, if I go around the equator which is a great circle also, one quarter around the earth, and I take a meridian back to the north pole, I leave the equator at 90 degrees. And I get back to the North Pole, because I've been a quarter of the way around the world, and here it's 90 degrees at the North Pole. So I've got 90, 90, 90 or 270 degrees. This is a typical spherical triangle, and the sums of the angles are NEVER 180 degrees!
That's one reason, then, why I went thru that model just there. Look at the spherical triangle you see. This same spherical triangle in the picture on your screen is now, those are 90 degree corners. Next picture of that. I can't seem to be able to get that series. There are three three such triangles in a series and they should now there is the same triangle with the 120 degree corners it's in the northern hemisphere; and the other one is in the southern hemisphere. And then one more the middle one where there is the equator. Because at the equator they all are 180 degrees. In other words, the angles can get up to 180 degrees. So, what we find then is, the larger the spherical triangle, the larger the sum of the angles. As it works towards I can make a just take the equator there and that's 180 degrees, we've got a triangle that is 180 at each corner.
Now, I bring those same three rods, then, up just a little here and the angles would be, say, 160 at each corner. Moving it up a little further and they would be l50 degrees at each corner, and they would, finally as the triangles get smaller and smaller they will approach being 60 degrees at each corner; but they are always going to be a little more.
The fact that we discover that what you and I were brought up on as a plane triangle as being normal, is a most extreme case of the most local tiny little triangle. That is very important for you to remember now. As we were taught at school, a triangle is an area bound by a closed line of three edges and three angles. A square is an area bound by a closed line of four edges and four angles, all equal, etc. All the geometries that we learned about were areas bound by closed lines. That's the way it was given to us. So all that we accredit about a triangle is what you see on one side of the line it's the little area inside here. Now, the fact is, I said you have to draw a triangle on something, because I'm going to be operational remember. There is no, even if I say an imaginary triangle, I'm going to be imagining one I scratched in the ground. I continually will imagine a special case. I talked with Sonny Applewhite a whole lot about this last night. While human beings are able to discover the mathematics of the generalized case, and though we are able to use the principle, we always have to use it, as I showed you the other day, objectively, in special case. And even though we understand the principle, when we are imagining it and I gave you conceptionality independent of size we will always however, when we make it conceptual, when we make the dots of something, we make the points out of something. We tend to very quickly associate. We'll make it pink or something like that in a blue background if we're abstracting it. But you'll find that the brain has to use special case. Brain is designed for special case. And only mind has it. So the mind can say to brain, "Think about conceptual triangle", but brain will immediately make it special case.
These are great nuances of exploration, but they are all coming out of operational procedure sticking strictly to it. So, when I say to a child, "draw me a triangle", he says "where?" And I say, "Draw me a triangle" So I say, "How about the ground?" So he draws it on the ground. And I say "You've drawn four triangles." And he says "No, I've just drawn one triangle." And I have to prove to him that he has drawn four triangles. So, we're in Philadelphia here, and so he's drawing on the ground here, a little tiny triangle. I say, "When you drew the triangle on there, you divided the surface of the you did it on the surface of the earth, and you divide the surface of the system that you did it on into two areas. You will agree that if I make a circle around the equator that I divided it into the northern and southern hemisphere, don't you?" "If I make the circle a little further north of the equator, I'll have divided the earth into two areas, a large southern and a small northern. If I get a little further north, it's a smaller northern and larger southern." So the little boy has drawn a triangle here, but it has divided the whole earth into two areas. And the, both areas are bound by a closed line of three edges and three angles. So I say, "You have drawn a very, very big triangle of all the rest of the earth here, and it's corners are you think you've got 60 it's corners, then are sixty from they are 300 degrees each. So the big triangle is 300 "He says "I'm not used to a triangle of 300 degrees." and I said "Well, because your school made you so specialized and so absolutely myopic, as not to pay attention to your environment. That's really we've got to really think of the reality, and the point is you have deliberately done something to our earth you have divided it into two areas." And he said "I didn't mean to be doing it." And I said "You thought up to now that you were not responsible, and now you are responsible, you're doing that whole earth." So he said, "Alright, you can give me two triangles a very big one, and a very small one. Where are the other two?" And I said "Concave and convex are not the same." And he can prove that by the reflection of light the diffusing of light on one side, and the concentrating of light on the other and so there is always there going to be a big concave and a little concave, and a big convex and a little convex. You've got four triangles, and you're always going to have four triangles." It's going to be our friend the tetrahedron. The accountability is there. This is a generalization of the tetrahedron as the minimum system in Universe the minimum structure. And it can appear as that kind of a "fourness". They will always be there. There is nothing you can do without it being there. So you can say, "I can hide away." "No, the Universe won't let you do this. Just thinking," I said, divides the Universe into an insideness and an outsideness you didn't mean to do that but you are. You are immediately dividing up the Universe. What right have you to divide up the Universe? Well, you were given this very special kind of capability of the mind. And so you can play with total Universe, and this gets to be quite exciting to feel this spherical triangles and understand that."
Incidentally, if you do any of the mathematics of plane trigonometry, are exactly the same for the spherical just simply because plane triangle is just an extremely limited case of spherical triangle. So the mathematics of the spherical triangle, really, and there is no such thing as plane trigonometry, there is only spherical. And it's dealing with total systems and the beautiful complementations of total systems.
Now, so we say all you have to do is learn the spherical and the plane is included. Give you the plane, and the spherical is not included. Again the advantage of starting to work from the whole to the particular.
Now, something else I was brought up with that schooling. I was taught fractions, and the teacher taught me that I could not have on top of the fraction, elephants, and peas below. You had to have elephants both top and bottom in a fraction. You could not fractionate dissimilar phenomena. That all felt fine, seemed logical, until we came to trigonometry, and they suddenly began to give you sign and cosine, so I said "What are those words?" "Show it to me." So they said "I can't really show it to you because it is a ratio between two it's a ratio between this edge of the triangle and this angle." So, a ratio is a fraction. So suddenly they were giving me elephants and peas and saying it was logical. One reason trigonometry has been difficult to people is because they insist on trying to equate, seemingly, dissimilar phenomena.
But, if we get into spherical trigonometry we have no trouble at all, because we then realize that the edges of the triangle are simply the arc of the central angle of the sphere. So you have central angles and surface angles ALL ANGLES. So your fractions are entirely between angles. That comes in all simple and nice, and gets to feel pretty good. In other words, the way that trigonometry is taught, you absolutely, automatically, cut the kids feelings right out. You say, this is something not like these are signs and cosines. They are exempt from the elephants and peas. This is when they said "Mathematics is something purely abstract forget about all those models." These are the disconnects that I talked to you about when I was trying to find, how do we get back to the conceptual and to our experience so that humanity can understand all of science? And they can!
The more you play with what I'm talking about, the more fun you're going to begin to have, and you're going to find it very easy to take ping pong balls and begin to try out great circles on them, and they're nice to write on ping pong balls, and the colored pens write on them nicely, and it's very easy to get make great circle rulers, so you find out what the diameter of the ping pong ball is, and then you get a little, like a napkin ring, that's just half of that, just the radius in depth, and you sit the little ball in it. And then you just draw on it all these great circles. So you can take the ball it doesn't make any difference, just get any two points and then just connect those points into a lovely great circle. You're going to find it a great deal of fun to play with great circles and have concentric triangles and see the way in which the angles begin to decrease.
The little man, then trying to start with a flat earth, and squares and cubes he said were just great. And just looking at the inside of the triangle inside of the square, looking at what nothing what we do there is teaching him to be absolutely biased. My side is right. My town is inside the wall here. This is, incredibly unbalancing to the little child to be exposed to such bias. I hope you feel more and more with me the sense of responsibility to the child. That little child starting out here, and how easy it was to give them misinformation. How easy it was for parents, just loving their kids to pieces, to say, well that rich man got this tutor, and he must know; so we'll get that tutor, and the tutor tells it his way, and it may be very ignorant. And how quickly conditioned reflexes develop about who is the authority about what.
But the minute you begin to do your own thinking and go back to the experiential basis of things, you can't get fooled. And you continually get better information. It's just so exciting the lovely, clean things you really, suddenly, every time you get understanding instead of something you memorize, some little local thing you memorize isolated from other things.
At any rate I, all my early the globes used to be always fastened to things. I always kept cutting them lose, and finally because you can always set a globe into a circle. As long as the circle is a lesser circle, it will always feel comfortable there. So it can sit on any bowl, any dish or so forth. so you can have your globes and really get feeling your whole earth. I suggest to all of you that you have plenty of globes around and get to seeing things this way.
When I developed my first map, it was in the 30's, and Life magazine published it in February, l943. There was a year or so getting ready to do it, and we went thru some very interesting experiences. The art editor of LIFE was a very good friend of mine. In fact I'd been on TIME and LIFE so I knew everybody there, and it was just after I left LIFE that they decided they'd like to do my map. The LIFE magazine didn't want to go into something like that without being sure that this wasn't something well known to geographers long ago and just sort of a rediscovery, and so they got two great experts. One was Dr. Boggs who was the Chief Geographer of the State Department, and the other was the man who was the President and Chief Cartographer of the American Geographical Society. And they got two mathematicians. All of the experts said to the LIFE magazine, "this is pure invention." I didn't conform to any of the well know mathematics, so obviously it was some kind of a fudging out invention meant just sort of fudged, so somehow I tricked everybody. So, I have a wonderful patent attorney. He was considered at that time the best in America. That was way back in the 30's.
And he said, "that's wonderful, we have the great experts saying, and testifying that it is pure invention" because the Chief Patent Examiner of the United States Patent Office, had ruled in 1900 that it is impossible to patent anything cartographic anymore. That all the mathematics had been exhausted, so that it was tabooed and never considered. So I got the first and only patent on a method of projection that has ever been granted in the United States Patent Office. Out of all these experts saying that it was "pure invention." It was just great!
So, at any rate, life took heart and decided to publish it anyway, and we published it in color, and it was a great hit. As LIFE and those big magazines do publish to get out a dummy a little ahead, and if they get if it looks like a good one and they'd make advertising from it they had eighteen pages of color, and you could paste the edges together and put it together. So, I'll never forget the Sunday, when it came out the War was right on, it was deep. And I went from Washington to New York that Sunday, and all around the street I saw kids going around with these globes. It was an unprecedented number for them. They went to 2 million. They had never had any such issue before.
