At our first meeting I reviewed what I could remember of my conscious input of what it is I am conscious of when I say I am thinking. Remember we came out then with the development of a conceptual set, and the process of the thinking itself generated a geometry by virtue of the fact that my conscious input was one of dismissing for the moment holding out irrelevancies in respect, irrelevant to the set of experiences which I had had, that intrigued me, and I wanted to understand what the relationship was. I, therefore, would have to hold to the thinking about that, and there was a tendency of intrusion of thoughts coming to me as a consequence of my probably having asked myself questions on something, and the brain had been searching and coming back with the answer. The point was that my conscious thought was holding off momentarily irrelevancies to the situation the constellation that I was concerned with. Having discovered that my conscious part was this holding off of irrelevancies, I found that the irrelevancies fell into two main categories all the irrelevancies all the experiences which were too large, too infrequent in any way, to alter or tune in with the magnitude of relationships I was considering; and all the experiences in my life which were too high frequency, too small in any way to be measurable at the magnitude that I am considering. So it was really a tunable set.
Because the experiences were inherently omnidirectional observation, because our earth is revolving, and we are revolving and we're going around the sun continual readdressing of our view in many, many directions. Then, this meant, then, the dismissal of the irrelevancies to a macro and a micro group was an omni-geometrical phenomena, sending them outwardly and inwardly, and there was then a lucid set of stars that were quite clearly relevant to one another. That lucid set, then, defined an insideness and an outsideness. They were between the outsideness of the macrocosm and the insideness of the microcosm. We then found that the minimum number of stars that could define an insideness and an outsideness would be a tetrahedron. That is, if the two points had "between-ness," but no "insideness" or "outsideness" three points had between-ness but no "inside-ness" or outside-ness," not until we have the four points. So as we came to a system, and we are looking for the fundamental number of interrelationships of somethings that provoked us we didn't know what other items might be in it if we only saw three of them, there could be very readily be a fourth one. So as we began to really dismiss properly we find out which side that fourth one might be, to give the "insideness" and "outsideness." We found that there were four points, but also the four points had six "inter-relatednesses." So we have a prime number "3" and a prime number "2" geometry being generated here by the process of thought.
Then I developed a great deal more with you on the idea of the tetrahedron then being the minimum system, because I defined a "system" as "an aggregate of events that divide the Universe into all Universe outside the system, all Universe inside the system, and a little bit of the Universe which is the system" which I said is the defining subdivision. The tetrahedron is the minimum system of Universe, and it turned out to be, then, omni-triangulated and we found that structures are always triangulated. Structure meant triangle and triangle meant structure, there were no other stable polygons. Therefore, tetrahedron turned out to be the minimum structural system of Universe.
We did a whole lot more exploring, reviewing about tetrahedra the cheese tetra, the cheese platonic solids, and the slicing parallel to the faces of them and discovering that all of them were made asymmetrical by such slicing, with the exception of the tetrahedron which simply became a smaller, but absolutely regular, symmetrical tetrahedron, if it was sliced parallel to any of its four faces. So we found that the only geometry that could accommodate, whose symmetry could persist as a symmetrical system in Universe and yet accommodate alterations aberrational alterations, uneven alterations. The other geometries could not coordinate this phenomena.
We find that everything was in reference to the four planes, that were subtaining the four vertexes of the tetrahedron, so that we were dealing in a basic four dimensional system. I've made many references since that time coming back to tetrahedron and structures, and as we get into then just recognizing late, late yesterday, that in the water people and the mathematics of the water people coming from the Indian Ocean, into Babylon, to Crete, and then to the Ionian Greece, the possibility that the king's symbol of the hexagon the six equilateral triangles and the household or the distaff side using the square somehow indicated that they got into the public domain, the general domain for the first time in history, the mathematics in terms of reference to x,y,z coordinates of a square, but not in relation to the w,x,y,z of the four axes of symmetry of the four triangles of the tetrahedron. So we find dimensionality suddenly then being identified with previously by society only with perpendiculars to the system. Assuming that you'd have to have rectiliniarity. But we find then that you can have four sets of perpendiculars of the tetrahedron to symmetry, and this is the only fundamental symmetry that is not altered. So that we find then this four-dimensional quality of the tetrahedron makes it possible to MODEL four-dimensional, five dimensional models. Whereas you cannot model with cubes anything more than three dimensions. Fourth dimension is not accommodated. So that we found that the tetrahedron's volume was one and the cube was three when the volume was one so that if you use cubes you are using up three times as much space to identify the number agglomeration, and the tetrahedron seemed to be, then, not only nature's basic simplest structural system an absolute limit case; but it also then seemed to be the basic unit of quantation. And we found, then, we were able to identify that then with the quantum mechanics with one unit of quantum.
So, I'm now going to develop some more with you today of the energetic-synergetic geometry. Now our book will be coming out, I was told today, approximately, will be published, I think, the 3rd of April. The editors of Macmillan are here with us tonight, out in the control room, and it would be a good idea for them to experience with us a little of the energetic-synergetic geometry.
I'm going to review quite quickly the hierarchy of values of the synergetic solids, in contradistinction to the lack of being able to have a hierarchy of values when you use the cube as unity and use the edge of the cube as your unit of linear measure. Then you find that the tetrahedron's equal-length edge, or the octahedron or the icosahedron all the other platonic solids their volumes are irrational numbers in respect to the cube. Whereas with the tetrahedron as unity and the edge vector of the tetrahedron being the diagonal of the cube rather than the edge of it because it is necessary, the cube does not as we saw have any structural stability without the triangulation, and only when you put in the diagonal, into the cube, do you have stabilization as a structure. So that, we see, then, it is the diagonal of the cube that makes it a structure. So if I consider only hierarchy of structural solids where the integrity of the form of the solid is actually guaranteed by being properly triangulated, then you find that the volume of the cube is three, the volume of the octahedron I showed you was four, the volume of what we call the rhombic dodecahedron is exactly six, and the go onto what we call the vector equilibrium. I gave you the pumping one that fills all space, which was the form of twelve spheres close packed around one sphere. It is the first nuclear array . There is no inherent sphere at the center of a cube. And you cannot get a stable cube made out of spheres until you get to a very high frequency number of those spheres. But we have the first fundamental nuclear system is then a growth of layers of spheres around a nuclear sphere. That gives you then the twelve around one, and the twelve around one gives us then the twelve vertices of the vector equilibrium this is just to remind us of the fact that the cube doesn't have any stability by itself the rubber joints give you a little bit, but that's where you put, triangular there is a little triangulation, like triangular gussets in the corners provided by the webs of the rubber. So it's triangulation that makes it stand a little. Here is our vector equilibrium, and here are then the three vertexes in the northern hemisphere, three down there, and we have then the six around the equator there are our twelve. Now, the volume of the vector equilibrium is 20 when the tetrahedron is one, and consists then of, you can see the eight tetrahedron there is a tetrahedron that goes from this triangular face into its center. There are eight such triangular faces so we get eight tetrahedra in here, and it comes from, each of these square faces if the cross section of an octahedron an half octahedron whose vertex is the center. And the octahedron has a volume of four, so half a octahedron has the volume of two, and I have six of those square faces so six times two is twelve, plus the eight tetrahedra with the volume of one each, makes a total of 20 the vector equilibrium is twenty. So that this, really, then is the maximum domain of a nucleus.
One of the things I started really to search for in the early days of the energetic geometry was the following: I said, "it could be that we might find some patterns in Universe in relation to something where there was really specifically a nucleus." I found there was no inherent nucleus in the cube if you just try if you take one sphere and put four eight spheres around it's corners, they just fall off, there is absolutely no structural stability whatsoever. So I want to have a nuclear array, and this is the minimum nuclear array. And that then began to really intrigue me, and I said "It could be that around a nucleus there is subtle pattern evolvements, as I have progressive layers, for instance, as I put on more spheres; where there may be a unique set of pattern experiences. But you may come to a point, where it suddenly repeats the earlier one there may be a limit set of absolutely unique nuclear pattern interrelationships." I found that that is exactly what happens.
I'm going to need to use my board tonight, and we'll have a red nucleus here and, now, I'm going to draw myself a hexagon and make this a little easier to do. And we have another, then, sphere here. (he's drawing on the board) Anyway, I have a hexagon suddenly showing up in here, and this is the in a plane, six around one. Then I find, that there's you'll see that between these three balls here there is a nest. Therefore, I can nest a ball on top of that. When I do that, then, I overlap this one too much so I can't get one on here. I can have one however, in this nesting this is a pretty bad drawing, I'm sorry. But there is a nest in here. So I can have one here, and one here and one here. Three balls can nest on top of here and touch each other. You may remember when I had three balls on the two could touch each other, three could touch each other. No question about it, that gives you a triangle. Then there is a nest on top of that and you can have a fourth ball and that makes a tetrahedron. Now, I'm really reversing that here now. There's one at the center here now and then three are on top of it, and sitting in those nests. I can get three on the other side and they give me then the twelve spheres around one.
But I want you to think about for the moment, just in a plane, and quite clearly this is a rather stable affair, it doesn't seem to be trying to do anything to us. I'm going to have another layer of balls though. Now, you say, I don't think anything is happening. Just put another row around, so... But, you find that I put six around here the first time, and the second row if you count them up, you'll find that there are twelve. And so you added, the first time I had six sets of one, now I've then six sets of two so I have really then a basic "sixness" around here because it is a hexagon. But, I find that there were six in the first row, and there were twelve in the next row, that makes eighteen around one. For six sets then, it must be six sets of three, because six times three is eighteen. So I want to collect these in threes. You'll find that those are turbining around you see the turbine action? The minute I put an additional layer then, the first layer didn't try to do anything, but put one more layer on and it's trying to go around.
