Quantum Topology & Quantum Teleportation

0:00 This is actually on teleportation and quantum technology, so I guess I changed the word I wrote this. This is more or less joint work with Samuel Monaco and Hillary Carter, which is to say that if you don't have a joint work on this one, you never have a book for planning. and what I'm going to do is talk about teleportation but there's a kind of nice way to think about it topologically and so I thought the best thing would be to just review everything, more or less review everything. Now maybe I should remind you while you're staring at this little slide what teleportation and we'll get to it later, the idea is that you would like to transmit a quantum state from point A to point B, and the people who are thinking about quantum information theory, and I haven't got my history straight, so I'll just say the people, because I'm not quite sure who invented the teleportation procedure that I'm going out to talk about. I should know. Maybe somebody here knows. I think it was about five people. So anyway, if you know the name Salem, I'm sorry, I didn't think to check. Anyway, teleportation intends to transmit a quantum state from here to there, but you can't copy quantum states, so you have to do something a little more subtle. And this invention is quite nice. The idea is that you take an Einstein-Rosen-Podolsky pair, an entangled pair, Zionists in this scenario are called Alice and Bob traditionally, so Alice wants to send a state to Bob, and Alice has one end of the pair, and Bob has the other end of the pair, and that pair gets used in conjunction with a measurement by Alice to change the state that Bob has, and then you could ask what measurement could Alice perform if she succeeded in performing it to make sure that Bob's state ended up being exactly what we want. But you can't always have that because you can't always predict the result of a measurement. But the simplest quest that you can do, and we'll do that eventually, is to think about what measurement should succeed for Alice

2:30 in order that Bob would have the right state. And then when Alice doesn't succeed, she could find out what she did measure and send a little information to Bob, and then the whole thing works. Okay, so that's the way this works. And if that made no sense to you, never mind, because I'm going to do it slowly. I'll ask an embarrassing question. I suppose the teleportation was paranormal transfer of objects, but you're not talking about that. Oh, no, no, I'm not talking about anything paranormal. This is the paranormal transfer of bits. At least, the intent to transfer a quantum state from one place to another. Yeah, they cop the work of science fiction. But they have actually done it. But you have to transfer classical information as well. So I want to remind you of basic quantum principles. So here's a quantum state in two dimensions for two possible outcomes, 0 and 1. and traditionally called the qubit, because it's like a bit, except it's probabilistic. And as a quantum state, it has coefficients that are complex numbers, the sum of whose squares is equal to one. And upon measurement, one state or the other appears, is indicated for the person who measures, and appears with probability of the absolute square of the coefficient in the state. And this is part of the more general situation of a quantum state where you may have some large number of mutually perpendicular outcomes which constitute all of the possible outcomes, and each one has a complex coefficient, the sum of whose squares is equal to 1, and the probability of landing in one of those states when you measure is the absolute square of its coefficient and when you measure in this notion of measurement you don't know which one you're going to get. You just have the probability. So that's what I was saying a moment ago that I could think of just the right measurement in some cases but if I'm being careful I have to think let's measure and then see what the outcomes could be. Okay, so that's the notion of a qubit and a bit of information.

5:00 I'm intending to kind of go slowly here at the beginning and just cover these basics, but the whole thing is not more complicated than the beginning. Here's quantum computing or quantum information in a nutshell. It says in a nutshell it will remove that since the slide. has a little trouble seeing them on the screen, but I just wanted to remind you of these things. I've already reminded you that a state is like this, or a cupid is like that. If you apply a unitary matrix, a matrix whose conjugate transpose is equal to its inverse to such a thing, then the length is preserved. I mean, that's the point about a unitary matrix, is that it preserves the complex length. consequently, the total probability is preserved, the sum of the squares, and so if you were trying to invent something that would preserve this sort of probability, you would think of unitary matrices, but in fact, in fact, the model in quantum mechanics is exactly that, that a process, a quantum mechanical process, is a composition of some unitary transformations, and a measurement consists in projecting into some direction, and the probability of getting that is the absolute square of that inner product. So in the case that we were looking at, where I want to project into the directions of 0 or 1, then when I take the inner product of 0 with V, I just get alpha, and if I take the inner product of 1 with V I just get the beta and the absolute square of that is the probability. And I'm assuming that these zeros and ones are perpendicular to one another like that. So that's the set. And to connect with AMPA considerations for a moment it's fun to look at this formula through Tom Etter's eyes thought we could just do that for a moment. Sorry, I keep eliminating the Tom. This is a Tom-Eder diagram of the probability calculation. If you just look at it algebraically for a moment, here's this multiplied by its conjugate. And when you take its conjugate and think about what