Well, Henry Luce, was very excited that they were doing this. And he there were two couriers coming from Australia, going to Churchill in England. They were two very especially eminent, and they were just coming thru New York, and Henry Luce was handling all the public relations for England in the United States so he was very privy these men came to him on the way to Churchill. And Henry decided that he would like to have my map taken to Churchill. He got the print outs before it was actually published. So the couriers did take it. So I put the so the couriers would understand how to show it to Mr. Churchill, I put the pieces together, and when I did this, I put the pieces together with Australia at the center. It was very much the water-ocean world as you see it here. But I made Australia actually the center of it, and the said "Gee, that's the way the world really is, isn't it?" I just want to point out the way things look to the local person that's the way the world is. Then I showed them the other way it could be, and that they ought to put it together for Mr. Churchill this way, and they did that too. But, all being exactly the same pieces just rearranged.
Incidentally, while Life was making up their mind about whether they would use my map, I had made many, many drawings, and I had them all in little cartoons and so forth, and Henry Luce asked me to come to his house in Westchester for the weekend. And he had a number of guests. And after dinner, they asked me to explain my map, so I was doing it on the parlor floor with people sitting around and so forth. And Henry was sitting over to one side , and I was explaining many things from the map, and grand strategies of different countries in history and so forth, the way the map tended to change with it. And Henry said "Bucky, every human being has an exact opposite, this is your poles, you have your antipathies on there you've been showing." He said, "You are might exact opposite." I said "That is very much of a compliment. How do you happen to give me such a high position?" And he said "Well, you seem to think there is something going on in this Universe..." he was an absolute freewheeler, he was sure, he said anything that I want to do I can do and you seem to keep introducing large patterns that are super to man, and he said "I find this a very disturbing way of looking at things." He felt that all the big patterns were purely man made. I was amazed that people in powerful positions can get to see things that way. He was used to enormous power what his magazine really could do to people around the world, gave him that feeling.
Now, incidentally, I liked Henry Luce very much. We were good friends for life, and one of the things that I had said at that occasion before the map was published was that there was a northwest spiraling of humanity going on sum totally really centers of population were moving westward if we took the sum total. That there had been a time when man had drifted east in rafts but since this sailing business, everybody is really trying to follow the sun. And, it's quite easy to show that northwest spiral, and then how Europe came into North America and didn't go south America, it was going north.
And if you go into the map of the United States, we have, you'll notice I have colors, and the color lines, I have the coldest is a blue and gets green, and yellow, and red is the warmest. There is a little line here where we go between yellow and green and the United States goes like that. And you can see it here in Asia, where ever it might be. You can see it going right through Europe here. That is the freezing line. That is where the average mean low temperature is 30 degrees. So North of that line you might get frozen if you didn't have clothing, for instance. If you are starting with naked man, that was a pretty, very formidable line at the outset. I found, then, that that line was really we get where, remember I've shown you pictures of where the people are, there's another one I showed you with little lights of where all the people are. Look around here, this is where enormous numbers are in here. This is where our population is.
I found that people tended to get up as close to the cold as they could because the ice was good for preserving things, but also you could freeze to death. And also, the cooler you get where there is less disease, and less infection. It seemed to be sort of the health area, and humanity had tended to find that.
Incidentally, when you get to mountains, there are rings around them. This is simply the sum total around the world. I didn't make separate rings with my colors going up to the mountains. But they do have all of those temperature changes, as you know.
So, one of the things that fascinated me was the censuses of the United States since 1790, it is quite easy then, demographically, mathematically to show where the center of population is. And the center of population of America has gone right from the very beginning was right on that 30 degrees low mean temperature. And so starting up here very much in the Philadelphia area , and then goes southward like that and then out into Indiana state, always right along the line. But, at the time of the Civil War, the population went just a little north of the freezing line. It had been just below it, following it, but the Civil War was an amazing moment. This is 1851 production steel. So for the first time, man had been doing his primary work really in the fields, and this was the beginning of the industrial where he is able now to have environment controls, and he goes in doors. So he is able, then, to be a little north of that line. So there is a tendency to have the industrialization north of the line. It sort of started with the beginnings being able to get north of the line.
At any rate, the tendency was still to get a little more northwardly, so I saw that the next population that I really could make predictions where they would be and they would get to working more and more northwest. So, I also showed this to Henry Luce, and I said, this, what had been called the British Empire, that I have talked to you about as being the British Empire, which I've also have shown you really wasn't the British Empire but really was the East India Company. That the East India Company were the actual enterprise the people who put their money in it, was limited limited liability where the individual risker could not be punished by somebody suing the outfit. We call that incorporated in America, but it was LTD., limited, in the English world. And the East India Company, itself, was moving west. The speculators began to move west.
And I said to Henry Luce, in due course this British Empire was going to be the control the economic controls were moving into North America, and I said that the English, whatever is English, that stays as English at all, is really going to move west. So, I said, they'll move the headquarters into Canada in due course. And Canada will begin to be strong. And you find this is where their investments are going. And he was very being then, as I said he was an Anglophile he was born in China went to Yale University. He was born in China in old China and he had a feeling about the English and Hong Kong and so forth, and as I said, he was the official public relations propagandist for the British Empire during World War II, and so he was concerned when I said that things were moving.
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.
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.
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.
And then it goes the negative five, the negative four, the negative three. It's just the same transferring between the northern and southern hemisphere that kind of idea.
Next picture please. Now here I've if we can sharpen this up all you can yes. I have here an array, and want to look into something on what would be the left hand column as you are looking at it. Up at the top there is one ball, and then below it there are two balls, and then there are three balls and four balls. What I am studying here are relationships between numbers of unique events. And up at the top one ball doesn't have any communication to anybody. It takes two balls to have communication or relationship. Like I said, no "otherness" no me. My awareness really begins with "otherness" because there must be some relationship. So, if I have two balls, I want to have the way we're going to say it is "How many private telephones do we have to have to talk between this "A" and "B"? You need only one private telephone because there is just "A" and "B". So two people have just one telephone. Now I'm going to have three people, so that there is "A", "B" and "C". So I'm going to have to have a telephone "AB" "AC" and "BC". I'm going to have to have three telephones in our telephone system. Any two people must have an absolute private wire. So now I'm going to have four people "A" "B" "C" and "D". I'm going to have to have and what you're looking at in the left hand column there are the telephone wires between little points. So between "A", "B" , "C", "D", I'm going to have to have AB, AC, AD that's three of them, and then I have to have BC and BD that 's five, and then I have to have "CD". It takes six. Four people have to have six private wires. This is our friend tetrahedron, it has to have six edges. The four requires six connectors. So, in fact you'll find, this gets to be a sort of fundamental model that way.
You find, then, the next there are two columns I have up there. One is the actual number of telephone wires you have, and then we put up the number, because, the first case we find we needed none , then we start with one between two. Three requires three. Four requires six. And five requires now I tell you, it is N to the second power minus N over 2. So you finally, you actually learn the equation. If I have 5, N is 5. N to the second power is twenty-five, minus five is twenty, so divide it by two is ten. You're going to find you have ten. So the numbers are going like one, three, six, ten.
Now the next one would be six. So six to the second power would be thirty six minus six is thirty divided by two is fifteen. (From the audience someone said "Do you want to draw that on here? Bucky "It would be nice if we could have this chart on here" From the audience "Because we can't get it much clearer than it is, and you could probably see it better if it were drawn but do it, do it your way." Bucky "Are you going to introduce this later then? or what?" From the audience "If you wanted to just draw it, the connections on the board you could see it a lot better" Bucky, "oh, oh, oh, I see. I think you're getting it, on your own, perfectly clearly." So, if I could go back to the drawing itself, the first one is all the people and their telephone wires, then the next was a summary of how many telephones are needed. It's a vertical line of columns, and I find that those numbers, l, 3, 6, 10, 15 are actually the total number of balls in a triangular collection. First you have one ball, then three balls, then you get six balls in the next triangular collection. Remember one, two-three, four-five-six-seven-eight-nine-ten. The next one is fifteen. So we find that the numbers of relationships are triangular numbers, as I call them. That gets to be pretty interesting.
Then we find that those triangular numbers are fascinating because, if I take three balls for instance, and I sit them on the what, the next is six. Three plus six equals nine. Or if I took the six and had it sit on the ten I get sixteen. Or if I took the ten and put it on the fifteen I get 25. What do we get? 9, 16, 25. Now these are second powers. In other words, any two sets make what we call the second power. I don't use the word "square" any more you notice. I always say the words "second power", "third power", "fourth power", I never get caught with saying "squaring" and "cubing". So we find that, here is a triangle sitting on top a triangle always one ball less, sitting on top of one, and it makes this second power number. You'll find as you take that top one and hinge it over, it lays over and makes a diamond. And when you look at that it's the diamond. "May we have the picture itself back here, because we have the, these pictures are " The diamond then, you'll see they're stacked up there. And then we get into the diamonds where, a diamond simply is a square, but remember these nest at 60 degree angles instead of 90. So it's a diamond. You can count up the you can understand it's the second power to see how the second half completes how the six completes the ten.
Now, then we find, if I stack these layers, two layers together, then I get tetrahedron vertically. So I find that whereas you and I need a private telephone, or any two of us want that private telephone, I have that in there. I could also call those the relationships between our experiences. You and a child have an experience, and have another experience. So it's a relationship with. If you begin to understand, you begin to understand the relationships between experiences.
So then, I find, if I stack the relationship of all my experiences together, in the end it comes out to be a tetrahedron of such and such a frequency. In other words the numbers of the telephone were triangular numbers, and the sums of all of which these numbers of telephones, then, were all the experiences I have had in my life. These were all the relationships between all the experiences I have had in my life. These are all of the understandings I have had of all of the experiences in my life. These are understanding relationships between those points. That's what those triangular numbers are the understandings of our relationships. And then those are individual experiences. Now, I keep integrating this set of understandings with a new one. I've just had a new experience so the final tetrahedronal stack up there is the sum of all the relationships between all of the experiences you have had. To really understand being a comprehensivist and the way I carry on with you, is very much relating then to all those relationships. Well, I'm really carrying on in a very "tetrahedronal" manner as far as the that it comes out in this beautiful, elegant, tetrahedron is just one more whatever it is always comes out a whole, rational tetrahedronal form a whole triangular form. I find this very exciting. I call this then the underlying orderliness in superficial disorder. Where the experiences seem to you and I to be very disorderly and random, but suddenly find underlying the whole thing absolute order. You cannot become disorderly. It becomes really a very exciting matter.
May I have the next picture then. I told you earlier about the two General Dynamic Scientists who were making experiments with titanium sheet. They were making experiments with titanium relative to re-entry problems in the rocketry. Do you remember that? And they had two hemispheres, one a half an inch less in radius than the other, and that they were concentrically arranged, and then they were actually sealed up, welded up at the bottom, and so there was a space between each one of them, a half inch. And then they clamped it into a frame, and this is what you are looking at in the picture. And the atmosphere is able to come in underneath the frame, into the inside of the dome. So the atmospheric pressure pushes the inner dome outwardly like that very normally.