I could have taken, instead of these three, I could have taken these three. If you do that then it wants to go the other way, but the minute there is a third layer, they have to go somewhere it sets up a dynamic patterning.
There is quite a little difference in the situation if I have a light these are transparent spheres, I have a light at the center here, and it's relationship to the nucleus it gets a very beautiful direct lighting just tangent. But this next ball here does not have the same amount of light coming through it, so that is a unique pattern I want to introduce. Different things are going on here as the second layer is coming in. If I put on a third layer, we've got the second now, I'll get into a third. You'll find the same thing happens. From here on, there is always turbining. Only a first layer does not turbine, or want to go anywhere. It is apparent really in neutral.
I then found, as I began to have the balls coming around, rolling around it in all directions, if I had the twelve on here, then this next layer I get it filling in, every time it comes out the same shape the vector equilibrium. Always has the eight triangles and the six squares, every time I keep enclosing it it comes out that way. I don't think we have any models of that here in the room. Now, I found then, that the first we have a ball at the center, it is "0." There are no layers so if I call these layers, this is the "zero" layer, and then we come to the next one, we have twelve balls in the first layer, the next layer we get there are 42 balls, and the third layer there are 92 balls. Now the fourth layer, it turns out to be 162 balls. By the fifth layer it has 252 balls. I found that no matter how many layers I had, it always ends in the number "two." When I then recognized that this is decimal system that I am counting in, congruence to modular ten it is called, I have a constant suffix of the "ten" so I take the two away, and that leaves me instead of the it gives me ten, forty, if I just take out 2 and say +2 to all of them and then this would be 90 and 160 and 250. Now each one of those are divisible by ten, so I do that, and I get the numbers 1,4,9,16, and 25 and you suddenly recognize that as first 1 to the second power, 2 to the second power, 3 to the second power, 4 to the second power so what's going on here is, I call this then FREQUENCY OF MODULAR SUBDIVISION. F is my frequency beautifully done! And we find that the frequency, I gave you also something the physicists have two kinds of acceleration when I talked about "precession," remember? There is angular acceleration when you're holding onto the ball that is going around in a circle, and linear when it is going away. The radials are going away, they are the linears, and this is the angular, going around like that.
Now, whereas in a square system, on an x,y,z system in order to identify any point in geometry, they always have to go follow the line, you can't take a diagonal you can't take a diagonal like that. In other words there is no short way, you can't go from point "a" here to "b" on the diagonal. You have to go thru "c" in fundamental analysis algebraic analysis of any positioning of any points. But in the 60 degree coordination, because you see then the hypotenuse and the legs are never the same so the angular acceleration would not be the same language as the radial linear. But, in here the linear and the angular are the same. Exactly the same size vectors, same energy vectors remember what a vector means a vector is a represents an energy event and it represents a mass times a velocity in a given direction in respect to observer there is an axis of observation, and we have a special angle to observe its moving. And the mass then times velocity gives you discreet length of line. Vectors are not lines that go to infinity they are inherently limited, so that when I talk about a vector these vectors in a vector equilibrium represent forces of the Universe in balance the tendency to explode I showed you the other day being exactly countered by the contractive forces. So that the hexagon has six radials trying to explode and six chords. The six chords are more favorably arranged because it is a chain, and there is mass interattraction, so they get into critical proximity end to end and they hold together, where as the six others try exploding, disintegrating, not helping one another. And the other six help one another. So that we have then, in the Synergetics accounting, a space between two balls in closest packing, is then a wave length, and you don't have a frequency until you have at least two wave lengths. So frequency doesn't really begin until you get to this layer out here. In other words, frequency doesn't occur until that turbining, the disturbing quality enters is trying to do something, is trying to go somewhere. So this is "frequency," this is, when you get out here is frequency 2 second power. So I'm going to then find a point that we have down here the numbers of balls that we have in any layer in the closest packing of spheres will always be frequency to the second power times ten plus the number two. That became a really fascinating kind of a matter. There was every layer had two balls assigned to the function of being a neutral axis. There were two extra balls for every layer to take care of the neutral axis of spin, so Nature provided for that. If any of you have ever thought about a Victrola record this part is going this way, and this part is going that way two opposite sides, but you get to the center where there isn't anything going anywhere. This is literally a neutral axis theoretically there, but you've never been able to demonstrate it in three dimensions. In the four dimensions you can. I'm going to show you that immediately now.
Come back to our model, the vector equilibrium here, and I would have then, I could get, there are two balls I say in every layer that can account for being a neutral axis, and I'm going to take, in the vector equilibrium like this, and I am going to I said "lower the top triangle towards the triangle on the floor. I'm not allowed to twist this is the axis, here, I'm not allowed to twist the axis. It simply contracts in length, it does not twist! I do so, but the equator goes around! Here you see then the axis absolutely neutralized and yet it is able to introduce the motions, the equatorial motions. So we are able to also make this model as you will see later on where we make these with wheels that are going, so it doesn't have to stop they can keep on and on and on accommodating. But the center axis is absolutely immobile. When you get into these four-dimensional systems, one system then, like this, can latch onto another on the neutral axis, without in any way frustrating the motions in which they are participating. It becomes an extraordinary kind of accommodation that we experience in our actual life, but we have never been able to accommodate in any three-dimensional model. But with a four-dimensional model it is right there!
So we find that "twoness" is a fascinating matter.
Euler, I told you, developing his incredible realization that all visual experiences were reducible to three main aspects: lines, the crossings of the lines and the areas bound by the lines never to be confused one for the other and that in a picture on a polyhedral face or a polyhedron itself, the numbers of vertexes plus the numbers of the areas will equal the numbers of the lines plus the number two absolutely infallibly. So if you make a donut I said put a cord thru it really where we've got that axis there, then the numbers he said are the vertexes plus the numbers of areas equal the number of the lines. The two had disappeared. I do not know why Euler did not identify that with the axis because Euler also made one of the what we call structural engineering analysis engineering analysis structural analysis goes back to Euler. He was the first to develop then the concept of a neutral axis of spin of all systems. And so it's a structural member and for us to find out what its neutral is for its dynamic and he knows exactly how you're going to get your bendings and so forth. We find then, why he didn't think of the "twoness" of his own formula as representing the poles of the neutral axis, I don't know but he didn't. But when I found he hadn't, therefore it became very exciting to me, and I said, "I am going to now always assign two of every layer of my balls so he didn't get into this kind of a ball-kind of pattern. He didn't get into these layers of closest packed spheres. And closest packed spheres. And closest packed spheres is the way the atoms are all packed, so it is a very extraordinary kind of pattern to be considering. And as we're dealing in atoms and we're dealing in nucleus, and it has an inherent nucleus and no other geometry that I know of starts with an inherent nucleus. It's the only one. Closest packed and has nucleus!
And this in every way conforms to all of our experience with the atoms. So, I found then, by taking two of every layer, they would always, then, take care of the neutral axis of the system. Therefore it would be able to latch on to any other system, and we can keep on accommodating all of the kinds of things we do. Now, this was a very exciting discovery.
I've spoken about an inherent nucleus. It is possible to get a nucleated cube, but it has to have many, many layers before a sphere tends to come into the center of the aggregation in pure symmetry. You can get a nucleated tetrahedron, and you can get a nucleated octahedron, and they occur very much more early, and very much less aggregation than does the cube. There is a hierarchy of behavior going on here.
Now, in this particular formula, we are then dealing with the vector equilibrium which has a volume of twenty do you remember? Also, I showed you that going from this closest packed condition where there is a nucleus, I took this thing, and I made it I made it contract symmetrically, remember? All the twelve vertexes worked towards common center at the same rate. Remember it finally gets to be octahedron they're all doubled up. As all the vectors came towards common center at a constant rate. The, as I dropped this then, lowered this towards the other side, notice the six squares begin to change triangles cannot change they are structural. Squares are unstable and they do change. We are now at a point where this thing has contracted slightly, but it is at a point where the short diagonal of those diamonds is exactly the same length of this chord, so you put this in there are six squares, become six diamonds, and you put in the six cross members, and you have the icosahedron so the icosahedron is a contracted form of the vector equilibrium. It still has the same twelve vertices, same balls, but because it is contracted, there's no room for the nucleus anymore. It becomes exactly the same phenomena except for one thing, it does not have a nucleus. Or, you have compressed the nucleus, but you say "You really can't compress that nucleus," so I have to really consider that it does not have it. So I find then, the vector equilibrium is in a sense a vitiated or an empty an inoperative one and I have an operative one which has the nucleus. This gets into very much the relationships, then, between our proton and our neutron. So, I find then, the icosahedron has a volume of 18.51 where vector equilibrium is 20.
This 18.51 is a very interesting number because the if you'll take the relative mass relative weight of the electron in respect to the proton, and it is 1/18.51. It's really kind of a familiar kind of a number in here. So this has something, when we get into the icosahedron level, with nothing at its center it has something to do with the electron. But, you cannot take the icosahedron and pack them with other icosahedron and fill all space. They will join to one another, and will finally produce a geometry they will come back to the octahedron but they make a very wide-open octahedron. But they will not make themselves. We find then, they cannot be multi-layered, because not only can they not have a ball at the center, but as you go from the outer layer in towards this they collapse and there is not room for another layer so it can only be a single layer. Icosahedron is always single layer.