7:30 computes that, you can reverse vector and evaluator and get a little, and get the U star in there like that. So that's the product of the complex number and its conjugate. But if you think of them being put together that way and make a diagram of it, you see that psi gets connected into a U and it also gets connected into a U star. And psi prime also gets connected into the U star and the U, so that the preparation and the measurement are connected into this circle containing the U and the U star, right? And if you were to remove either preparation or measurement from this picture, then the U and the U star would multiply each other and nothing would happen. So that's a kind of an insight into the structure of the amplitude and the structure of the probability. Very small. and the insight as I've tried to write it here is that without both the preparation and the measurement there's no event, the quantum event interlocks the preparation and the measurement, the past and the future in that sense, and without that divide of past and future there's only the unitary matrix and its conjugate which multiply each other, leaving you with a void circle or a circular eternity and Thomas said things like that are similar things in his paper with Pierre which you'll find in Peir's book in a common place. Can you comment on that? That's neat. It's neat, right. And diagramming the matrix multiplications is also practical, and we'll be doing some of that. But I think it also helps, in a way, to think about that in relation to the meaning of the quantum theory. But that's a prejudice for that. In any case, there is this idea of computing And if you're thinking of computing, then you'd like to have naught, which changes 0 to 1 and 1 to 0. And that isn't any kind of unitary transformation in itself that you would want, necessarily. But, well, actually, let's see. I mean, you could think of that. You could think of that all by itself. But a particular kind of gate that people like is the control nod, and that's also unitary. If you decide that on zero, with another input, the gadget will not do anything, but on one, it will act like not.

10:00 And you write down the matrix for that. Now I'm dealing with two qubits. Then you see that you have a little unitary matrix there. Quite a useful idea. So that's control nod. example of this notion of building computers by using unitary gates, which constitutes this attempted field of quantum computing, which is perhaps going to actually produce real computers. So, and I've also introduced here the fact that we might want to think about two qubits, and I will use the usual notation of putting the two in a little bracket here. But you can also think of this as lines like this and the lines correspond to the elements of a tensor product, right? This is one vector tensored with another vector. So lines parallel to one another correspond to tensors. I'll use that again later. And similarly here, these guys are tensors also. So if I write 0, 1, you may wish to think of it as 0 tensor 1, which is just, tensor, of course, is just a way of combining things into other things that look like products. But when you're doing a linear algebra, saying tensor is a way of explaining how, if I want to apply a certain pair of matrices, say A and B, A tensor B, A will act on this one and B will act on this one, over one another. So it comes up. It's a formal product. So you can take formal products of matrices in the same way, letting one act on one factor and the other on the other. One more part of this sort of little background lesson in thinking about this sort of thing. These are all well-known things, but I just thought it would be good to go through them a little bit. Here's what's called the Hadamard transformation, this little unitary on the left, whose square is equal to the identity. And this has to do with the relationship between matrices and thought experiments, really. So in the thought experiments below, I have a half-silvered mirror, which H is talking about.

12:30 If 0 comes in, then it will reflect and turn into 1, or it will go through and turn into 0. Or if 1 comes in, it will reflect and turn into 0, but it changes phase when it goes through. So there's a phase change there. And these, of course, represent the two columns in the matrix. This represents 1 times 0 plus 1 times 1. And this represents 1 times 0 minus 1 times 1. So the matrix language lets you pack up the possibilities gadget. And I have another gadget here, a half-silvered mirror, which doesn't transmit, but reflects. And when it reflects just like this one, it changes the state as well. So that's a 0, 1, 1, 0 matrix. You put in a 1 put in a 0 and you get out a 1. Where the 0s and the 1s that I'm speaking are a little confusing because they're not the same as the 0s and the 1s in the matrix, of course. But this 1 stands for second second vector which is one and this zero stands for zero times first vector which is zero. Everybody clear on my zero one speaking. Yeah? Is your own notation like Dirac bra ket notation. So these are the little kets are the prepared states, and a bra, like bra zero, is thinking of evaluating the state. So it's a function on it. And really that little matrix in the square brackets is a representation, isn't it? Right, so this little matrix is a representation of the action of the half-silvered mirror, and the Hadamard is a representation of the action of the, I'm sorry, it's a representation, the Hadamard is a representation of the half-silvered So this guy here is the representation of the mirror. And if you then put these together into this little gadget here, then you can use this instead of the double-slit experiment if you want to discuss elements of quantum mechanics. Here's the photon zero coming in with this spin. And then what happens, right? Well, it goes into this superposition of being transmitted or reflected. And then I put the mirrors in so I could do another Hanamod over here on the right. So then we have these possible pathways, reflects, reflects, reflects, or transmits,