But then they exhaust with a vacuum pump the air in between the two so the atmosphere operating in the outside one pushes it in towards the other one and it dimples in, I spoke about "dimpling in", in the same exact icosahedronal this happens to be, it turns into a four frequency geodesic tensegrity structure. In other words, I had also been giving you the way the molecules of gases operate and so forth. And so they always want to get the most economical, which means always A great circle. So they insist, not on lesser circles, they get beautifully into whole great circles. And when two great circles, remember, crossed the disk you've got three great circles triangle, and now with all this triangulation in there, there is an enormous amount of action in there, so they average out an equilateral they keep trying to get absolutely equilateral. So the whole thing just makes itself get orderly. Time and again I get so excited to find how beautifully life is really carrying on, how the Universe under all these things going on around us, and there's this lovely order.
Next picture. It's fun to see what's going to come up here! Now, we see three lines crossing one another. Now one of the great differences between myself and the mathematicians is that all the mathematicians assume you can have a plurality of lines going thru the same point at the same time. As you get into geometry where they get into the Non-Euclidean Geometry, get into the hyperbolic or they assume, still, a plurality of lines going through the same point at the same time. This is exactly what physics finds can't happen! And I say the line HAS to be an action. They have their lines going thru the same point at the same time is the point. And, I simply say to you, then that the lines, if one could go through it and then another if you had a machine gun and then another machine gun and synchronize them so that one went thru, then the other, and it might look like it was a couple of lines crossing but they're not.
Hold the picture a minute. I want you to notice then how those three lines are really crossing. And what they do then, is that one has to be superimposed on the other. So they, just automatically, do what is going on here.
May I have the next picture please. And in this next picture here, then, we see what the physicist is saying. The difference between the mathematician and the physicist is that we find that when we have two events because a line is an event there is just no question about it. You cannot have just an imaginary I proved that to you yesterday. We have an event and there is already an action taking place. Therefore, it's what we call an interference. And with an interference it could be a glancing blow, and be what you call then a refraction just a little ticking here, and it changes it's angle a little pssssss that brings about refraction of life, incidentally. Just exactly that.
There can be a very tight blow like that, may I have the picture back again please, and we get then a bounce-off. Then there could be another one where they go in first, and through; and then a fourth one where we have a smash up. Now these are all the things that go on in the cloud chamber when the physicist is bombarding, or sending a neutron or whatever it may be. You can really see these "bouncings around". And they are all to do then with refracting, reflecting, refracting-reflecting-smashing, or the one fourth one there where they are going almost the same direction and get in what they call critical proximity, and they get one of those mass attractions and they become one. If the angles of convergence are close enough you then really can see that mass attraction taking over, and see them pull over.
We had a very interesting experience in the navy in World War I. I told you I was in a transport service where we took these l30 men over. First were the German submarines, an enormous hazard. They tried running at night without lights. Of course they didn't want any lights, because the submarine was laying off there watching for silhouetting against those lights and so forth, lights were an anathema, so you didn't want any lights. So these enormous groups of ships were going together and they had a they wanted to stay together, and yet you had the enormous danger of following the other men at night there. This was a tortuous kind of game. On one occasion two of the big transports found, just in avoiding trouble, the end of one got overlapping the other one like this, pretty close. They were two big ships the Mount Vernon and the George Washington. Both big ships up in the 30 or 40 thousand tons big. And they get where, being in water, and being big ships, their mass was enough so that they began to attract one another.
That's one of the when you get big ships at sea, they really begin to show that mass attraction that you don't see of two apples sitting on the table where the pull of gravity is so great and the friction of the apple on the table, they don't try to go towards each other. But these two ships at sea and the acceleration, found that they were being pulled together. And as they got the seas were heavy and they just chewed each other up, and lost I think it involved on that particular case, there were some many were lost, about 20,000 human beings!
The captains of these two ships tried to see what they could do, because the mass attraction went if you tried to get too fast the bow would come over and so forth. They tried to figure out how one could accelerate a little faster than the other and they made trouble. They finally found that all they could possibly do was to open up an angle between them, and just keep them apart, the idea that they could pull apart until they could get out of that critical proximity, because of the second power business. They finally did, but they were actually a whole day pulling apart.
So, when I spoke about lines coming together one, you could have then light refraction, another one you could have reflection real bounce off. You could have a smash up. Or you could have angular convergence be so delicate that you would have critical proximity and a possible fall in. Those are all the things that happen in the cloud chamber this is all the physicists deal in. And this is exactly what the mathematician has completely lost. He has lost all the privilege of the thing, because he says the lines are going thru the same point at the same time. This is typical, again, of what these false assumptions that superficially say "It's obvious they said, two lines go through a point, anybody can see that two chalk lines on the blackboard." But the actual fact, if I do it on the board, here, chalk or whatever it may be look at it here, you will see it literally cross exactly like two, let's go in the snow, and you have this tire go this way, and the next one, the top one is, quite clearly, separate. It always is.
So, being completely experiential, completely operational, these are the kind of very exciting informations we get. So now we're playing a geometry where we don't get deceived. And this is very, very important when dealing in all those, because it uses vectors, always, my geometry is vectorial.
Now this brings me then to explaining to you a little about my grand strategy, when I was young saying, "I don't think that Nature has any Department of Physics, Mathematics," I said that to you, "Biology, and has to have department meetings to know what to do when a leaf drops in the water." I said "I think she has only one department, and I would like if I can to find what it is. Because, the chemistry of it says it is very simple. It says H2 O. And that's the way it associates, in a very beautiful low number. So if I'm going to do my geometry, and I didn't like, I said, at all the geometry where the teacher said, 'This is a ghost cube', and it didn't have any longevity, it didn't have any heat, it didn't have any weight, "I want to get those qualities in."
Now, I'm not the first one who has wanted to do that. The scientists going really back to Babylon were trying very hard to do that when they chose the 60 minutes and 60 seconds as a fraction, hoping they'd be able to correlate it with the circle and so forth of trigonometry and we have then, the scientists having x,y,z coordinates, and then they needed to have those qualities of mass; so they couldn't find, really, anything. So they very arbitrarily, getting into the centimeter as nice as it is, it is o.k. in relation to decimal system, if decimal system is what nature is using, but the arbitrary thing is decimal because we've got five fingers on each hand. So they insist, then, that it's going to be decimal.
So we have then the centimeter one cubic centimeter, and it is really a cube, cubic centimeter with water they said, now we have what we call a gram. And we know that the weight of that water is the basis of weight in relation to volume. We want the weight and volume coming together. Then they found that the water changed its volume with temperature. They hadn't thought about that and that became very upsetting thing, so they finally had to add that 4 degrees centigrade in the temperature there, then it fills one cubic centimeter. That gave you then, the official gram. So then we have, I spoke about dealing in weight, lifting a given weight against gravity, a given distance. So, lifting one gram one centimeter in one second CGS this is the centimeter gram second, and in relation to x,y,z coordination, this became the built-ins. But even then, they didn't have time in there. So the longevity, it didn't say how old the water was. There was nothing in there to really identify time. So I became interested, and I said "Nature does have her time, because I find that you are just so old, and I'm so old, etc. There was a time dimension. I'd like to have that in there.
So, here is the way my own strategy began, you might as well just know because this goes really, way way back. So, in physics, mathematics in my preparatory school for Harvard, I became very interested in, for instance, the fact that Avogadro oh, I liked vectors. I said that if I used the vector, the vector is an experience. And does represent an event. And does represent an amount of physical Universe as mass, going in a given direction at a given velocity. Because velocity then has both has time in it. So this is very satisfactory so I got both frequency and time all of these things nicely in there, and those kind of vectorial lines are beautiful because they don't go to infinity. Again, the mathematician kept telling me about infinity and I said, I remember when the teacher said "This line goes on to infinity," and I said "have you ever been there?" She said "No". I said "how do you know it goes to infinity then?" At any rate, so I said "Well where does the other end go to?" She said "infinity." I said "which way is infinity, then?" So she couldn't tell me which way infinity was, and I didn't like this infinity very much. I liked something , because I also had never personally experienced infinity, and I'd like to have something that went along with my experience. So I like vectors because they are an absolutely discrete length of line. They do not have inherent extension they are just exactly what you see there.
So I thought, I wonder if I can't get up a geometry out of vectors. Because that then would have then the time quality, and would have the velocity, and the velocity and mass impact converted to heat, so it would have all the elements of experience in it. So if I could only get a geometry of vectors, that would be great. Then came the moment in my learning about science that we were learning about Avogadro. And Avogadro had a very extraordinary intuitive awareness, I spoke to you earlier about the human beings, and the five lights in the sky and becoming interested in it. I also spoke to you about Priestly making his experiment with fire under a bell jar, and how Lavoisier identified he said why the products of the fire added up to more than the weight of the things he put in it. Which was because something else had joined in, and it came out of the "nothingness" which was the air, out of this then we get for the first time that elements were gases. And this was so terribly important as to really open up as I said, thermodynamics and everything. It is not surprising that the next five elements to be discovered were all gases, and it was because we had the enormous competition of who was going to run the ocean world, so the French were putting up money for their scientists, and the kings of France and the king of Spain, and everybody was putting up money for the scientists and so we have Lavoisier is French; and incidentally, one of the most extraordinary things that society ever did that was blind and short-sighted was that in the French Revolution they cut off Lavoisier's head. Of all the heads to cut off! I can't think of a worse choice. At any rate. He was so excited, he introduced then this gas business, and realized that it was the very essence of the understanding of steam.
Therefore, the English, who did want all the Great Pirates had headquarters in England, so they were putting up money, so Cavendish the next five chemical elements were all gases were all Cavendish's. Now we have enormous preoccupation with these gases Boyle and others, and amongst them was an Italian scientists, Avogadro, and Avogadro very astutely looking at all, comprehensively not to being too specialized, looking at the total idea of total gases as his patrons came and said I want you to catch up to these boys, and we'd better have better steamships than those other boys or what ever it is. Avogadro then said, "it looks as though that all gases under identical pressure and heat would disclose the same number of molecules for given volume." "Boy, I said, this is something!" He then went onto prove it. Suddenly then we have volume and number for a plurality of gases which are elements. And you know how that elements are elements because they are unique were coming together volumetrically and number-wise, which is very much better than just putting water into a cube. Nice! So I said "It could be because elements go through their liquid, their gaseous and their crystalline states there seemed to be that kind of inter-transformability. And the only reason that certain things are crystalline in our planet is its relative conditions of this part of Universe. There has to be this set of heat. And in the sun they're going to be incandescent, they may be plasmic. I see then, because the elements can go thru liquid, gaseous and we might then think about all the elements under some identical conditions. So instead of just saying under identical conditions of heat or pressure. That's what he has said about the molecules of gas. I said, it could be, you could generalize that, and all elements under identical energy conditions, which means under either heat, or pressure, or any of them, might disclose the same number of somethings per given volume. I don't know what is going to show up. But I thought, that should be more or less the nature of the generalization. So I said, "If then, I want to have a geometry made of vectors, and all the energy conditions are the same, then all the vectors will be the same. That's wonderful! " So I say, it not only means they are all going to be the same length, but they are going to be converging at the same kind of angles. I said, can you make a model where all the vectors are the same length, and they are all converging and also, but they have to take care of the actions and reactions and resultants so they must be angles joining angles at both ends of the lines. Can I get a system where all the vectors are in a closed system? All the same length? and all the angles are the same? And that turned out to be exactly what you are looking at up here.