But, it has other qualities very close to the same as the vector equilibrium. And those the vector equilibrium has a characteristic of "twentiness," and the prime number five is in it. Whereas in the octahedron it has the prime number six there are six vertexes, and there are four faces and so forth the prime number two is in there, the prime number three is in there. But in vector equilibrium we first come to the prime number five is in there. We find the icosahedron the same prime number five, see? Pending five around each corner and so forth. It is a very fundamentally "fiveness." Now then, that "fiveness" is in here as a basic characteristic of the either the vector equilibrium and by the way the vector equilibrium I now write this way "VE" because I have to keep saying it all the time, so I use that symbol for it. And so this here is really two times five times frequency to the second power plus two. There is a multiplicative two showing here and there is an additive two, with the prime number five and frequency to the second power. Frequency to the second power is a very intriguing matter, because we have now layers something growing around, absolutely symmetrically, like waves. It's an omnidirectional wave phenomena and every way characterized by the great electromagnetic fundamental wave phenomena omni-directional wavings. Propagation.
In respect to it we have, remember Einstein's equation for energy, how much energy is locked up in a given mass, and I went into the knots and so forth here it's self interferences. But, Universe is, the physical energy is, the physical Universe is, the physical is energy and energy is either energy as radiation, unfettered, or mass brought together. So we have energy = M, that's the brought together side, times, is modified, how much energy is in there by it's relationship to the speed of radiation to the second power. See, the speed of radiation to the second power, as we said that is the rate that a surface wave grows this is the second power. Then we come down to the gravitational constant, and we come back again to our friend the second power which I spoke to you about, the exponential two that shows up, which is something apparently then, to do with surfaces, and we find out here is a system growing, rationally, beautiful rational number, absolutely in relation to this frequency to the second power! Which characterizes both gravity gravitational constant and the radiation constant. It gets to be very, very intriguing. As we go on in these kids of numbers the 12-42-92, I'd like you to add up those numbers 12-42-92. Why am I interested in that? Because, incidentally, I am going to stop for just a minute and double back on myself for just a little (turns the page of his drawing board).
I'm going to get into a little more discussion about nuclear phenomena. I have one ball is not a nucleus by itself. And I'm going to take start triangulation of balls, and here is one ball, and then I'm going to have two balls tangent like that. There's no ball at the center of the group. Then I have another ball, and another ball. No ball at the center of the group, is there? Now I'm going to have another layer of balls. For the first time there is a ball at the center of the group. It's going to be red nucleus. Now, let me have another layer. This is the center, there is no ball at the center of the layer. Now we'll have another layer. There's no ball at the center of the layer. Now I need to have one more layer. I don't know if I can really work this or not we'll try. And, suddenly there's another ball at the center of the layer. So it went, No, No, Yes No, No, Yes No, No Yes. It's not Yes, No, Yes, No at all. It's a very interesting kind of periodicity.
So it was not until we got to the here's a frequency phenomena. Just pay attention to this triangle in here. We have, while you see four balls to the edge, it's a three frequency one, two three one, two, three one, two, three. It's a three frequency system. Three frequency then has a ball at the nucleus. Therefore, as I begin to build up the vector equilibrium, and each edge of the vector equilibrium shows four balls, it's a three-frequency system, and at that level there is a new nucleus showing but it is not a nucleus because it is just showing on the surface. There is a nucleus at the center of the thing, there is a nucleus on the surface, but it's not a nucleus until it, too, is equally enclosed with the original nucleus, which apparently always gets two good layers of its own. So this one is going to have so I could get a four frequency five frequency, after five frequency I suddenly am really enclosed and have a new nucleus vying with the original nucleus.
So we find that the this nuclei idea is one which the first nucleus shows up at the layer the first layer was this 12 42, no nucleus not until we get to 92 do we have a new nucleus showing. But we say it wasn't one it too would have to have another layer on the outside of it here. Still isn't a nucleus. It's not really a nucleus until it gets to this one, and then it suddenly has, now it is really enclosed. You can see here this 92 isn't an enclosure point. This number 92 becomes a very important number, and I'm going to take the 12, 42, and 92 and I want to add them up. This is not including the ball at the center of the system it's just the layers. We find it has six. He adds 146! That's the number of neutrons in uranium which is the chemical element number 92! That gets to be very impressive suddenly. And then you find that you add to the 146, this is 92, there is a matching, always 92-ness there is always a twoness on the outside of the system, so there is another 92 that comes out of here, that gives you 8 uranium 238. So if you want to make it fissionable you knock out four. So, these numbers suddenly get to be very intriguing, there is compatibility with both, accommodating both Einstein's radiation and the Newtonian gravitation, which was what Einstein hoped for very much, this is your unified field theory suddenly showing up.
I want you to realize, I'm concerning these with you in really a very "kindergarten" kind of way, but this is the truth . So you can imagine how excited I became about energetic geometry as I began to get into it a great many years ago.
Because, I 'd just like to recite once more, as a little boy there were things that I was only being told "Never mind what you think, pay attention to the teacher," and I was trying very hard. But the teacher would say things from time to time, that I couldn't help but have some reservations about it.
Now, another thought, as we get going to learning first fractions and we learned all about how you manipulate fractions, and everything was going great. And one day the teacher said "I'm going to show you a better way of doing it." I wondered why she didn't give us a better way the first time "it is called decimals." So she had a .125 that's l/8 and .25 that's a quarter. .333 goes out the window and over the hill. Every once in a while things would go out the window and over the hill, and other things would stay in the room I wondered if she really knew what she was talking about. It didn't seem to me a very good classification.
So I began to really ask myself a lot of questions, and particularly where it came to geometry, because I loved geometry. And she had a point, she put it on the blackboard, and said it didn't exist. So she went and then she said, now I'm going to take a number of these points and put them side by side, and that makes a line, and that doesn't exist and she wiped that out. Then she took a number of these lines that didn't exist made out of points, and laid them parallel to one another, and got a plane. And said that doesn't exist. She wiped that out. And then she stacked a number of planes that didn't exist made out of lines that didn't exist made out of points that didn't exist, like this and says it's a cube and now she says it exists! (Audience laughs). So, I wonder how you get existence out of non-existence to the fourth power?
So I said, "If it exists? How old is it?" And she said "Don't be naughty?" And then I wanted to know how much it weighed, and what it's temperature was? because the word existence has something to do for me with existing. And, of course she couldn't identify it was an absolute ghost cube of her imagination.
Now, then I want to come back to something else I call the dilemma of mathematics and it's imaginary phenomena. We find the mathematicians, then, talking about lines, and lines that go to infinity, and from the Einsteinian viewpoint that doesn't make much sense because, I say, he is entirely operational he's never been to infinity, so he doesn't say that. He finds that all energy is in finite packages and seem to be an aggregate of finites. Einstein is not talking about any kind of infinity at all. But you find the mathematicians have what he calls a beautiful straight line I said, "Well draw it," and he takes his ruler and goes to the board; and I say, "Well, it's really quite crooked look at, see that chalk going up and down there." And he said "You're not in the spirit of mathematics this is just an imaginary straight line absolutely pure." So I said, "I don't know how the word imagination comes out of experience, and the word line was invented by me for an experience I was having either the trajectory of leaving some smoke behind, or leaving some chalk behind, or I've taken a chisel and am clearing something away. I've left a tracery of my action, and that is always going to be very crooked." Anyway, the mathematician said, "I mean a line of sight it's straight. Get yourself a surveyor's transit." and I said, "Alright, we'll put the surveyor's transit on the sun, just kissing the horizon, in the evening, and we find that the sun hasn't been there for eight minutes, so you're looking right around the curvature of the earth." And the mathematician said "You apparently just don't want to get into the spirit of mathematics here. You'll never understand it." So I finally came to a discovery which I find begins to work fairly well.
I'm going to take, I spoke about Boole the other night, Boole developing his Boolian algebra when you can't find the logical way, take the most absurd way. Take the most absolutely absurd you can get, and get something a little less absurd, and you'll gradually get working toward something that might be reasonable by elimination of absurdities. At any rate, I'm going to take a deliberately nonstraight line, and instead of saying I have a straight line, I want to be invariably sure of it, so I take the rope which is obviously curly it's all twisted, and I'm going to take this and one of the definitions of a straight line is that it never returns upon itself, so I'm going to take the ends of my rope and deliberately splice them together, so I have a most clearly deliberately non-straight line, either along the local service or the ends coming together. It is a closed circuit and then I'm going to take that piece of rope, and I find that I'm going to take any two parts of it and put them aside like this and put a clamp on it then I'm going to massage the rope from the clamp on like this, keep massaging it very evenly, and I come to where it turns around very sharply, back on itself. It's a tight little radius, and I keep it pinched, and I make a mark and put a little red ribbon on that turning point, and I go to the clamp and I massage the other way, and come to where it turns around again. Anyway I get just as close as I can to the middle of that arc, and put another ribbon on. Quite clearly I've now divided the rope into two parts that are fairly, reasonably each is just about half of the rope. Heisenberg makes it absolutely clear we can never be exact, so that we just only struggle so far. I'm really content that I've taken unity and divided it into approximately two parts. Now I'm going to take each one of those parts between the ribbons and pair them up the same way. And I'm going to get a quarter point. I'm going to do it the same way. And I'm going to get a quarter point. I'm going to do it the same way keep halving the distances between points. We're not increasing or multiplying here, we're simply continuing locally, halvings so that each one is a reasonably good half. And we get down to sixteenths, and thirty seconds and sixty-fourths and so forth. All nice clear marking distinctions so we know which point we're dealing in. Now we're going to go to the wall and put up some nails, and I'm going to put a nail on the wall this is the wall we're looking at over here. And we'll put a nail here and I'm going to ask you, I'm going to take the piece of rope and put the first marker on here. I'm going to ask you to really hold it nice and tightly, and I'm going to go over to the quarter marker, quarter way around the rope and I'm going to pull it tautly from you, so it swings as a radius here, and I can put in another nail, anywhere I want on that radius. So this distance between this is, I know, is one-quarter of my rope. Then with this same thing I'm going to come down here, another quarter point and I can swing it anyway I want in here, so I say, let's put it here; excuse me it should be about over here so then I'm going to get to another quarter point. There's I've got somebody holding it, and I've got holding it here there's a slack piece in here now, and I come down to the marker, pull it tautly and put in another nail. So now my rope is stretched over that; and the rope is we've got a diamond, and it's an equilateral parallelogram. You're very familiar with that, and there is nothing you've learned in geometry that as far as just playing games with lines, that I will not go along with. It's fine. This is an equilateral parallelogram.