15:00 reflects, or transmits, and transmits. And you have to take the superposition of everything that happens from beginning to end, assuming you make no measurements in between. So I draw it all in the chart, right? All possible pathways are drawn in the chart. These, as you see at the end, these pathways add up to zero. So there's a complete interference at the top. And this is a little reinforcement down here. So that says that if you send a photon in here, it will come out the bottom and it won't come out the top unless some measurement happened in between. And what does that look like in matrix? calculations. It looks the same, only I've done it with matrices. The matrix calculates the concatenation of the three matrices is the same as adding up all the different paths in all the different ways. So that when we multiply these three matrices together, we found 1, 0, and 0 minus 1. This first column corresponds to what happens if ket0 goes into the system, and it says that ket0 will come out. And ket0 was down. So that's a nice example, which can be discussed at more length. But all of that gives a picture of pretty much all of the principles involved, right? The idea of summing over all of the different possible pathways, the idea that things can be represented by matrices, and so on. Comments? Okay. Then there's entanglement, and this is the most complicated slide, and then I'll get into things that are a little less condensed, but this one is admittedly an extremely condensed slide, but let's look at it for a moment because it's got a lot. A little more. A little more down. There we go. There's a moral at the bottom, but we can go this far. Here are some operators. They have eigenvalue. The eigenvalue is plus and minus 1. And if they were classical, then you would expect to just see plus or minus 1 each time when you made the measurements. And as quantum, then a measurement will have eigenvalue plus or minus 1 as well. But you have to compute your expectations and probabilities using the rules of quantum theory if you want to do it as quantum.

17:30 And then this very particular expression lets you see something about the difference between classical and modern observations. So this is a nice exercise of a few pages of little matrix calculations. But if you do them, you get some nice results. And I'm just going to tell you what the results of the exercise are. So I'm starting with this state, two-cubit state, with coefficients A, B, C, D. and I compute the expectation here if A and B and C and D are real then I don't have to worry too much about squaring things and that's the calculation I actually did here you could carry it over the complex numbers just as well so this is the square root of probability that you would expect for this and this is the expectation and what you get for it if you compute this I just want to tell you how to do the exercise. You can't sit here and really do it. But this means Q acting on the first tensor factor and S acting on the second. You see, so that means that if you went up here and applied QS to phi, it would mean that you took the first one, Q applied to 0, well, that's 0. And then S applied to the second factor, which is a 0, and that would be minus 0 plus 1. Sorry, minus ket 0 minus ket 1 over root 2. And then you move on together, and you add all this stuff up and compute this. And that's the expectation, and the expectation that you get in the end is this quadratic kind of expression here. And then on the other hand, what is the expected value, expected eigenvalue if everybody was classical? Well, then either you're going to get eigenvalue 1 or you're going to get eigenvalue minus 1. Everything is very simple in the classical case. So you're going to add up, and you're going to get 1 and 1 and 1 minus 1. you get 2, or you'll get minus 1 and minus 1 and minus 1 and plus 1, you get minus 2. So the expectation classically is directly plus or minus 2. The expectation is less than or equal to 2. And so what you have is that if you compare the classical expectation with what you will get for certain states, you find that it's not the same. So, for example, if the state you started with was 0, 1, minus 1, 0, over 2,

20:00 that means that B is 1 and C is minus 1. And you put that in here, and A and D are 0, and B and C are 1 and minus 1, yeah? So that's, this is, I'm sorry, 1 is 1 and 1 is minus 1. so so this is one and that's minus what's there's this too I wanted two and 4 is 6, but it looks like I made a mistake in my arithmetic. That's correct. I think it's right. Because you get 2 plus 4, it's often 6. Yes, if b times c was equal to 1. Sine is here. Minus 1 times minus 4 is plus 4. Yeah, plus 2. Okay. Alright, alright, so it comes out. And if it didn't, you could change a sign in some way. And in any case, you see, the point is that it ended up greater than two, and on the other hand, it was classical. So it shows you that you can get quite different kinds of expectations out of the quantum situation than you would out of the classical one. And this inequality situation is what people call Bayless inequality. And then you can ask, well, what if I had a general state and I wondered whether I could or could not violate the inequality? Well, where are we here? This over the square root of 2 greater than 2 is the same as this greater than 2 root 2, which is what I wrote here. So you want this expression to be greater than 2 root 2. And then you play with this a little bit. And you find that it's the same as saying that root 2 minus 1 over 2 should be less than this. This is squared and negative, so that's pulling it down. the determinant of some little matrix, right? So what you found was that if you went back up to the state, there's a matrix associated with the state that you should consider, and it is the matrix whose entries are A and D and B and C, yeah? And that matrix is telling you something about when you will get these violations. The matrix is A, B, C, and D. So this is one reason