Again, a very fortunate thing happened in my life. Often what seems to be misfortune turns out to be fortune. I find that life is highly compensatory. Because I was very, very short-sighted when I was born. So short-sighted that, so very far sighted, that, I can't see when I have my glasses off, I see exactly now what I saw when I was four years of age. The correction has not been changed at all in all those years. This is my 76th year of these lenses. I see an absolute blur of faces I cannot see human eyes, I can actually make out some darkness where eyes are, and I can see more or less a shadow of noses, I can see the two sort of dark colors it's purely a color matter, there is nothing the matter with my spectrum of colors, so there is a pinkness this side of your face is a little lighter than that side of your face, and I get really just a sort of shadowiness that are the eyes. Very much like a Lorenzian kind of a painting. And there is a little bit more of a pinkness in here. I can make out a little color differential. So I didn't see a human eye until I was four and a half years of age, when I got my glasses.
Now, because of that, I tended to try to get my I didn't know, how would a little child know? what I see is not what everybody else sees, so I assumed that everybody else was seeing the same way. But my problem of understanding was really quite a different one from the people who could see the details. And I had a sister three years older than I was, and she was continually telling me the things she could see. And I thought she was making it up, so I didn't want every child has imagination, so I thought and I had been read fairy stories these are just fairy stories here, so I'd invent what I could see. And I could see some very extraordinary things, and I would always get laughs. And my sister didn't seem to get any laughs for her description of what she could see. So you can imagine what happened when I was suddenly four years of age, and I saw that she hadn't been telling me stories at all. And I suddenly saw hairs. Now, I spoke about, there is a compensation here.
I went to kindergarten before I got my glasses, and in the kindergarten, the teacher had some dried peas semi-dried peas, and she had toothpicks and she told us to make structures. To stick the end of the toothpick in the pea, and we found that they joined the tooth picks so that you could make structure. All the kids that could see, the minute they were told to make a structure, immediately tried to imitate houses. That was the first thing they thought of. So they were all rectilinear. Now what I did, because I couldn't see at all, I wanted to feel something that feels good, so that a square and those forms didn't feel right, but when I got to triangles, they felt great. So I could really feel that was nice and stable, so I made, literally a structure like that. And I remember the teacher calling in other teachers from the other school there to come over, and they were all very surprised as to why I had made this strange thing. But it was purely a consequence of my not being able to see. I was not trying to imitate something, I really she said, make structure, and I just got to where it felt right. You can understand that, somebody going around groping their way it's purely a matter of feeling. And anybody working in clay would just have that kind of feeling whether it's going to tip over or not or if it's cohering.
At any rate, I did do that. So it was something deep in me, also about that so when we have the moment of my being excited by Avogadro, he seemed to be giving me some Universal condition, and I had wanted to use vectors, and I felt this I said, I think I can make this. Now, this is called in mathematics an Isotropic Vector Matrix. Isotropic everywhere the same. Isotropic Vector Matrix. So I found I could make an Isotropic Vector Matrix, and that was just great, but then the Isotropic Vector Matrix turned out to be simply spheres in closest packing. Remember, the "two" the "me" and the "otherness", and then we suddenly come together, here, and suddenly we roll around on each other. These little Styrofoam balls are great to play to get the feeling of that rolling around, and from where I am, the fact that I'm rolling, you don't really notice that the profiles stay just the same. And so then I get three of them rolling together, and I get the one on top nesting in it the four. And then, there are your six vectors and. so spheres in closest packing, all the same radius unit radius spheres, closest packing, they simply, automatically come out the Isotropic Vector Matrix.
Now, we were interested in atoms and how the atoms behave and the volumes of numbers per volume and all those things. In every kind of a way this is a very satisfactory matter. It was at that point that I also said, I would like to see about a nucleus. And that's what brought me then, into finding the twelve six around one in a plane, and three on top and three on the bottom, giving us then, the vector equilibrium.
I would like, we have a lot of these balls, and I really would like you to do some experiments with them yourself. And I don't want to slow the picture down too much here. And you all are getting pretty well versed here. Now I've also given you tetrahedron, and I've also given you the idea of Euler's topology that vertexes and the numbers of the edges are not the same as the vertexes, remember. For every vertex we are going to have three edges and two faces. There is an absolute relative constant abundance of those in Universe, in addition to the polarity "twoness" which has to be taken care of, and that was what was not recognized by the topologists. They didn't realize that there was a hierarchy. It was never really understood that there is this hierarchy I am finding completely my own discovery to come then where the tetrahedron was unity. Which, you can see how absolutely logical it was. It was the minimum omni-triangulated vectorial model. And the cube just didn't work. It just was very uncomfortable, so you can see how quickly I really got into, sort of spontaneously, in here.
I want you to see that this is how a child carries on. I'm just going I think of all the things that were, I find important I don't think anything is quite so important as naivete. Just cherish your naivete. Don't let anybody try to belittle you because they say you are naive. This is the most beautiful thing we have. So I think I have been very naive many times, and the as a sort of off character, seeing the wrong way at first with the teacher and everything; and then, I didn't wear my big glasses. I was a kind of ugly looking character anyway, and I NEVER was any teacher's favorite, I assure you. And often, my friends discovered that to such an extent they found that they could play tricks and I'd be the one who'd get the blame. It was sort of a standard matter in every class I was in. I was continually having to stay after school and so forth, and when you stay after school and the teacher is rather nice, and got to saying, you really like mathematics very much and I'll teach you some more. Actually at Milton Academy where I was the mathematics teacher said, "I think you might just as well go on into college of mathematics ." So, I went through a whole lot of mathematics while I was still in school. Simply, this was afternoon time when I was being penned in anyway, so, it came out fine. As I said, there are compensations that go on. And the kids didn't know what a favor they were doing me. I didn't know either.
I don't want you to think at anytime that I'm being something abnormally smart here. Everything that has happened in my life here so far, as far as I can see it, comes really out of the physical circumstances. And, of all the things I think my not being able to see properly nothing was quite so important. Because at four and a half imagine if you had never seen a hair and you suddenly see a hair. And you'd never seen a dew drop before. Can you imagine. I hadn't seen eyes particularly eyes. And particularly eyes of creatures, and eyes of those kittens and eyes of the snake. And I seem to be able to talk with people's eyes. I just love snakes and toads and they like me and we get on, and I fill my blouse up with them and people would holler about that. But I want you to understand, I don't think anything went on with me here, that wasn't just a very, fantastically normal, average character, with certain physical deficiencies at the outset which get fixed up. Cause at four and a half I really started life all over again. Imagine getting a second chance. I'm sure that through the years when things went badly, I would say, because I couldn't see at first, I'd say "When I don't see or I don't understand what's going on here, I think all I have to do is to wait a little." There must be something in me psychologically that came about through that delay and that second sight. That second take where you really suddenly understood. I think this has helped me to hold on and to hold on many times to the total package. And certainly that business of not being able to see details I was having to put together smelling, and hearing, and touching. Which were very much less effective than the seeing. So I was using the three-way system to sort of zero in.
And, personally, my grandson says to me now that he's not sure whether I'm really being effective with human beings when I say this thing, but I am confident that the only thing that is important about this particular character me is that I do represent an average character, more or less getting peeled off by something wrong, like the kids fooling you so you get to peel off for the afternoon. I found myself getting isolated more and more but it gives me more and more perspective on the show.
And tending then to see long distances, to put together big patterns. When I say, then, as far as I am concerned, I am very clearly a demonstration of what any human being can do if they are disembarrassed of the game where society is trying to make you a specialist no question about society doing what it did in great love. The grownups really feeling an enormous affection they are sure they are doing the right things for their kids. There is anything but malevolence in here. I don't think, again, that you can see anything if you assume bad or good in here, you have to understand how people got caught in the picture they are in, why people do what they do, and you may find out something.
Now, I have some more slides I'd like to go on with. Oh! there is something I'd like to show you here, because I find there are five of these balls and five sets of the tetrahedron have been put together. You remember how when we had a tetrahedron, just look at it in the corner, you can get four tetrahedron and one octahedron in the center. You remember that. So that's getting clearer and clearer to you. But I also pointed out to you that vertices and edges are not the same. So sometimes you're looking at tetrahedron and they look like this. And when you're doing that, these are entirely vertices. The spheres are the little points enlarged. This is vertexial topologically. And then when I see it as a line, then I can make it out of solids, and those are areas. They really are, these are three different topological phenomena, and the counts come out differently. It's very important to realize this. Then we have two balls here, but there is only one edge between them. That's one reason why we came out where four balls had six inter-relationships. They are not the same number .
So, I wanted to show you how what you are very familiar with now, putting a tetrahedron, four tetrahedra in each corner, and one on top. So here is a tetrahedron. Another tetrahedron here. Another tetrahedron sits here. Now, we've learned it in areas where it looks like solids. Then you put a tetrahedron on top and you put an octahedron inside here. That's not what you do at all. You say the octahedron I'm going to take a tetrahedron and put it upside down. Excuse me. This tetrahedron I'm going to lay it in upside down, and each of these four are going to come in the middle of those like this, it fits right down there. Now, I'm going to take one more and it fits right on top there. Now we've got a three ball, or a two-frequency tetrahedron made out of five tetrahedra. Gives us the number five showing up quite interestingly, where you've got an octahedronal kind of four that's a prime number difference in there. It is very, very important not to get fooled about see it is very neat.
So vertexial associability comes out differently from edge associability or face associability. In fact, vertexial associability is the universal joint. This is the way the gases are, remember? And edge associability is liquids. Still flexible, distributing loads. And face associability is triple bonded so it's single bond, double bond, triple bond. And then we get the greatest tension, but no flexibility. No distributing loads any more. And this then is crystalline, this is liquid and this is gaseous. This is the way where I suddenly found out how to integrate Willard Gibbs' phase rule dealing in the chemistries of the liquids, crystalline and gases with the topology, so I find they are all the same.