But I have very clearly already marked on here other points, so I'm going to take the 1/8 points, and I'm going to take the rope off of this nail here and pull it down firmly; and so the rope is now going to come down to here. I'm going to take it off of this nail here and pull it firmly up like that, and it's going to come up to here. We have now two parallelograms, and I can eliminate this one out here. This is the same piece of rope, now, but it is two diamonds end to end. And it's very easy all the way through it is equilateral parallelogram, holding absolutely true. Same length between "a" and "b" all the time here. Now it's no trouble at all to do that again. So each time I'm going to half it here, and we eliminate this, and then eliminate this, and now we have four four diamonds in a row. There it goes. In no time at all when we get to this 32nds and 64ths and so forth, we get down where every time I do this I half the distance.
So what we have now is this baby. And those vertexes again closer but it's always the same length, it's always the same non deliberately non-straight line, but it keeps getting straighter and straighter. By the time you get to the 64th and so forth, in almost no time at all it begins to look like a straight line, and it gets straighter and straighter. And I have a way of getting it straighter and straighter which the mathematician didn't have, at least I've got a progression towards straighter he didn't really have anything except making kind of "get a line of sight or so forth" he didn't have any really methodical way of getting there, but I know it's non straight. So I'm going to be able then to get a line for the mathematician which I can probably prove mathematically is a little finer than any straight line he's ever used, but I know that it is deliberately non-straight.
Now the physicist when he wants to get the student feeling wave we're getting into quantum and wave we really want to feel wave. One of the first tricks he does is to put a nail in the wall and fasten a rope to it, and stand over here, and throw a whip into the rope, and it goes to the wall and comes right back here and stops. It's a fundamental characteristic of a wave that it comes back where it started. Beautiful thing. What's going on here, is then from "a" you whip here from "a," it goes out here to "b" and it comes right back to itself. It's a wave. See it? Now when he said, "I meant a line of sight," that is always a wave phenomena. This is the line of sight. Now I'm really able to show him what his real line of sight really was, it is a wave. Physics has found NO straight lines; ONLY WAVES, ONLY CURVES.
So I say, "Mr. Mathematician, now I've given you a tool that is much more reliable for you, but never kid yourself again, never call it a straight line. You are simply dealing in wave phenomena, and now we can go on and do all the geometry we ever did. This is the first time the mathematician really began to be a little friendly with me, because he did not like really being excluded from the experiential club of the physicist. And his salary is very much smaller than the physicist.
Now, this brings me then to I've given you some agglomeration of spheres, and I, there are many things that I would do if I had more models around me; but I'm fairly limited in my choices of the things I see because I see one frequency and so forth and not agglomeration. I've already done octahedron and tetrahedron with you, and you've felt those and you've felt those grow.
And the you'll find if we take tetrahedra, I do get four tetrahedra together but no nucleus right? If I, however, then, make another layer I can have a tetrahedron where you see three balls on an edge where there is actually frequency two. And it's number of balls is four on the top and six on the bottom. There are ten balls. Six-ten Then if I have another layer here's another ten it gets to twenty. In this layer, however, the twenty layer no, when there are twenty altogether, this is the ten. And you have one, two, three, four, five, six, seven, eight, nine, ten that's where the nucleus begins to show up again, on the "ten" layer. So, on this surface of the tetrahedron on each of the four surfaces, you see a new nucleus A nucleus beginning to show for the first time, because the tetrahedron did not have a nucleus of its own. I'll then have to put one more layer, and the next layer will have 15 in it so it goes to 35. One, two, three, four, five, six seven, eight, nine, ten eleven, twelve, thirteen, fourteen, fifteen there are fifteen balls. When you do, then, for the first time we have a nucleated tetrahedron. So there is a nucleated tetrahedron, the same way we can get then to how do you get an octahedron with a nucleus.
Whereas this same formula then for the number of balls in the outer layer of the nucleated tetrahedron in contradistinction to the vector equilibrium where it was ten times frequency to the second power plus 2, it comes out four times frequency to the second power plus two. But the four as the ten was two times five, the four is two times two times frequency to the second power plus two. That's the number of balls in the outer layer when it is tetrahedronal.
When you do it octahedronal, the number comes out four. We have there is a multiplicative two here, and I take that out, so there is a prime "oneness." We find tetrahedron coming out the prime number "one." The octahedron comes out the prime number "two." And the cube is a prime number "three." And the vector equilibrium and icosahedron are the prime number "five." These are the first four prime numbers one, two, three, five of all numbers. And we find as we're going to go on here, some very interesting things, the number really goes up only to four. So it's like the four of the vertexes of the basic structural system of Universe. You get four positive and four negative, we get to the number "eight" and I'm going to try to show you that. I'm sorry we don't have the good pages and models and everything all printed out. We will come back in our video WE HAVE ALL THESE PAGES IN SYNERGETICS, SO WE'LL BE ABLE TO TAKE PAGES FROM SYNERGETICS AND REINTRODUCE THEM INTO THE VIDEO.
May I have your chart then. I wonder if I could sit on here would that? in that very, very bright light there and everybody can see it a little better. You'll find, this is the SYNERGETIC HIERARCHY OF TOPOLOGICAL CHARACTERISTICS OF OMNITRIANGULATED POLYHEDRAL SYSTEMS (See pages 46 and 47 of SYNERGETICS I). And you must remember when you are talking about the cube, in order to have a cube you must put a diagonal in its face. It always must be triangulated. These are structural systems. In other words, they are absolutely stable in doing what they are doing. And, there are a great many other items on here, but this is where we begin with the vector-edged tetrahedron, with a volume of one. The octahedron has a quality of always doubling on itself. Which, you may remember as I pump this down here, octahedron seems to occur in double bond always. You see two octahedra congruent one with the other. The more you get familiar with synergetic geometry, you'll realize that this is fundamental for the octahedron so it occurs twice, keeps showing the number "four" when it really represents the prime number "two." This doubles itself, and we find then that this is this hierarchy, and I'll go through then the vector-edged tetrahedron and the vector-edged octahedron, and the vector-diagonaled cube and so forth, and vector equilibrium. We find then that the vector-edged icosahedron, combined volumetrically with the vector-edged cube, where the cube likes to be edged this way, it's number comes out to the two come out together altogether they come out the number twenty-seven. And we find all the vector-edged octahedra and so forth, these are all beautiful, rational numbers.
Now, what I found, I spoke to you about, that Euler didn't think to do, was to identify that the "plus 'twoness'" of his equation with poles. So I find that every system always every system is inherently as he himself knew, is rotatable in other words there is a neutral axis of spin of the system. So that you have to have two vertexes have to have the function of being poles. So when I take the Euler formulas, as nobody had done, and automatically subtract two take out let's go through some of these (he's still looking at the above-mentioned chart on SYNERGETICS HIERARCHY). The tetrahedron has four vertexes plus four faces, equals six edges plus the number two. Four plus four equals eight, and six plus two. In the octahedron we have six vertexes plus eight faces equals fourteen which is twelve edges plus the number two fourteen. Or we get to the cube, and it is now triangulated so it has eight vertexes, plus, instead of six, I have twelve faces that's eighteen, equals then, it's eighteen edges plus two (= 20). We keep coming out all right.
Now, what I did was to take all of the formulas as given by Euler, and no topologist looking at this recognized some further order in it because they didn't take out the two vertexes for spin. I now take out the two vertexes for spin, and that leaves me for the tetrahedron two plus four equals six. Remember, I've got two taken out for poles. This leaves me on the octahedron is four plus eight equals twelve; the cube is six plus twelve equals eighteen; the vector equilibrium is ten plus twenty equals thirty; the icosahedron is ten plus twenty equals thirty. Now each one of these, then, is coming out in even numbers.
I find then, because they are all even numbers, I can divide them all by two. So I try that. So I get, instead of, for the tetrahedron 2 + 4 = 6, I get 1 + 2 = 3. That's a very simple kind of relationship: 1 + 2 = 3. Then, the next was the octahedron, and that had been 4 + 8 = 12, so I divided it by 2 and I get 2 + 4 = 6. Let me write those down. The first one I got 1 + 2 = 3: now I'm getting 2 + 4 = 6. Then I get to the cube and I've got 6 + 12 = 18. So I've said, I divide those by two and that gives me 3 + 6 = 9. I wish I had done it 1 + 2 = 3; 2 + 4 = 6 and 3 + 6 = 9 so what's the next one, the vector equilibrium or the icosahedron which was 10 + 20 = 30 and I divide that by two and I get 5 + l0 = 15. Now these are very interesting numbers because you find 1 + 2 = 3, you couldn't have something simpler. But the next one 2 + 4 = 6 is 1 + 2 = 3 x 2! And the next one 3 + 6 = 9 divide that by 3 and it's 1 + 2 = 3! And the next one is 5 + 10 = 15. Divide that by 5 and you get 1 + 2 = 3! So we have then, we have in every case here 1 + 2 = 3 times tetrahedron is by 1, octahedron is by 2, multiplied by 2, and cube by 3, and icosa or vector equilibrium by 5 those first four prime numbers.