22:30 to think of this matrix associated with the state, not having anything to do with anything other than this subject at hand, that if you're thinking of a two-cubit state, you're going to just think about this little two-by-two matrix, and if its determinant is non-zero, that could unbalance this equation and create the violation of the inequality, which is, in fact, exactly what happened here in the case we were considering where the determinant was minus one, or one, it was in one, and ended up unbalancing the inequality. So if this determinant is non-zero, then it's possible to violate the Bell inequality and have a quantum state which behaves different from a classical one, okay? And as I said, this is a very concentrated slide, but I recommend it to you if you want to try to do the rest of the exercise because it contains the lore about the Bell inequality. If you just do the rest of the exercise, you have all that for yourself. And the key point is, what does this mean, this determinant of non-vanishing, okay? Well, it turns out to mean something algebraic that's quite clear. You'll see on the next slide what it means. Suppose that I take a state that's a tensor product of two single-cubit states. So now I'm using that tensor product. I take the tensor product of two single-cubit states. I multiply out formally, and I see I get AC, 0, 0, AD, 10, 0, 1, BC, 1, 0, B, D, 1, 1, right? There it is. And then you could ask, well, does that have determinant non-zero? No, it doesn't, see? Because when you multiply the endpoints, the determinant was the product of these two minus these two, right? And when you multiply those out, you get AC, B, D, minus AD, BC. determinant is zero. So this state is said to be unentangled, and a state is said to be entangled if that determinant is non-zero, and equivalently that means that it cannot be written as a tensor product of two individual states. So people often just say states are entangled if they're not tensor products, but I thought it would be a good idea to actually tell the story, tell the whole story. The reason why people say that states are entangled

25:00 is because if they're entangled, they could violate bell inequalities for some choice of operators. Not necessarily any choice of operators, but it's possible. And so that's where this strangeness comes in. And the other moral is that one should think of this 2x2 determinant, sorry, this 2x2 matrix in relation to a two-quidate state. Okay? So now let's turn back to matrix algebra and topology. First, a little matrix algebra in diagrams. Well, everybody knows the formula for multiplying two matrices. It's this, right? The sum on the second index of the first and the first index of the second, you multiply them together, and you get the product of the two matrices. And if you draw simply two boxes, one for the, in this case I'm drawing vertically, so this is the first index and that's the second index, and they're tied together, the second index of A to the first index of B, then the concatenation of these two boxes can represent the product of the two matrices. And the only convention you need if you want to play this game is that you have to sum over the index that corresponds to where the two boxes were tied together, And if you have a free end, then you don't have to sum over it. Okay? And then you can see various aspects of matrix algebra from this point of view. For example, the trace of a matrix is obtained by tying the bottom to the top or the end to the end. And you see here that the trace of AB must be the same as the trace of BA because you can just push one around to the bottom around the loop. or if you want to play games with the quantum mechanical formalism, you can. Like, here's a famous formula for the expectation here, as the trace of u rho, or u as the unitary transformation, and then I'm evaluating back in alpha. and rho is the so-called density matrix and that's this guy here the Kemp-Bra

27:30 product. So that's a matrix because you see when you just form this you haven't summed on anything and you've got two free indices so that's a matrix and this is a well-known formula. So that's another example of this. And then here's a general idea, which is not, strictly speaking, part of a talk, but I wanted to put it in because I think it connects across the way I'm thinking and the way Tom and Dick Schaub are thinking also, and Pierre, about these networks. And I forgot the terminology for this. But if you have some network that really corresponds to matrices, you see, then going into that network and putting in a density matrix like this corresponds to computing an expectation. And so you would think that you should go in somewhere and cut one of the links in the graph and put in a density matrix like that that will correspond to doing a measurement. And so you can think that way about spin networks and other objects like this. where you have a network that represents the quantum process and measurements correspond to doing things in the network, whereas evolutions correspond to just allowing the network calculations to happen or possibly putting in a unitary somewhere. Anyway, you can think that way. You can make the diagrams into a kind of space, and the physics is happening inside the diagrams, if you like. Now let's shift all the way back to simple topological considerations without any quantum concepts for a couple of minutes. So I'm thinking about very elementary topology. I'm thinking about things like one-dimensional membranes, like Dirk was talking about this morning in two dimensions, and the fact that I may want to take a curve like this and deform it into a curve like that. and I want to think about how to turn this into algebra now the reasons why I would be motivated as a topologist to turn such a thing into algebra has to do with the fact that on my side that it isn't just the curves