And that was, again, a breakthrough. There have been very many breakthroughs in my life, that where you say you don't understand...for instance there is something, I'm sure you've heard of it the four color problem. Why do you need no more than four colors to do any mapping, so that you'll always have different colors between two areas of your map no matter what their shapes are. And this is, simply, because Nature does work then in tetrahedra. And a system a map is always going to be on the surface of a system. There is no plane all by itself. No interminable plane. It's always going to be on a surface, and it's always going to turn out then that, you've got a triangle, because you get down to triangles for the net structure, and the triangle on the outside of the sphere is the base of a tetrahedron whose center is the center of the sphere, and the tetrahedra are going to come apart that way. So the tetrahedron has four colors, and the four colors you have a red on the outside, and the blue, green yellow on the inside blue, green, white on this side.
You find, then, these act like gears, and every time you just have those four colors and for one outside you have three buried. Because the Universe is a three-way gearing. And they just can't come out wrong. So you can really make a model. This then proves for very, very many years, it has been said for over a century, nobody ever proved the four-color problem, show why, but this is fundamental because you and I are now dealing in FUNDAMENTAL systems. I have been able to say absolute limit conditions all the way through. We are at THE SIMPLEST and here it is. O.K. I now would like to have the next picture. This is really quite a simple one. I wanted you Remember I was talking about pulling could you eliminate me for a minute? you have on the left hand side, "pushing", and we find then it's tending to be a sphere. This is the precessional effect. You push on the two ends and the whole thing begins to increases in girth. And you pull on it and it contracts. I just want you to feel that. Now, before we go on to more pictures, I'm going to go through something that relates to a statement that we made here just a few minutes ago about, I just think about children and human beings apprehending.
So it is the touching and the smelling and the seeing and the hearing, and I am sure that sometime we're going to learn about this electro-magnetic-telepathetic interrelationship. But the only ones we know about are the touching, smelling, hearing so far, seeing. And, I find, as I grew up the Insurance Companies had equal indemnity for the loss of the hand and the loss of an eye, for instance. There was a sort of as I grew up there was a sort of feeling, you were being told by grown ups, you know you may lose your hearing, but you've still got your sight. They were sort of alternate faculties, that's the way they were looked at. They were not really evaluated very tightly. And I felt a great necessity to make an experiential, operational assessment of our senses. Having been, then, myself not having had the seeing, at the outset and having had primarily the hearing, the smelling and the touching. Because, as I was young, I can't tell you how much I smelled. The smelling was very, very important. I really was almost like a dog, and I knew people by their smells. To such an extent, my father said that, just in the last few years my sensitivity of smelling is going down like my hearing. It seems to be nervous, the nerves are breaking down, but through my life smelling had been so I always really smelled people. That was the first thing about them. And if they smelled wrong to me, they were not going to be my kind of a pal that's all. There was no question about their smelling. When they looked great, and didn't smell right, I learned to just turn away get out of the way. There'd just be trouble.
So, now I'm going to go into this assessment of our nerves. And I'd like to make an experiment on a good scientific basis now. And I'm going to take my body, and I'm going to bandage up my mouth and my nose and I'm going to get myself an oxygen tube so that I won't be asphyxiated. And I can't smell anything I'm not going to use it but I just know that the oxygen is very prominent in there, but that's not what I'm dealing with in here it's not going to give me any information. I'm going to cover up my ears, I'm covering up my eyes they're all blind folded. I have only my body. I'm naked, and I need to get some information. And certainly, in any unknown territory, I'm not going to step over here. I have no experience to tell me that there is a step there. And, I keep my balance on this foot, and I'm going to try it very, very much before I ever throw any weight there. It's like a kid testing ice. And, I'm also going to be, I don't want something run my head into things too, so I'm going to be doing this. As I move along I'm going to be very, very tentative, and, in every way, I'll be acting like an insect. And, under these conditions, I find that I'll also begin to get a little bit familiar with the floor, and I'll get to know it. I have a good angular sense, and I start feeling an orientation like this if I start turning like that, it's going to be like that. I can really feel what comes next here, because there really is a pivotal effect. And you begin to learn exactly what it is like under your feet.
I don't know if you've ever done this as a child, but often in the country is a place where you can just enjoy yourself tremendously a little kid, and they let you get out at night a little. And you find that there are paths, and you get so familiar of the feeling of the paths with your feet, that you can run down that path at night, really quite fast. Even though you can't see foggy night, you still feel very very comfortable with your feet. So I would say then, if you just had the touch, you gradually would learn about a certain amount of territory. And, it can be a fairly large territory, and if you feel very comfortable, within that territory, you would dare run. But with just my legs, and short as they are, I'm not going to cover very much of my planet earth. That's obvious.
The information that I would get, because I'm also going to be watching this as I run through the night, and I learn that there are other things that fall in there once in a while, so you can't be too sure about the spaces, so you're going to be doing this all the time as you are running around like that. So I see then that the limit of the information that I can reach, just standing still, as I start off, is the limit between one toe of one foot and the finger of another. This is the total. This is considerably more than my height. I'm going to have a very tall basketball player, and I'm going to give him ten feet stretch from his toe to his fingertip. Keep it a simple number. I simply say, then, that under the conditions of just touch, you have a static range of 10 feet. But under the dynamic, you and I can run possibly up to five miles an hour, and the amount of time in a total lifetime is, say, about how much? You and I would have a dynamic thing, where I could run five miles an hour to get information, plus a static reach of ten feet. So I'm going to put that on the board here.
We've got start on a tactile basis. I'm going to have a static range and a dynamic range. The static range is going to be ten feet, and this (the dynamic) is going to be 5MPH. A mile is about 5,000 feet. So we'll divide 5,000 by 10 to get down to 500. This is 1/500 mile. This is, then, I'm going to put 1/500 of a mile per hour. See, I want to get it relative to miles.
Now, the next thing we have, I'm going to cover up all my skin so that I can't get any information from my skin. So during that running I could get information all over my body to tell me what to do. And I also felt heat. But now I am going to cover my whole body, and I would also, with just my body, not come too close to a fire I would feel that heat. I'd feel the cold this way. You get a lot of information alright.
So now, I've covered up all my skin, and I can't move anymore I'm not allowed to do that. And I open up my nostrils my eyes are covered, and my ears are covered. That is all I have, and what can I learn. We have learned that sailors time and again in history sailors coming in from long sea voyages have been able to smell citrus groves and pines, at sea, with no wind, in the calm, a mile off shore. You can smell pine and citrus a mile away. So I've got a static, in this new one, this is now the olfactory, and I have in that a one mile static against a 1/500th MPH, and you can smell without that was one mile static; but if the wind was blowing it could bring you a lot of smells. And how fast can the wind blow? Well at earth's level, we have 400 MPH in some of the stratosphere, but the level at which you and I operate, the best winds we have, the fastest is down in the Antarctic, 180 mph. So with 180 miles an hour smells can be brought to you. And often we get forest fires, information about them really considerable distance, but I've taken the static, the airs dispel, they get thinner and thinner, so that the mile is as far as you can get with the so there is a dynamic additive to the smelling I think I can give you a 200 MPH dynamic added to the smelling. So, I didn't really do this all columns very well I'll leave these, these are all MPH here anyway. So we had 1/500 and 5. This we had 1 mile static, but I'm going to give you 200 MPH in the dynamic.
Now I'm going to cover up your nose and all your skin, and open up your ears for the first time. Your eyes are covered up. What do we know about this? We know that humans have heard the atomic bomb blasts in the desert 100 miles away. That's not usual. If it's just you and I trying to shout to one another we can hear an explosion considerably further than we can shout to one another. So we'll have to use the explosion. And the atomic bomb blast is the biggest. So we have to use the static here is 100 MPH. This, we're now dealing in hearing. And then, the speed of sound in air is 700 MPH. But if the wind is blowing your way it makes it come 900 MPH. It actually can blow towards you or blow away from you. Because it is in the air. There is, then, a dynamic additive, so this gets up to be 900 MPH.
Now, I cover up all of touch, smell and hear and open my eyes for the first time. And as I have been going into with you, you can see a galaxy a million light years away. I find that the range is so incredibly large, as I try to put down what it is that I am seeing. I'll have to take that's a good one, 100 million light years away. So we take our million and we multiply it times 6.5 trillion miles per year. So 6.5 trillion, what would we have here? We have your 6 zeros in 1,000,000. And in your trillion you have nine zeros, so you have 6.5 times one, so 6.5 x 1, and you have 6 and 9, you've got 15 zeros, you've got 10 to the 15th power. This is the number of miles you and I are seeing: 6.5 x 10 to the 15th power. And, if you want to say it out, we'd have then, let's see 15 zeros, so let's put it down that way so we're saying this is millions, billions, trillions, quadrillions, sextillions. The thing is, we can see a sextillion miles. So this is the sight. So it is an incredible figure so you just have to write it 10 to the fifteenth power. It would be 6.5 times that.
O.K. then that was the static range! Then to that I have to add really the velocity. Because that was the actual number of miles I saw. So the velocity which I see is also then 700 million MPH. That's my velocity. Now, if I tried to make a chart to plot these things, you'll find that these first three are really quite close to one another. So, I can get them on my chart alright, but for the fourth faculty, the seeing, I'd have to take an airplane, and I'd be going quite a long while before I'd get to a point where I could plot it. So that I find that the first three senses are really very closely coordinated, and the seeing one is very remote from them so the fact that I had those first three in the beginning, put me really in really quite a pedestrian kind of a way, and you can imagine the excitation of seeing this one secondarily. I think that kids probably using the first one primarily, and not really realizing what a jump-up it is. Again, I think this is why circumstance happened to take a very average character, a fairly efficient average character, and get it into all this trouble!
You'll find me continually then trying to find hierarchies of experience trying to get total experience all onto one chart. This is my proclivity all of the time. Now this tells me a lot more.
When a child is born, I'm sure many of you all of you have been near a newborn child. Your sister's or your mother's and so forth. And a little child is lying there, and if you notice it's hands seem to move rather deftly. At first they can move their heads, and suckling and so forth but their hands, and these motions are kind of spasmodic the legs and whatever. But the hands seem to close. You'll find yourself tempted, and I've talked to so many people who do, because I know immediately I had the temptation to put my finger into the little child's hand. And sure enough it closed on it just as deftly as can be. Such an amazing thing. I feel as though this little child has started talking through hands. And, I'd like to take my hand away, and it immediately opens up! So I started to pull away, and it opens right up accommodates. Put it together immediately couples up. Now, this is not to surprise us, because the child has been in tactile communication with its mother for months. It can't see, it can't hear it can't smell in the womb. But it feels, so the tactile is there. The tactile is already operative when the child comes out, highly developed, very sophisticated way of feeling. So it is considerably later that the next thing comes in outside of the womb which is then the olfactory smelling for the other and so forth, going after milk. Then, considerably later, a child begins to hear. And the very last thing he does is to see.