We have, then, I found there is what you call a multiplicative "two" and an additive "two." There was an additive two of the poles for EVERY system in Universe. There was also a multiplicative two because there is a concave and a convex there is inherent duality of this congruence of an inside system because concave and convex are not the same. You just have to realize that you have a fundamental congruence of the macrocosm and microcosm. There is negative and positive simply congruent there, but you can't separate them. But the concave radiation impinging on concave, converts concentrates the radiation, convex diffuses it. So, and energy-wise you find that they are absolutely not they are just not the same, yet they are congruent, you can't separate them, so this is what I call then the "duality twoness." So you find every system has a multiplicative a duality twoness and it has a plus twoness of poles for axial rotation. When I take that out, then the constant there is a constant relative abundance for every vertex, two faces and three edges. And the only difference there is a prime number, that a tetrahedron is a "one," and octahedron is a "two," a cube is a "three," and a vector equilibrium (VE) or an icosahedron are the number "five." Now this gets to be very, very exciting.
Then I gave you frequency the other day, and then I showed you a series of triangles, the edge reads two, then you have four triangles the edge reads three edge is frequency. So I have there frequency to the second power and you remember it came out then alright as triangulation. So as we get into any of these, we find that they all are triangulated, so simply increase the frequency, so then in addition to the duality twoness of every system, a polarity twoness of every system (that is the plus twoness) (the duality twoness is a multiplicative twoness) a multiplicative twoness, an additive twoness then there are the four prime numbers, and everything else is just frequency to the second power times that frequency, whatever it is. This tells you all about all the structural systems in Universe. Which is very, very exciting, because then you find, because there is a duality, you do have to have the multiplicative twoness therefore you find that for every positive one vertex, you're going to have a negative one in the system, or the opposite. So I said, 1 + 2 = 3, but instead of that I've got to say 2 + 4 = 6.
That is, quite clearly, all the numbers or points in Universe will be divisible by two, and for every point in Universe there are always going to be three vectors, because there are always going to be pairs of points, then you are always going to have six I said the other day, then, there are six basic because there are six vectors always with every event in Universe you have six vectors. And those are the six each one is a positive and negative, so there are my twelve degrees of freedom I gave you the other day. You want to see how beautifully these things begin to prove themselves up and there is a very swift simplification of a great comprehensive accounting as we get into SYNERGETICS HIERARCHY. Everything coming out rational and whole.
I was really so terribly impressed when I was a kid by the fact that whereas that chemistry was always associating in whole, rational low-order numbers, associating and disassociating in beautiful, whole rational numbers physics was always coming out with irrational numbers. And I felt that what was really causing it was that we were really using yardsticks that were not the logical yardsticks that we came in the attic window and were trying to measure all the rest of the windows by the attic window or something. So it just was an unreasonable unreasonable story, so I feel that man, then, being fairly monological, thinking of it as a flat earth I can understand his making cubes and cubes were nice, and they seemed to fill all space they were building blocks. Tetrahedron wouldn't, all by itself, so you had to cast that out. But it was a flat earth anyway so you might as well plan on cubes, and that's the way to divide the Universe. The minute you get into the spherical you're going to realize that they are not going to work very nice, but you could have a triangle on the surface and then it went to the center of the sphere and you get a beautiful tetrahedron right there all the time.
Now, this chart goes on to get into really these complex forms that we get into here, they are all superbly accounted you'll never get in trouble, because all of them are some combination of those first four prime numbers. That's all you have to have, and the minute you get a three you know you're dealing in cubes that's all there is to it, it's always going to come out that way.
I'm going to run a few slides now that confirm some of the things I've talked about earlier, but I must ask you to imagine. The ones I'm going to use now I'd like to have first, Bob that, the half octahedra. You see two one-half octahedra and a whole octahedron. And you remember, the octahedron does have a volume of four so that each half octahedra has a volume of two. And each one of those you remember nests very neatly into the square faces of the vector equilibrium.
May I have the next slide? Now you see a one-half octahedron cut into four one-eighth octahedra there on the left hand side. The gray ones each one of those are one-eighth octahedra, and they have an equilateral triangular face on the outside, but at the center they have the 90° angle and subtended by two 45 degrees on the outside.
Next picture. Now you can see that one-eighth octahedron extracted from the octahedron coming out from the center of gravity.
Next picture. Now I'm going to take, there is a round tetrahedron and four one-eighth octahedra. I wonder if that picture couldn't be elevated? At any rate, addressing the four one-eighth octahedra the equilateral triangular faces of them which would be their outside faces when they are an octahedron to the equilateral triangles of four tetrahedra's equilateral triangular faces, and together they make the cube. Next picture. Can you see this coming together to make the cube?
Next picture. Now this time, I'm going to cut the picture out, just hold onto that for a minute. You see a great circle. I'm going to remember how I like to be sure you have a limit case, you come to the end of things I like to deal in where there is no question about our dealing in unity. And here is a circle, and it is very interesting, that a circle, you can take any two points on that circle doesn't make any difference, any two points, and it will always, if you make the edge there, it always goes congruent no trouble at all. And then you fold it and you have to half circles alright. This is a very simple kind of a folding.
I now want to do something I'm going to try to divide these in thirds, so can you see how I am taking this part and making it match as two halves, alright? Then I fold back on the other side in just the same way. I've now divided my circle into approximately six, sixty degree equal parts. Now I'm going to do that for several more great circles. Here's a half, and again, I'm going to try to make that just as even as I can between the two halves, and this fold back,, the other corner. Now this way. And do that four times all together. I'm taking four great circles. I'm taking four great circles because of the interest we really have in that "fourness" and four great circles of a plane... I want you to remember what a great circle is. A great circle is a line formed on a sphere by a plane going through the center of the sphere. I think I had mentioned to you before that the great circle is the shortest distance between two points on the sphere. Remember how I took the latitude of eighty degrees North latitude and superimposed it on the equator, crossing the equator do you remember that, and it was a shorter distance between "a" and "b" where the little circle crossed the bigger circle, much shorter distance to stay on the equator than to go off on the detour of the little circle. This is typical of the great circle being a shorter distance.
The word geodesic in mathematics, SYNERGETICS, means "the most economical relationship between events." One event would be a bird flying in the sky, and the other event might be you, and I don't know why you would want to do it, but suppose you wanted to fire a gun at the bird, which I am sorry to say many people do, if they want to hit the bird they don't fire the gun at where the bird is, because the bird is in flight. They fire where they figure it is going to be. And they find, while there is not much gravity effect, there is always a gravity effect. So that the firing is pulled a little like this, towards the earth. It may be infinitesimal to your eye nevertheless there is such a measurement, and in due course it is going to go right towards the earth.
So we have, then, the bird is in flight and there is always some wind. There is also a little inequity of the surface of this bullet and so one side has a little more drag than the other. If you take, which they often do, during World War II there were a great many photographs taken at night of two airplanes in a dogfight, where they were using tracer bullets and the picture is usually taken from another plane, of the two. And it doesn't make any difference if it is taken from one of the planes, or another plane. What you saw was absolute corkscrew fire. That is the shortest distance, most economical distance between these two was a geodesic line. And they are not straight they are always curves, waves whatever.
There would be for instance the earth revolving before the sun, very rapidly. We have a vine growing on top of the earth. And this top of the vine, growing each day. And it is very flexible, and it wants the sun. So in the morning the little stem will come out and the leaf opens toward the east to get the sun. And then as the day goes on the earth is revolving the earth is revolving but the leaf keeps growing apparently towards the sun and so in the afternoon it seems to be reaching towards the sun. And then tomorrow morning it's over here again. That's why they are spiraling the reaching this way but this leaf was always much near to the sun than was it's roots. And if you really take a total picture of it go around the total sun, revolving, it describes a line very much nearer to the sun than the rest of the earth. So these are geodesics they are interesting things.
So, the great circles are the shortest, most economical distance between the points on a sphere. Therefore, great circles are called geodesics. Now I'm taking I made four of these great circles, and folded them up, you saw me, into thirds. And I'm going to put them together using bobby pins. I'll put one to they get two tetrahedra here. And, another one. There. We now have our eight tetrahedra of the vector equilibrium, in pairs. I'm going to take just two of these you see when they sit like this they tend, really to come together in in sort of a natural way. A bobby pin there. And another bobby pin here. It's quite a neat form it gets to be. Then, put two more of these together. Then take those two and sit them on the top of here. Get some more pins. Another pair. Now these are absolutely perfect they are whole great circles and there is nothing extra in them, and so there begins to be a little tension as you begin to pull them together. There's quite a little gap there. So another pin, and sure enough the slack is in there. Now suddenly I took four great circles, and you see four great circle planes all over again. Here's one, here's one there they are. The four great circles have been, then, folded locally, so in local energy holding patterns, and we have a very extraordinary thing here where we can either go completely around, or we can go around locally with the same amount of energy. You remember those six moves that you can make; a very local holding pattern that can go on and on.