You'll find then there is a hierarchy of relative magnitudes of effectiveness, but they are employed in that order. The first one that is used is the tactile, the second is the olfactory, and the next is the hearing, and finally the seeing. This, again, when you find such an agreement of hierarchies, of rates of use and so forth, it makes you think a little more about it. So when I began to think about things I might do on behalf of the new life, in the way of developing environments, that will have available within them the means for the child to acquire what it wants to acquire in the way of information because that's what it's going after all the time information. It couldn't be more after information! And, absolutely nothing can interfere with it, with getting that information. And so, I'd like to have it get it safely, yet not I'd like it to know it can hurt, you have to have some feeling about what that gravity does. But I'd like it to know it gets hurt, but not the feeling that you're going to be done away with that you're going to get damaged I don't want the equipment damaged. I'm sure all those things are designable. And they must be thought of in this kind of intimacy.
Now, something I've just gone through very simply here, but this brought me to an awareness, because the touching is already operative, I find all the other faculties are rated in the terms, relatively, of the touching. So you say it's "feet" away touch. Distances are "feet". "Feet" or "hands high" or so forth. That's how man began talking. And, I find then, that all the things he does then are rated back again to the tactile that's it. That's the big one. To such an extent here comes to me, really a shocker. I find that you only see what you couldn't touch. You don't know me by what I'm saying, you don't know me by what I'm hearing, what I'm smelling. You know me by my touch. The dog tends to know me by my smell. But we just absolutely take the absolutely limit case that you only begin where I can touch you. Which is exactly what is not you. Now I say, it may be, it could be, that Democritus is sitting here with us, but every time I say the word "atom" Democritus is here. He invented the word. He had the conception. Democritus is just absolutely immortal. As long as you ever hear the word "atom" there is Democritus. So I say, what we really are, if you had been really paralyzed, on your tactile, when you were young, then you probably would begin where you smell me. And maybe a shorter time in the womb of a dog make him want to be much more smelling, you understand.
But the point being, also if you had been tactilely paralyzed, and you had no smelling, then I would be where you hear me that's exactly where I'd be. And if you had had the other three paralyzed, then I'd be where you could see me. And what you could see would be very, very different I'm sure. You'd probably see my emotions rather than, what we'd really be I don't know, but this really gave me insight into how incredibly, at the very beginnings of things we really are in our apprehending. We continually come back to the tactile. To such an extent, we put on different clothes, or whatever these make you look different. We put on masks and a little child comes around, and there is no question that masks are very powerful and the Africans learned it long ago. A mask is a mask. Boy!
I'm going to I'd like to come back to pictures. I ought to have something I would say, after I say "Come back to pictures." Now, we're looking at, you remember, I gave you degrees of freedom, where, there was myself and Universe. I think if that could be put sideways could that be put sideways so we because you're only showing two and I think there are four frames. The top one is where you are just the tether ball, and the, now we'll turn sideways and you'll see where the there it is the tether ball is way on the left. Then in the next one you see yourself in a plane where the three make a plane. You can find that you can move. You're looking at the degrees of freedom, and you are allowed to make you can make a plane. And, I'm sorry to say, the tension line gravity, is to your right. So gravity is pulling the thing sideways, and the top ball you can see is free. And then in the next cartoon you can see the ball moving around in a plane, as it were in the middle of a music string. And the third one you can see it moving only in a line, see what it does in the middle of a drum head; and the fourth one you see it localized as far as locality goes, in order to prevent it from wiggling locally, and rotating, then we had to have finally the twelve restraints.
Next picture please. I've got here a water spout. The upper one there is a tornado or water spout. Where we get the airs get disturbed and they get rotative you get a thermal, and all air, like all ropes, they always twist. The thermal is twisting, and gradually it gets to sucking so much that it pulls the dust into it. It is absolutely invisible until it gets the dust into it. What you see of the tornado is everything it's sucking up there. They are fully loaded with debris. I have also, then, taken that funnel twister, and put a knot in it up there, on the upper right-hand side. And in the next one I have three twisters, or three water storms and they are twisting to make a piece of rope. I want you to realize then what goes on with this twister is that it finally, it sends out and it comes back on itself like this. It's trying to be an apple shape. It's our friend "Involuting and Evoluting". It's evoluting at the top and its involuting at the bottom. Because the air is being exhausted this way and automatically pulls the air in there to satisfy it.
So, what you're looking at up there, now at the lower left you are seeing the beginning of the Bikini bomb where you can see it spreading at the top, and the second one you see the rolling donut, and then the third the middle one at the bottom is the electromagnetic field the magnetic field around our earth gets to be this kind of a form.
Now, we're looking at the reality of those electromagnetic fields here. A very extraordinary space picture of it.
Next picture please. Now I'm seeing. This is the atomic bomb. It's starting to spread the mushroom.
Next picture. This next picture was the cover picture on LIFE of the first Bikini atomic bomb exploding and it's amazing what comes out. It is a geodesic dome! And you'll find it, again is coming out, as one of the regular geodesics. It is a three-frequency geodesic. As this thing finally gets really shaping itself towards a sphere, then suddenly these are the least resistance forms. Very, very exciting.
Next picture. What we're looking at here you can see is highly geodesic and here is the model made for me by one of the early virologists. This is the polio virus. And it is strictly the icosahedron. This man was the physicist at the Boston Children's Hospital, and he now works primarily with the Cavendish Laboratory group.
Next picture. Now you are looking at something else that looks kind of familiar kind of geodesic, with the hexes and the pents because we get those kinds of spaces. But what you're looking at is the actual fibrous web in, this is an eye, the eye of a bull. Also, incidentally, the bull's testicle comes out in exactly the same pattern. But all the eye structures are this way these are eye structures. It's fantastic how these things suddenly show up, and as I get more and more scientist friends who are interested in my saying this, this was sent to me by a Viennese biophysicist, and he realized how interested I would be. Next picture. Looking here at pictures, if you could remove me again from the picture, what we are looking at are pictures made of the radiolaria. This is when the voyaging of the British scientific ship that Darwin went on, and so forth, these are pictures of the sea life. And we have these particular ones are radiolaria, but we have also the diatoms tend to be in these kinds of forms.
Next picture please. And incidentally, what you're looking at are the central angles of the tetrahedron. This is the one I found I COULD NOT MAKE WITH GREAT CIRCLES FOLDED UP. And incidentally, I'll just remind you about those great circles, that there were seven great circles that were foldable, and that they all had to do with the energy going thru space or cutting it off in local holding patterns and there are no others that can be folded. These are a limit case. There are only seven, and they all have to do with this basic symmetry.
Now, next picture. Here you are looking at some more of the radiolaria and the algae these are not the algae, and the diatoms. Look at the octahedron, and the really, really, very beautiful thing would you remove me and get me out of the way of this array because it is very, remove my picture. I want you to see in the lower left hand corner there is a dodecahedron, and there is the octahedron up in the upper left hand corner, and so it goes with tetrahedron up in the upper right. They are all there. We get then sea life absolutely the simplest things coming together in the simplest ways, and this is where they go. Again, this confirms what I want you to do get the very simplest and here we are.
Next picture. Now, we've got a stack of ping pong balls with a red pingpong ball at the top. And I spoke to you, I'm going pretty much through what we spoke about yesterday, but there is going to be more added to it each time. I want you to be really very familiar. There is one red ball on the top. In other words, it is the only potential nucleus that you and I can be looking at. Let's take that next picture please and it would be a good idea if I were out of the way for these pictures. I have just taken off the top tetrahedron and you can see the six balls of the top with no ball in the center.
Next picture please. And eliminate me again that's fine. So then you see I am now down to four balls to the edge, it's a three frequency, and there is the new nucleus showing for the first time. There is the first red ball on the table to the left, and then the two layers that have no ball at the center, and then the fourth layer has a ball at the center.
Next picture. Now no ball in the next layer. Five layers.
Next picture. No ball again in that one.
Next picture. And there we suddenly come to it. So I say it is a yes-yes no no, yes no no, yes no no all the way through. And these are the fundamental distances that make the fundamental distances which are between nuclei. And that yes no no also has agreement remember when I was spinning the great circles yesterday and they read yes, no, no that happened very often. And, where we get to Nature's taking one out remember my giving you going from the octahedron to the three tetrahedra face to face. I think I'll do that again because it is worth our doing at the moment.
Here is our octahedron absolutely symmetrical. It's generalized. It's very much in the middle because tetrahedron is a minimum case. The icosahedron is the maximum case of structural system. It's (the octahedron) the middle ground. And we find it has the tendency of being doubled up. It always really has a storage. It is that fundamental "twoness" of Universe all in one here. And, so we have our two extremes but it is in the middle. So it's two or something. And I'm going to take one out one vector out, and just taking it out momentarily but then I'm going to put it right back in but I'm going to precess it, you remember, precessing goes like that. There's a precessing effect on it, and so it goes in here, and it goes in here. And what we have now are triple bonded tetrahedra. One tetrahedron here, the second tetrahedron here, and so the volume is three and the volume was four when it was an octahedron, so you literally have annihilated but all the energies are here, because the energies are the edges are the vectors.
So, I want you to understand, that this is all there is in Universe and physics is about annihilation. But the interesting thing about it is that it also gets to be the beginning of the tetrahelix. So we've gone from the generalized immediately going into the special case. It's a very, very exciting realization.
O.K. Next picture please. Now you see up to the upper right the two red balls and there are always two layers between them. And what that makes quite interestingly, is two tetrahedra, but two tetrahedra made of balls rather than so this is behaving differently see if they were planar they could come face to face, but these don't do that. The balls nest. They go at 60 degree precession. So there are the two red balls, and what do you have in between? Six balls, and those are the octahedron. So these two red balls is the octahedron. There is a basic symmetry between two nuclei. Two potential nuclei. That's all that is saying. As interesting as it is not that kind of tetrahedron it is this kind of tetrahedron. It is a nested tetrahedron. Now if these two were joints, they would come together and you wouldn't see them. Because they would become congruent as they would bond together.
O.K. Next picture. This is a repetition of the one ball, the three balls, the six balls, the ten balls, and so forth.
Next picture. And this is a little review of thinking about instead of calling it a nucleus I called it the ability to nest. So that in one case I am talking about it positively, this is a new nucleus showing up, and then it has the advantage of being a space where you can nest.
Begins with the tetrahedron, second is a one-eighth octahedron, the third is a one quarter tetrahedron. So these are fundamental increments of Universe, and it is really, actually spelled out by fundamental disassociability. You feel much more comfortable about using quarter tetrahedron and really looking at things that way once you have discovered that's so.