Now these, there are twelve points here and when spheres are closest packed around spheres, these points are where they touch the next sphere, so if energy were traveling through space through atoms that were in closest packing, you find that energy follows a convex surface not the concave. It's very easy to understand. Just take a piece of paper and just bend it. The exterior this goes into a little more tension on the outside doesn't it it tries to resist you. So tension, high tension, and energy follows the higher tension. We have a great copper sphere hollow sphere, maybe 20 feet in diameter Van De Graaff generator where you simply keep loading electrical charges to it, and they always stay on the outside. You can get up to a couple of million volts and they are used for making artificial lightning. But people can walk around on the inside with absolutely no trouble at all they will never be short-circuited because energy always stays on the convex side. For this reason, when you're trying to plate, silver plate, any plate metals, the convex is very easy to plate. The concave is almost impossible. You have to get your anode almost in practically touching, in order to get it to flow it on there at all.
So we find energy is always following the convex. So that energy, going from here to there in Universe, following the convex would follow the outside of the sphere, where we came to the point of it could only get to the next sphere through a point of tangency. And it could get on such great circles as these, and so these begin to be the beginnings of railroad tracks. All the great circles that go through these twelve points. Just sort of fundamental symmetry are going to the way in which energy can get from here to there in Universe thru closest packed vector equilibrium (VA).
Now, it gets terribly interesting in this particular device when we begin to pay a little more attention to it. This is interesting, these are the same four planes, and I want you to see remember this is our friend vector equilibrium how you could pump that around. It is an extraordinary thing. You can flatten it down to get all the planes congruent or it opens four completely different ways, and you can flatten it any one of those ways. It comes out a different looking pattern altogether. These are typical of the intertransformability, starting from our wonderful vector equilibrium . And, I have mentioned, the other day, the vector equilibrium really was the limit domain of the nucleus, and everything that goes on within the vector equilibrium is unique to nuclei and to atoms, what goes on outside of them when they join up with others, is unique to chemical compounding and molecules and so forth, where things join up. Joining is outside, this is the domain of the non-joining, inside. It's very fundamental it's the basic patterns.
Now, I want to talk about other great circles. And this one is very easy to make because they are all the same and you can do your own improvising really quite easily. But I have slide pictures of other great circles. And I want you to think about what they might be. As, for instance, we have the tetrahedron, and I'd like to find symmetries in it. For instance, I could, you might say, it doesn't seem to have a pole there. But I take a mid-edge and a mid-edge, and suddenly it does have symmetry. Tetrahedron, supposing I were to take a knife it's made out of cheese, and I cut parallel to this plane here, but up here. I could truncate this little corner couldn't I? And leave a little triangular unit, can you see that? I could cut off this corner, I could truncate each of the four corners and get little additional triangles on here. If I did that, having cut here, you can see where I've cut into here you find you have a hexagon. So I get four hexagonal faces plus four triangular corners. You also see that figure showing up. Then, suppose I wanted to take a knife, or a plane, and I slice parallel to this edge itself, in other words I cut off, truncate the edge, can you see how I do that? My lines would look like that parallel to the edge. So I cut the cheese off so I've got a little flat plane on each of the six edges of the tetrahedron. And so, that will leave me still four flat faces out here and I could then truncate these corners. Sum totally, I could get facets on here I could get up to the four faces already there, plus six facets if I truncated the edges makes ten plus four facets at the corners that's fourteen, and these always they are opposites they must be in pairs, so there are actually seven axes of symmetry in the "fourteenness" of the four, plus four, plus six. And that fourteenness shows up as seven sets of the great circles.
And these seven sets of great circles have very interesting properties. We're going to look at those, and they are really all the axes of symmetry of all crystallography. There are seven fundamental symmetries. And the let's come for instance to the, may I have the first picture now next picture. We're looking at the vector equilibrium again remember the four great circles. Now you're getting familiar with it all of a sudden, and we're looking at it made in colors. Next picture, next picture again. This is one where I get what I call a concave and a convex one and you're going to find those very interesting as I said Vector Equilibrium was the limit case. And if I take the vectors edge of the VE I could bend them and make them into arcs. This means that all the vertexes go inwardly a little . Or if I bent the exterior edges concave, it would give you a shortening of the lines, therefore the vertexes would have to come in. In this seemingly straight condition it takes the most room in Universe. And those concave and convex qualities you see in that picture, relate then to the first like knocking out the central ball and it becomes an icosahedron. These are the first degrees of contraction where you have to follow the hierarchy of forms that begins to generate.
Next picture. This is a little difficult to see. That is a transparent four great circle.
Next picture. What we're looking at here now is I've tried to make just take two great circles and cross them. And they really become unstable, they just look like this. They have a common axis but they flap, and I try to make, then, the
Next picture please. There we tried to make the central angles of the tetrahedron what we call one hundred and ninety degrees and twenty-eight minutes, where , that doesn't work, you'll find that the one hundred and ninety and twenty-eight is what each one of these arcs are and they don't come out in whole great circles.
Next picture. Here is the octahedron and you'll say, well those are 90 degrees, if you try to make those in supposing I try to make a bowtie the way I have here-90,90,90,90 what do I get there, four times 90 that's alright, that's 360. But then you find that you can't make a, you have to take two whole great circles.
Next picture please No, you take six of them! You take six great circles folded to make the three great circles. I've told you this before, octahedra always appear double, they always appear congruent, so to make the octahedron in great circles, folded great circles, it has to be double., again. So it's really six great circles that look like three.
Next picture. There you are looking at the octahedron. No, that's the attempted central angles of the tetrahedron and they do not work.
Next picture. Now we're looking at the six great circles. And the six great circles you will like to know where they come from. Let me then take the vector equilibrium itself, just let's see what it's got. It's got those six square faces, eight triangular faces. It has twenty-four edges, it has twelve vertexes. So if you take twelve vertexes they will then have six equators they are opposite each other. The twelve vertexes are in pairs north and south. There are twelve vertexes that are opposite from each other and you have six great circle planes as I revolve it it goes perpendicular the perpendicular bisector triangle goes square, triangle, triangle square, triangle, triangle and that gives you, that is the axis of symmetry that gives you six great circles which I have been showing you.
Next picture. This is looking at the same six great circles.
Next picture oh, incidentally, you get the six great circles if you want to by, take a cube and put both sets of diagonals you have the two tetrahedra crossing one another inside the cube and that gives you the six great circles. You find that, six great circles have four times six twenty-four triangular faces. They are not equilateral, but they are isosceles.
Next picture please. Now we're looking at the twelve great circles. Say, where did those come from? Well, remember, there are twenty-four edges here in the icosahedron. So, if I take the mid-edges of the twenty-four edges it gives me twelve axes. See that. That would give me then twelve axes of spin, so this is really quite a complicated one. You see it goes through mid-edge, corner, mid-edge, mid-edge corner, mid-edge, mid-edge, corner, mid-edge, mid-edge, corner. So there is a symmetry about it, but it makes it quite a complicated one. Look behind me there and you'll see it's quite a complicated form.
Next picture. There is another of the...
Next picture. Another of the twelve great circles.
Next picture. Now, I come back again to the Vector Equilibrium. I have here, how many ? There are six faces square faces aren't there? If I take opposite the mid of each square face; the six of them would give me three axes, and this will revolve, go vertex, vertex, vertex. It has a square section in there, can you see that? I can't put my finger here, but, I have to hold onto it to do it, but this is how it revolves. This is, then, what they call the three great circles.
How do we get the four great circles? I go to the, there are eight triangular faces, so I take the eight triangles, and take their mid their centers of gravity, and there would be four axes between the eight faces and so I revolve it on those, and there you see the four great circles. See that great circle? Here is a triangular one again. As always that's what gave us this beautiful form here. Those come out of the triangular faces, so the three great circles of the square faces; the four great circles of the triangular faces; there are what other features do we have here? Then we have the twelve great circles of the mid edges. There are three four six, we have there were twelve of these vertexes so there are six of these three, four six, twelve. So three and four make seven and six make thirteen and twelve makes 25. There are twenty five great circles on the vector equilibrium. They are 3, 4, 6 and l2. They are four of the seven of the basic symmetries of crystallography. And you can see why how absolutely simple and fundamental ...
Now, I want you to watch each one of the ones that I have just done with you. Go back to the three great circles which were square faces. It goes vertex, vertex, vertex, vertex. There are four vertexes involved, right? And then next we go to the triangular faces, and I have six vertexes all the time. So they go through many more vertexes the four great circles go through many, many more stations of tangency more spheres it can this is a railroad track, it could get you into more stations than the three great circles. And then we take the six great circles where the, where we get vertex, no vertex, no vertex, vertex you get two vertexes on the six great circles, but none the less they do transfer at the main grand central station of tangency to other spheres so that the energy can travel over the convex surface of spheres the most the shortest distance, because all great circles are the shortest distance they are going to travel, so they can travel on the six great circles. Then look at the twelve great circles, mid-edges, remember? Sure enough it goes thru a vertex, mid-edge, mid-edge; vertex so it goes only thru two again on the twelve, these are really very fascinating characteristics, but this is part of the main switching of energy in Universe, and every one of the ones I have just given you the three, four, six and twelve, are all foldable out of whole great circles. You have to do your spherical trigonometry to know what the central angles are, but once you have you can fold this up very neatly and you will literally take twelve great circles, fold them up, and make the twelve great circles, and come out the continuous great circles out of these bow tie forms which come together. So it tells you that everyone of them has a holding action where you can go around locally, or go on and travel, but the fact that some of them have two stations, four stations, six stations means that they really are quite a different set of options for travel on those different sets of great circles. But and one of them has more of them than the other. The one that has twelve has only two stations, so that is really twenty four cases there; the one that has six great circles we had two again so you see the twelve opportunities there. So the things are not coming out the same number I want you to realize.
Now, the next thing we come to is the icosahedron. May I have the next picture. Here is our friend the icosahedron. You see some pentagons and right away you say, this is due to that "fiveness" something to do with the icosahedron. So, what do we have here? We have the same twelve vertexes, so it has six great circles. It was interesting, there was one the six great circles of the twelve vertexes, but also don't forget, where you get the three great circles was the octahedron and it took six of them to do it. So it really is a six there is six appearing in here twice in the vector equilibrium. There are also six on the icosahedron. Now look what it does. I spin it and it doesn't go through any of the stations! So, suddenly, there is a cut off. Then, let me see, what other features do you have here? The other one has squares and triangles, this one has only triangles, so I have twenty triangular faces I have twenty faces, therefore I have ten axes of spin. This is the ten axes of spin here, and you'll find that it is a very amazing thing on the icosahedron it keeps missing the vertexes. And then I have the what else do I have? I have thirty edges which gives me fifteen and this is the only one where they transfer. It goes yes, yes, no; yes, yes, no; yes, yes, no something like that other kind of pumping you ran into. The yes, yes, no shows up quite often in basic series here, and makes it possible to do yes, yes, no; and a yes, no, yes, no so that you don't have any interferences. So, icosahedron has only two chances.
Now we find the icosahedra, they do not carry on and fill all space, therefore they are not what you get in closest packing closest packing you only get with the vector-equilibrium. Then it contracted in order to be the icosahedron, so it doesn't have the contacts. Time and again I want you to feel this kind of neutral condition, like a neutron without vitality; and the one that does have the nucleus you go into the proton. Same number system, but just a little bit contracted. And the difference in the contraction is just the difference of an electron. So, we find that this thing cannot have many layers. In fact, it tends to act only as an electron. It really has to be a free space actor.
And it does have only one way in which it can actually ever make contact with things and get something out of the system. And that was this one, of which there are fifteen of those, great circles, thirty edges.
So, let's look at the slides again, and everyone of those are foldable, out of whole great circles. That is the six great circles.
Next picture and the six great circles again in transparency. I've done them in quite a number of different ways different opaques.
Next picture please. And this is showing the mathematics with...
Next picture. Same thing again, still six great circles.
Now we are into the ten great circles.
Next picture. That's the ten great circles.
Next picture. And there are the fifteen great circles. This is a very beautiful one. Fifteen great circles are as large as we get, and the Babylonians discovered this long ago, that, I gave you you remember structural systems where I had tetrahedron, omni-triangulated, inside and outsideness a system. Octahedron, inside and out. Icosahedron. But tetrahedron has three triangles around each corner. Octahedron four. Icosahedron five, and you couldn't have six because they would add up to 360 degrees and would not come back to themselves. They could not be a system. So there was absolutely a limit of three possible cases you remember that.
Now, in the, I've lost track in coming back to my picture. Can somebody give me a help? (From the audience: "fifteen great circles" ) Right! So the most equilateral triangles the tetrahedron has only four, the octahedron has eight, but twenty is the largest number of equilateral triangles you could possibly have in the system quite clearly.. Because otherwise they would add up to more than 360 degrees this is a limit case. Now, each one of those equilateral triangles, quite obviously, you can divided an equilateral triangle by a perpendicular bisector very nice symmetry, so each triangle has three perpendicular bisectors, which will then divided it into six right triangles. There are three positive and three negative. Yes. six of them. We have six then times twenty faces l20. May I have the l5 great circles back again?
The Babylonians, mathematicians, discovered then these l5 great circles. And I want you to realize the difference between a spherical great circle triangle than a chordal. Because, look at the right and lefts in those if they were flat edged they would be hinges, but they are arc edge, so they will not hinge to the side. So you find that the concave and the convex cannot rotate the one cannot take the place of the other, on the flat they can. The positive and negative right triangles. One is red on the inside and white on the inside. You have a red and white seemingly congruent this way but in the concave-convex they can't do it, so there are 60 positive and 60 negative right spherical triangles into which you divide unity. This is a limit case of similarity of subdivision of unity. IT'S A BIG ONE! So the Greeks the Babylonians discovered that, and therefore, this is where they came to trying to coordinate time and circles. The two kind of unity. So they came to the sixty second, sixty minute. This is where the sixtiness comes from. This became, then, to them, really the top necessary number, and they included the prime numbers l, 2, 3 and 5. So I just wanted you to know that the Babylonians show this figure in the old things and it is very exciting to see it.
Next picture please. This is the fifteen great circles folded, and you find that they are folded, there are fifteen of them but you will find that they make a total of l20 triangles. Each one gets folded into divide l5 into l20, what do you get? Eight. Yes. Well, each one of these has to be in a special fold. You'll find that they are not these are each very nice and symmetrical they are bow ties, just as neat as can be. This is the model I have used different colors, I've used yellows and blues and blacks and so forth, and the model these are strange kite-tales, where the one tetrahedron is edge to edge with the next tetrahedron, and they come together to make tetrahedron spaces outside of themselves and inside of themselves. So each one of them has four, and each one of them has two no it makes up four on the outside four inside and four outside of these strange things, and they do not come together in a symmetrical manner. It is absolutely impossible to make them symmetrical.
The icosahedron has these very interesting, very independent properties where it seems to peel off. And Vector Equilibrium is where everything really is passing through Unity and from thereon everything that goes on is some kind of an aberration a folding up, or a skewing, or whatever it may be.
Next picture please. Looking at the fifteen, same.
Next picture 120 triangles.
Next picture please these are revolving the icosahedron on the ten and the fifteen I just want to and the six.
Next picture. There is the icosahedron showing all of it's what does it have it has six, ten and fifteen thirty one great circles. O.K.? But the first one's where you use the same twelve vertexes that you had in the icosahedron in the vector equilibrium. And those twelve vertexes gave me this very nice great circle where you did have two vertexes you went thru, so it was contact, but the sixth great circle on the icosahedron does not, it is absolutely pure equator a great equidistant from all things. It would not conduct at all. So we take the, remember, I had twenty-five great circles on the vector equilibrium. There are twenty-five that really match them that are taut or twisted on the icosahedron, and then there is a sixth additional that goes around, but does not touch anything, so each one has one has thirty-one and the other twenty-five, but twenty five plus six is the thirty-one, the extra six which does not go thru any of the grand central stations.
Next picture, please. Now I am going to, see if we can make this bright enough for you to see it, I spoke to you a little while ago, when I had the vector equilibrium, remember, pumping up and down, and the equator was rotating, but the axis was not rotating. That is the big thing, right. Now, I can make this same kind of a model I have eight triangles, you can see them alright. Then I have four axes to the eight faces those would be the same as the perpendiculars to the faces of the tetrahedron the four axes. You can find those four axes if you want just go to a cube, and there are eight corners, and they are symmetrical to one another. And take the diagonal from this corner of the cube down to that one there, and there are the four diagonals between the eight corners, and they are the same lines and the same central angles as the perpendiculars to these square faces here. Then I could take this vector equilibrium and put a one-eighth octahedron on here, and the whole thing it becomes a cube. So it's just coming from this center. Now, because that's so, between vector equilibrium there is something I call each one of these triangular faces has a one-eighth octahedron, so if eight of them come together, they make one octahedron. So it's what I call an exterior octahedron, and inside, when I bring vector equilibrium to vector equilibrium this square face touches it there is an interior octahedron and there is an interior. Two types of octahedron that keep showing up interior and exterior to the nucleus. And they have to do with the loanings and the joinings of molecules, of the chemistry of atoms coming together. How you can loan so many charges one to the other. And this is what is done in here.
Now the I'm going to, instead of I'm going to put eight and four rods coming thru a common center here, and weld them together nice shiny rods, we'll say a quarter inch in diameter. And now that they are welded together I'm going to take, instead of eight triangles, I'm going to take eight little automobile tires. I'm going to have this rod, then, it's diameter will be the size of the hole thru, get little toy automobile tires with the little metal wheel in the center, and then it has a little hole for a journal going through so we can slide it onto a rod. And I'm going to slide the eight automobile tires, toy automobile tires onto these rods, so that the plane of the tire this is the wheel, it's over like this sliding in thru its hub at the center of gravity where the triangle would be here. So each one of those wheels will be touching another wheel at three points. Can you see that? There's one here, there's another one here, and so the automobile tires slide in on the rods until they keep meeting each other because they are converging so they begin to push very hard the rubber` on one another. So I bring them all in a equal distance and in tight contact with each others surface, and then I put a little journal on the outside of the rod so that they can't slide outwardly we'll use a little metal washer, and then some tape on there to hold them where they are, so they are held in tight friction with all the other automobile tires. That is a model you see up there behind my head. Once you have it on your minds maybe it will be more clear. There are then these eight wheels, and I find then they are absolutely independently journaled, free on here, yet they are touching one another. So if I take and put this if I hold onto this as a system, these rods then stick out and I can hold on to these rods independently, if I rotate one of these wheels here, then this one has to move they all move. If I rotate one all eight rotate reciprocally very beautifully. I can try anyone of them and I found all eight of them absolutely beautiful to go round and round, so this motion that you saw, I want you to suddenly realize this could be the same motion I say they're rotating on each other, but this top one here is staying put and the ones around the equator are rolling along, can you see them? This one could go like that and then keep on going. Can you really feel them going around the equator? Well, alright. Now, for the first time, then, this has a limit until you come to the end of the hinges, but the one model I give you now there are no hinges, so they keep rotating one way or another and the whole thing is reciprocal. Then you'll find , going thru these four pictures, I have up in the top left hand side a little white marker, and what I do now is to take a hold of one of the wheels with my fingers like this, so I immobilize that one wheel. And I take a hold of the wires that are sticking out and pull the system around the one that I am holding onto, because you'll find the three touching they just roll nicely around they roll around, they're ball bearings. And these ones are rolling this way. The three are rolling the other way and there is one at the top.