Next picture. Now I've got two balls on a wire. And now there are two balls on the wire, and I want you to see what they are doing. They are remote from one another, the pair. May I have the just picture you leave oh if you want to put me in this picture, fine. What happens to these two balls is that they are near each other, and the effect of the motion is that they precess, so they do this, and they do this. I want you to realize how one half of the tetrahedron is really precessed to the other half the pairs.
Linus Pauling does a great deal with these kind of spheres, and as I said he is the Nobel Prize Chemist. And we will see some of his models after a little bit here. And he said that because the numbers of the vertexes in Universe do come out evenly, all the sphere agglomerations in Universe can be divided into pairs they are twos. Universe can be divided up into a fundamental "twoness". There is a multiplicative "twoness" and the additive "twoness". This is the multiplicative "twoness". So this one sphere it has "insideness" and "outsideness" and you can see both of them at the same time.
Next picture please. This is where the two balls are just coming together precessionally.
Next picture. Making the tetrahedron.
Next picture. Now you see three balls in a row, and three balls and four balls. Now I'm going to make a model of this. And don't go any further with your picturing for the moment, because I would like to make this model up for you with my pins and the spheres. Now here's three in a row. I was talking about nuclear phenomena, how different patterns obtain at different levels, so that I get the two balls that are precessing and you say, that's very simple, everything else must do that. But I got a three ball, a couple of sets of three balls there and I am going to have to make a pair, and you make another pair there are two pairs, but this time I am not going to precess them. They are not in the precessing part of this story. I am going to run my pin perpendicularly there, and another one perpendicularly. And I am going to make them into a square which would not be valid if they were all by themselves, because the square wouldn't hold it's shape.
Now, we have, precessing is something you do tetrahedra tend to precess, so I find that what happens here is, this wants to do this, but it just is very wrong in here it wouldn't be stable. So what we have then this one is here, and this one is here, and now it goes like that. So then there is a three ball tetrahedron. I want you to see how this precessing has been accommodated by this square in the center. That square is also then the cross section of the octahedron which is at the center of the four tetrahedra. So, it's getting into an octahedronal kind of a condition. In fact, if I took the four balls away from the four corners, you'd find you have the octahedron sitting in there. That's really quite different from when I put a four-ball edge there, that way, and there were five tetrahedra. But this one has the octahedron in the center. So things are coming out quite differently in different layers here, as the frequencies exchange, something unique is going on.
Now the next one I'm going to do for you, you're going to be able to see in the picture. May I have the next picture after the one that I have there. So there you can see the two of those that came together so a three ball edge, which is two frequency remember.
Next picture. Now you've got four balls in a row at the upper left, then four balls in a row at the bottom right and on top of the group at the bottom you see six balls, and you are going to see that there are six balls the one at the left. Just keep that and I'm going to make another model. Maybe somebody will come here and help me make this model. We'll just take our pins, and we could do it, it's good to leave that model there. Make four in a row darling like that with a pin. Look out that you don't get hurt. Now, you make up a set of three-three pairs. And you've got your three pairs. Alright now, where I made this square, pin them together in parallel. It makes a group like that. I need another "expert". Will you be an expert then, will you make what she's made four in a row. Now you've got your six here dear put together fasten those to your four . Now, I have had a very interesting time with these particular pieces in the past. I have made this model many times, and well, you know at top Universities like Dartmouth or whatever it may be, and I have had the top mathematicians and so forth. And I have given them this to a mathematician, and there's another one just like it. You've been with me now so much, you know what to do. But I just give these two items and particularly if I made it out of paper, in fact, I can't make it with paper I can only make it with the balls.
So here are two of them exactly the same. I say put those together. And he says, he's got to find something symmetrical, so he finds six and tries that. Or he'll try this like that, and it doesn't seem to do anything. You just, because the six are a rectangle with completely different dimensions, there is no reason in the world why you would think of any way to put them together except by the sixness. The way people think. They think ninety degreeness. They don't think sixty degreeness. Sixty degreeness is always convergent, and they are thinking parallel motions. So what you do, again, is cross precess, and here's this lovely thing! So six met six alright, but they converge! And I find that the human eyes just don't think that way.
I've really had very distinguished mathematicians and they never catch it. My grandson and I, he has accompanied me for a whole year around we went to fifty different Universities last year. And we were at Rhode Island, and there was a little girl, the professor's daughter. And she was getting quite good at school, she was about 10. And I gave her a model of this. We had to leave, and she couldn't get it, so I showed her. My grandson said, "Grandpa, you cheated that little child." That's just what you've got to find out for yourself! If she were given the chance, she would have found out, he said. You cheated that child of the right to find out. And if you did find out then you would feel the 60 degreeness you have to explain it to yourself if you found out for yourself. But just telling it like that you lose the whole beauty of it. So, this is one of my most beautiful lessons I've ever had. It's still part of that how to get on with that child. I was amazed how my nonsense of wanting to feel gratification really, you show off to the kid. So the big thing is here, the 60 degree convergence. Everything is converging and diverging. That's the way nuclear things are they converge and diverge. They don't parallel in. And all of humanity keeps working on parallel lines and cubes and squares. That's not the way Universe is. So when you begin to think "convergence" "divergence" then things really fit.
Now, next picture please. Now, there's the one I just put together. Next picture. Now you see a really very big one. This is a very exciting one. There are, count the edges, 1, 2, 3, 4, 5, 6, 7, 8. That would be seven frequency.
Next picture. Take these two apart. There they are. And the two parts are hollow. They really are very strange. This is even more provoking. When I make this one up it's so big I don't want to try to stop here to do it, but there are in each half there 50 balls. So the two balls coming together have exactly l00 balls. Because they are hollow. The do really surprise people. Now what they have inside is exactly room for a four ball edge which is a three frequency just goes inside. And it has, we may remember, exactly 20 balls when I put this together. Four times five. What fits inside the one that's up on the wall is "twentiness" and the enclosure is 100. So I could get, and we know there is no ball in the center of this group. Remember, there is a tetrahedron there. It's center is absolutely space. So twenty ball is absolute space in the center, so I can get 100 on the outside, I can get 120 balls around a common vacant nucleus. This is the largest number of balls I really can do in a Then, next thing, if I want to have a nucleus I'd have to put one ball layer on the 120 and then you would suddenly have a nucleus.
Next picture. Now, you see something else rather interesting. You remember that I showed you the nestability of the tetrahedron. I can make a three ball edged triangle, alright. Now, that is nestable. So fasten that one in the center. I'd like to make another one of the same to make another one just like that. That is a second case nestability remember? I told you it is a one-eighth octahedron. And if you're making a one eighth octahedron, you'll find, sure enough, the angle is exactly right. Then I'm going to ask you to do something logical with these two pieces. Something that feels good to everybody. We have been talking about precession. How could you precess? What could possibly match there? You match the little faces of the tetrahedra, somebody else want to try? What else could you match. The triangular faces. I have something to tell you. When you finish you'll know you're right. You got so close to it, it makes me I bet she sees it. Precess. There it is, put it down on the floor. It's a cube. You got it. These pins are bad. Come near me, I'll fix the pins. That's a cube a beautiful cube. Just put it down on the floor if you can. (The young woman who had been experimenting with Bucky says "I didn't even recognize it." That's because those angles, you remember were the right angles. It's a quarter-eighth and one ninety degrees in those corners. It's one-eighth octahedron. So two one-eighth octahedron give you the cube. They will not do it just by themselves in planar, so they've got to do it with only in the balls, with vertexes.
Next picture. That's going to be up in here in the show. Now you see a big cube, so there are big cubes, and you'll find that on each corner of the big cube is one of those one-eighth octahedra. And in the big cube those corners have been put onto a vector equilibrium.
Next picture. There is the vector equilibrium. You are taking a one-eighth octahedron off of the corners of the cube and there is your vector equilibrium, and it is a, count the number of balls at the edges, it is 1, 2, 3, 4, 5, it is a four frequency vector equilibrium, and that really is the limit case of the nucleus. I'm quite certain as we get into the post-uraniums, because you get outside of the 92, but this arrangement still is the nucleus.
Next picture. Now you are looking at an icosahedron made out of balls, but you have cut the balls so that they are down to the planar side, so that you can see what the balls look like together. That is the icosahedron where there is no ball at the center.
Next picture. You'll see when you do the counting whether it is the vector equilibrium, or it is the icosahedron. You'll find if you cut the balls away like that, the poles, the plus "twoness" is quite clearly, is a different color than the others. And the other faces get together where I gave you that three come together with two, or with one, or whatever it may be but you'll see all the triangles coming together and giving you the second power area, where the numbers of balls in the outer layer will be frequency to the second power, and then the poles, plus two. And you'll always see the balls are right there. The count is there every time. It is really a very beautiful thing.
Next picture. These particular models are supposed to show it to you but some how or other, they have faded away. You see the icosahedron there. This is where you get the multi-frequencies, and I want you to understand how well they work for the single just the plane tetrahedron, or where you see then twenty face icosahedron, they break into ten diamonds. You remember how the diamonds, then, compliment. You are looking at the ten diamonds grasping each other here. The blacks and whites. But it's always one extra north pole and one extra south pole. They are lovely things to make the count.
Next picture. There it is, that's vector equilibrium. And, the poles have been identified there. This coloring is really quite badly faded. That is the same one you saw as vector equilibrium becoming the icosahedron. But it cannot have any layers inside or it will not be able to collapse to do it. No matter what the frequency is, the icosahedron closest packed surface it can only be one layer. And I'm quite certain this has to do with it's "electronness", but the icosahedron's electronness cannot have the nucleus, but it has the same count as the vector equilibrium which is the nucleus.
Next picture. Over, could you put that over, I guess that's alright. You're looking at pictures made by Linus Pauling, or rather models that he made from his Nobel Laureate book paper. There you see, one of the things he does, is take the vector equilibrium, and he takes the top three balls and rotates them, because remember there are six nests on top here and we only use three. Three alternate ones are what you do. The minute you rotate, the top then becomes polarized. It's absolutely omni-directionally equilibrius. Until you take the three top and rotate them it becomes polarized. And when you take the three top, so you've got 12 balls , three balls at the bottom and three balls like that, then you get around the equator you get pairs of squares, pairs of triangles, pairs of squares, and then a triangle and triangle on the top. So that it is completely polarized, and when you make a section through the polarized what used to be the vector equilibrium , it's no longer equilibrius because it is polarized, then a cross section of it, is the chemical hex. So I want to bring you into proximity with other phenomena that you All of hexes have to do with polarizations where things, I said you never will catch nature in that vector equilibrium, she is always going to be in the polarized, she is always going to be offset one way or the other.