As I hold this one fixed and I roll them around so there are out of eight of them three of them in the northern hemisphere, three of them in the southern hemisphere are rolling beautifully. But the top one is absolutely immobile. If I immobilize the bottom one, the top one is absolutely immobilized. And that is what you can see in this picture as I go around.
Next picture. You see the marker will stay up there at the same position all the time.
Next picture, I'm sorry, I seem to be so much in the way of this thing.
Next picture. The hands had to really stay fixed at any rate.
Now, what I have shown you is the I've given you an independence of the axes that you can fasten onto another system, yet the rest of the system can be carrying on. So, I said, every system I find always has axes, it always has an isolatable axes this is four dimensionality.
Now, four axes of the basic symmetry. So the next thing I wanted to point out to you is that those rubber tires, I could have made them a distorted donut, to be a little triangular can you see? So they just look like a cam. Here's the circle and I begin to make it go like that, so this is a little shorter radius, and this is a little bigger radius on the side, I could make each one of those a triangle. And if there were springs holding them in towards each other as they rotated, they would go into the position of the octahedron when they simply get down into this closest position of the sides of the triangles versus being on the corners of the triangles. So as I rotate the system everyone of those triangles is going to be pumping like that by just their own friction, and around and around they go.
Now, the next thing about it that I am going to say remember I had an involuting and evoluting donut? Rubber donut? So I'm going to make each one of the triangular cam rubber tires into also involuting and evoluting so, when I hold onto one triangle at the bottom end, I'm holding onto it, which makes the one at the top do something, you'll find this whole things goes through now I'll take a hold of one edge and start to move it around force doing that, and the whole thing pumps like this and continues involuting and evoluting . And when you see something called turbulence this is what you are looking at. It's a very, very beautiful thing. When we begin to really study what is turbulence, this is the big show!
I find it fascinating that with just a relatively few models, begin again to be able to do this in your imagination.
This is a very fascinating pattern, because the first time the scientists ever made photographs of the atom with a field emission microscope, it came out, you could really see the whole atom and it's operating and it was this vector equilibrium. It's that picture you see right there. I think we have that picture in a set somewhere. We'll find it for you and we'll put it on for you tomorrow. But it is really spectacularly there. The square is a little larger, it's sort of that kind of an aberration or distortion, but you can really spot the whole twenty-five great circles.
Next picture. And there we are looking at the icosahedron and its thirty-one great circles. And there is the icosahedron in the spherical with the venetian blind straps. Now I say there, you have seen now, the symmetries, actually, visually, the seven great symmetries of crystallography. You're a crystallographer you spoke about it yesterday the normal way... This becomes very exciting to see!
I found one that the crystallographers were not very familiar with were the twelve great circles, for some reason or other, of the vector equilibrium.
Next picture. Now we are looking at them both, and this is the end of these particular slides that we are going to use. There are other slides, Meddy, that we had put aside, reconfirming some of the things we have been over here today; but we might as well let that go for the moment.
I'm sure you are beginning to feel with me the interrelatedness of everything. I don't think there is anything that I have talked about in all these hours now, and we're getting pretty close now I think we're about to sixteen hours that everything is continually interrelatable. And think how really different that is from all of the specialization and the times when I was young when biology didn't seem to have anything really to do with chemistry and chemistry didn't have anything to do with physics. The UTTER interrelatedness appearing!
I'm going to bring you back to C.P. Snow and his book TWO WORLDS. And his book about the two worlds meant the two worlds of the humanities and the sciences, and he was absolutely convinced that there was a chasm building between them that absolutely would never be spanned. It was going to get worse and worse. He felt this was really a very great warning and that humanity must appreciate it.
He, then, in his book, if you read it, he attributes the chasm beginnings he goes back to about a century and a half to the middle of the 19th century the first half of the 19th century. And he points out then, for instance in America Emerson and Thoreau, he felt manifested antipathy to industrialization. Snow says that. The actual fact is that I think that is a very bad example and I'll give you good reason for it in a minute, but then he gave a number of authors in England, because he said, the literary man, the humanists just felt he didn't like the smell of the laboratory, he didn't like the feel of the factories that the labor was being cheated and so forth. It just felt wrong.
C.P. Snow asked me to come to visit him, just for an afternoon in his apartment flat in London, England when I was there. And I went over the energetic-synergetic geometry with him, and I went back to the point where I've said to you that scientists, starting with the beautiful Priestly-Lavoisier set of events of identifying steam, and combustion metallurgy out of it came the steam and the ships and the great wealth that was made by the people who put steam in their ships and they didn't have to wait for the wind in their sails. Brought about enormous patronage of the scientists and these great funds to the Royal Society out of which came thermodynamics.
And I said to C.P. Snow, as long as it was steam, the humanist could then go to the scientist and say, "I see just what's going on there you can see the steam, you can see how it goes you can turn it into pipe and things and you can see exactly what it does. You can FEEL it. It was no trouble for the humanists to describe that in a book. But when he got to electromagnetics and he couldn't see what was going on, then the humanist said "You've got to tell us Mr. Scientist, what IS going on? you must give us a model so we can describe, it. We always have to describe what goes on. " And that is the connection between science and humanists. And the scientists said "We can't, it's something invisible" and as I told you went into that the other day. And they felt a little guilty about it, but they suddenly felt great when they came to discovering in energy studies that black body radiation had a fourth power, exponential 4 rate of change, and they said, quite clearly then, nature you can't make anything but a three dimensional model because to them dimension was perpendicularity. And they said "You can't find another perpendicular system it's just parallel to a line that is already there, therefore you cannot have a fourth dimension....but, "the scientists said " Nature quite clearly is using a fourth-power inter-relationship, therefore, quite clearly Nature is not using models; therefore we are now excused and exempt from any requirements so we are justified in the position we did take, we're very lucky we took the position!"
As a consequence, Science, then, in the mid-19th century, what you and I, then, then would call "flying on instruments" they started flying on instruments and were not looking out the window anymore. And, they have been really flying for a century and a half on instruments. And this has really in the meantime when I was a kid, I was being told then that "no model and so forth" and I felt there was something probably wrong about that. Again, it is really interesting, the kind of strange suspicions I had that I'm not hearing things quite right, like the fractions and the decimals and so forth, and all the geometry arguments. What seemed to be self-evident to the geometer. I felt, then, that the we'd just get a more powerful microscope, and every time we'd get a microscope we could see something, because if Nature, then, really had a threshold, and she doesn't really have models, she'd get to where she didn't have any models. But when we got into that area, the people were saying there were only mathematical equations, then suddenly there were still some models. But the models were not easily explicable in the terms of x,y,z coordinates. So they say, the scientist used to say to me, that nature is just facetious, pay no attention to those pictures you see there.
That was a very strange attitude but it still was quite strongly in the time of say World War I, and between World War I and the great crash. Thank you. At any rate, it was, then, my feeling that the scientists were in some way making bad starts and bad assumptions when I saw that Nature is continually using models and something went on very tantalizing that seemed to be more or less orderly. And, so that made me persist as I have here.
At any rate, with C.P. Snow I showed him energetic-synergetic geometry, and I said to Snow "I don't think it was antipathy of the writers for the smell of laboratories and factories that made them into I think it was simply the scientists saying to the humanists "We can't give you a model. And C.P. Snow said "I really think you're right." So then I went over with him the energetic-synergetic geometry which he didn't know about. And I said "It is my hope that we really do have conceptuality returning and the conceptuality comes because I can make the fourth-dimensional models. We're not using up all this space around an omni-directional clock. I've got room for twenty hours and you only had room for eight, so with the twenty hours we can get in the fourth power no trouble at all.
So, when we get to two frequency, for instance, vector equilibrium two frequency we get to where the volume is twenty times eight 160, and you find that that is two to the fifth power times five you can literally make the model of it! Alright, it gets to be very exciting that you can make At any rate I showed him the models and then I said, all the things that made them give up the idea of models, because they said Nature could not, she was using that mathematics but you could not make it into a model, but I said "You can make it into a model." So any kid could really do nuclear physics here.
That meeting of mine with C.P. Snow was about six years ago, and he, that New Years, when scientists are often asked to make some statements, he made a statement in England which came out in the New York Times that he was convinced by an American architect, that the chasm between the sciences and the humanities could be closed. That he would like to change his position.
Now, I hope you begin to feel with me, because I feel a deep responsibility to have you feel with me that we really do have a potential coming up here, and that this is a great option for humanity, and I feel very committed to being sure that young people get a good chance at it. Because, I'm not going to go much farther into the energetic geometry today because I would like to keep sorting my models, because the models in the pictures we have are extremely informative. And, I do like giving you the pictures, and I've done it several times in the past, really make it in your own brain, but I think it's better this way.
Incidentally, there are the six great circles of the icosahedron. And there are the twelve holes in here. And those are our friend the same twelve of the closest packing of spheres there.
Now, I'm going to ask for a break, because in the conducting of what I am doing with you doing everything spontaneously, I do not have something I really feel immediate that I want to get at, because I don't have the tools and I feel a little bit of impotence about it. I would like to have a break, and it's no where nearly time to stop, so that if you don't mind a little more break we'll try to get a few more slides.