Next picture. These are more of Linus Pauling's pictures. Next picture of polarized sets, how he could bring together "threeness" in various positions all polarized. Now the top one is a vector equilibrium and it has a red ball inside, and you knock the red ball out of the center and it becomes the icosahedron. You can make a rubber model like this, and have it fastened together with rubber bands pull the middle one out and it would immediately snap back into the vector equilibrium the icosahedron.
Next picture. Now we are looking at the three on the top which you do rotate.
Next picture. I want to show you how to go from the polarized to the vector equilibrium, just rotate them.
Next picture. Now they're beginning to look at a little larger vector equilibrium of a there are four balls to the edge so it is a three frequency vector equilibrium. And notice, then, there is a triangular face towards us, and there is a red ball at the center, there is a new nucleus beginning to show in the eight faces, but it will not be a nucleus until it has it's two layers around it. But this is the one that has 92 balls in the outer layer that I gave you yesterday, 42 in the inner and l2 in the innermost.
Next picture please. Now I've opened that up so you could see all the different layers as they come together and these balls have different colors when the amount of light that they get from the nucleus differ. Because I talked to you about trying to find a nuclear set of events that would repeat itself, and so I get absolute uniqueness with those first three, four, up to the four, and then the fifth we get suddenly repeating. But these colors, relate then to the amount of light or radiation available or the attraction from the central nucleus to any given layer of ball.
Next picture. You can see on the top one, the twelve corners are always in direct contact, so the light goes right through them.
Next picture. This is a picture taken through one of my models. I have used the beads which Meddy found over there the other day. They are lovely beads that were developed during W.W.I, very uniform radius, and they are glued together. And they are all transparent, but some of them are colored. You are looking at the vector equilibrium. Would you remove me now so that I am not in the way of the picture. And, I want you to see, what really kind of extraordinary thing, the bright white lights of the twelve corners, you can see how they go, is simply giving, there is a red at the center, but it gives you a little sense of what I mean by the relative amount of light that can come through the different balls in a different position. And this particular model, we get where you'll find enormous agreement with much of the light emission microscope kind of things of atoms.
Next picture. Now, I spoke about spaces between spheres. And here is a tetrahedron. Wait a second, will you have the picture still. I'm doing this so you can really look at the picture on the wall because it is well done. There is a space, then, inside here. And what do we know about it? Well, it's got four balls around it. If I made this out of pingpong balls and glued them together. Then I took a safety razor and cut away everything except between, we'd find then there is a little triangular concave triangle up at the top here nesting down, touching three others making would you remove my picture from here over in the left hand side there you will see them coming together. There are four triangles, and I have four spaces, therefore it is the octahedron. But it is a concave octahedron. So at the center of the tetrahedra there are concave octahedra.
And now I'm going to make an octahedron, here are three and three, our friend "precess" and it becomes the octahedron. If I would have done that before you did the cube one, you might have thought of it. But there is your octahedron. Now the octahedron, then, has six balls. And you see a square section. And you remember then, how when we made it a three ball it had a square section, so you really feel those things. And, so there are six balls, and they make a square section so six balls touch each other, each ball touches each other with a square section. See this top one here if we glued in the ping pong ball and cut away everything, it would leave me with a concave square. So there are six concave squares where the six balls are. But that also then, there are eight triangular windows, because it is the octahedron. So what you have then,
Next picture, will be the vector equilibrium. The concave vector equilibrium with eight with the six concave squares and the eight triangular windows. These are all the spaces there are. There are only two kinds of spaces between closest packed spheres. Concave octahedra, and concave vector equilibrium. And they are pretty interesting because you start with the vector equilibrium and it goes down to that octahedron where the things double up, so it looks like it could be that the openness doubles up to itself to it's octahedron in its own space in here, something to do with that.
Next picture please. Now I am going to take. There are other pictures of these. Here we're doing that in a really rather open frame so that you can see the vector equilibrium with its triangles the octahedron on the left, and the concave vector equilibrium on the right.
Next picture. Now we can see two vector equilibria coming together with one another. I've showed you the square faces come to one another, and that there is an octahedron between the two. Remember that? That then left a space on the outside, so that there is an external and internal octahedron in relation to the vector equilibrium.
Next picture. Now, what you are seeing there are a number of the actual ping pong balls. In the lower right hand side are the concave octahedra, in the lower middle right side are the concave vector equilibria. So you put the little triangular windows of both together, making the edges match, and together they come to create, then, holes that fill all space. But you get an aggregate of balls, where you see the convexity of it on the outside, you're seeing only the concave side. This looks very much as if, I don't know if you you must have done it, picking up fossils where clams have been fossilized into clay. And the clam died in between because the two clam shells came apart, so what you see is the concave side of the shell in the clay matrix. That's what it looks like. But you keep putting these together and they keep filling all space, but the outer group will always be in the concave side. So what we are now seeing is really very interesting. There is some relationship between spaces and spheres. And remember that there were nests that you didn't use because the aggregate of the three only let you use one set of the nests at a time. There is an alternate set of nests which are also then these spaces that are in there, and there are two kinds the octahedron and the vector equilibrium.
Now, next picture. I'm looking at an aggregate here of vector equilibria plus octahedra. Where there are octahedra, the external octahedra are put on the outside triangular face of the vector equilibrium, and the interior octahedra go in between the two square faces. You're going to see some more pictures that will help.
Next picture. That is a vector equilibrium with the four removable spheres in the four faces which are very important to chemical compounding.
Next picture. Now you see red vector equilibria and you see a white octahedron which is, I said, this is an external octahedron. Where it nests down between any four, because there are triangular corners in every vector equilibrium, and when you bring when eight of them come together more or less cubically, and they have an octahedron between them.
Next picture. Now you are looking at red I'm sorry to say there are red and yellow ones, and there are white ones. Every other one of those, one is where a sphere is going to be and the other is where a space is going to be. They could all be actually cubes and you can stack a bunch of cubes together, as you know you can, close packing, but if you then realize that where the corners of the cubes come together are where the external octahedra are and where the faces of the cubes come together are the internal octahedra. That is a pack that you are looking at right now, but I want you to realize then that you're going to find in closest packing that the arrangement of centers of spheres and centers of spaces is this arrangement. Where you are suddenly going to discover that the vector equilibrium, I gave you originally the vector equilibrium flat, and then I gave you where it curved the edges, where it became convex, or it could be concave. The same vector equilibrium can become concave or it can become convex. And so can the octahedron. So we have, then, spaces that suddenly become spheres, they blow up and the spaces contract. So there is something terribly exciting going on here.
Next picture. Now you're looking at. Would you remove me again? You're looking at, I made a steel frame, it's a cubical frame, and there are brass rods or wires that run with the cubic frame I told you that the perpendiculars to the faces of the vector equilibrium are the same as the perpendiculars of the four faces of the tetrahedron which is our basic system of all. If you run there are eight corners to the cube and so if I run a line from one corner of the cube, diagonally down thru the cube to the opposite corner down on the floor, I get then four diagonals for the eight corners, and they are the lines which are perpendicular to the faces of the tetrahedron, or the eight triangles of the vector equilibrium. I have now mounted in there, you remember how I made the jitterbug. And the jitterbug, then, remember can go from being open it's a vector equilibrium. It can become octahedron. And what you're looking at I've made, I've put little transparent Plexiglass red triangles and white triangles. And I've mounted them on the rods. Could you go back to the picture itself now? You are going to see that there are eight octahedra showing there. But if you look very carefully, you're going to see some white or clear sheets. Those are the vector equilibria. There are vector equilibria and the octahedra those are the external octahedra. Now, each of those triangles has a, we put a stove pipe rivet through it a journal made of brass. We are able to mount those triangles on the rods. The triangle's corners are tied together with just a little Dacron thread so that they are vector equilibria. So the vector equilibrium is open, and the red ones are vector equilibria that are closed into the octahedronal state like this. We found on that frame, we put carbon dust so that everything would slide it's very best, I took one pencil and pushed one face of the white, clear vector equilibrium that is open, just push on one face, just one force operating the whole system, and the vector equilibria collapse, all the vector equilibria collapse, and all the red octahedra open up. But it is a very three-way kind of affair I assure you, because due to the internal and external octahedra. But what happens when I push on there every sphere becomes a space and every space becomes a sphere. Now when you come to this kind of an aggregate, for instance in a liquid, you begin to see how you can pierce thru a liquid. Because the spheres keep getting out and keep becoming a space. This to me is a very extraordinary matter, because now this is made symmetrically. There are the eight octahedra that you see showing and there are the I think there are the same number, yes, there are the same number of vector equilibria, and they simply interchange.
What you're in this model because they are all mounted on those wires, as remember this thing rotates as it opens therefore the corners of the triangles take a little more space. And there are a number of other models I am going to be showing you tomorrow and Monday in which you'll see then tetrahedra rotating in cubes. It's a fascinating thing but they do. Float absolutely beautifully through cubes. But anyway, you'll find that the way the tetrahedra rotate in cubes, make the cube's sides bulge out every way like that. So when I make one sphere become a space in a system, and have the space become a sphere, the whole all the wires bulge outwardly symmetrically. Pulse outwardly like this. They are changing from sphere to space it makes it do this. You see for the first time, remember when we dropped the stone in the water you see a wave. This is the first time you see electromagnetic waving propagated. Actually, the model does it.
Can I have the next picture please, and you'll see it happen. Now all the red octahedra opened in the vector equilibrium form, and all the little white octahedra nested. I have this model in our Cambridge office. It is quite old, it's l951-71, it's 23 years old and it's getting a little poor, but to me this was sort of the supreme moment of Synergetics. When I realized you were really seeing electromagnetic wave form in the eye.
Now, I think this is a very good place to stop for Sunday. It is now almost half past three twenty after three. I would like to keep myself fresh. Can you tell me how much time we have done today? (From the audience "We've got about 4 1/2 hours') We got something then worthwhile do you feel? I would think that I might go a little slow on you now, and I'd like not to do that. So let us break up. I would like you to realize we have enough tape for sixty hours. I don't think we're going to make the sixty hours. If we did four hours today, and I think we had let's see, we're about at twenty, and we have just about the same amount of run ahead. I think we may get up to forty, and it is my suspicion that I have learned to say things more compactly. I know that I am really covering a whole lot of territory today, where I used to go quite detailed following, I'm exploring myself and now I'm so much more familiar that probably I am compacting the sixty hours into the forty. I feel that way about it. So that I think we're going to have the total experience. If we get to the end of the time, I'm quite certain that I'm not going to be withholding from you some of the things that I feel are all this important interrelatedness, because I do come into you time and again with new kinds of thrusts, and yet you find everything getting back into the same fundamental world. It really gets more and more thrilling. Thank you.