30:00 but I can also represent knots and other entities by this but this is the base topology that lies underneath the knots and I'm not going to talk about knots particularly, I'm just going to talk about the curves And the curves have an enormous structure in and of themselves, as you can see, as soon as you doodle some Jordan curve in the plane, like that. And all of that structure of the curves in the plane comes really from the fact that you can have maxima and that you can have minima, and then one very complicated Jordan curve like that guy is some kind of composition of forms of maxima and minima going round and eventually connecting back up with themselves. And in fact, let's just digress for a second and get a clean sheet here. There's the famous Jordan curve theory, you know, which says that if your pen has ink in it, is that visible at all? Let's see if we can do something better. Well, if you draw a curve that is void in appearance, then there's no need to do anything about it, right? So there's some palestration for you. and the Jordan curve theorem says that any curve that I draw on the plane like this divides it into an inside and an outside and in fact that the inside and the outside are each topologically simple let's say just worry about the inside it's a disc the outside is a plane with a disc removed simple disc removed and and of course in this case you can just look at it by i-n-c-s-n-d-dot-com. I see the inside disk and the outside disk, the formed disks involved. Or you can even do some counting to figure out where you are. For example, it's quite clear that if you cross the curve, if you're outside and you run a line through the curve, then you will run it an even number of times. You see, like, you go from outside to outside and you cross through twice. So outside is an even intersection number and inside is an odd intersection number. If I'm here

32:30 where I am, inside or outside, I send a line out as far as it'll go, and I see that I cross it once, even an odd number of times. So odds and evens tell you whether you're inside or out. And the theorem says that this can be deformed into a simple circle. And one way of understanding the theorem, or even proving it to yourself, is to understand the following, that there is this elementary deformation like this of a maximum and a minimum cancelling one another. A nearby maximum and minimum will undo one another. You can just give it a little pull over that string and it goes away. And so you could look into the curve and search for cancelling maximum and minimum. And you see various examples of them. For example, here. there's a cancelling pair and so I can just locally replace that with a straighter line without eliminating one maximum and one minimum. And I go along like this, clipping off pairs and eventually reducing this to something which has exactly one maximum and one minimum and then it will be a circle. So that's the proof of the Jordan curve here from this point of view. But it could go wrong, or it could give you some problem. Like, let's say you went to this one, which is a minimum, and then you hit a maximum. That's all right, that cancels. But suppose you went from this maximum to this maximum, they don't cancel, right? So you could go from maximum to following maximum and find that this happened, and then they wouldn't be canceled, right? So if you're searching for it, you might start searching here, and I'm looking for it, and that doesn't pair with that. one oh but this one did pair and you see what the only way that this process could go wrong if you start searching for maximum and a minimum pair is if you spiral you see you might it might happen that you started here and then this didn't pair and then this didn't pair and this didn't pair and this didn't pair and this didn't pair and if you spot if you were able to spiral to infinity in there then you wouldn't be able to get out you wouldn't be able to cancel but if there are only a finite number of of these maxima and minima then you won't spiral to infinity and the whole thing will unwind like that. So I just thought I'd digress and talk about the Jordan curve here.

35:00 But if we want to make algebra out of this, then we need to make an algebraic image of the maxima and the minimum. And that's what I started doing on this slide that I pulled. You see, if I make a matrix for the maximum and I make another matrix for the minimum, might do in this kind of situation, and then we'll get back to the quantum information theory. So I make a matrix for a maximum and another matrix for a minimum. That's two indices. And then if you concatenate in a maximum and a minimum like this, this is supposed to be the same as this, which should be the identity, right? There shouldn't be any change as you go from here to here. So that should force the lower index to be equal to the upper index. that's going to be an identity matrix so maybe I should write that that this here is going to be the identity matrix delta AB which is equal to 1 when A is equal to B and 0 otherwise and then we have a matrix equation corresponding to a topological deformation M lower A I, M upper IB is equal to delta AB And, of course, you have to sum on I when you go over to matrices. Same as our rules before about matrices in terms of diagrams. So then this would all work very nicely. And I have a way to represent by matrices what happens with the curves. And if you had a more complicated curve arrangement, you could have a more complicated matrix product, like this. Here, I had a little spiraling thing happening, and I'll just translate it into matrices. M-A-I, M-K-J, M-upper-K-V, and I guess we missed one, M-upper-K-J, M-upper-J-I, sorry, M-upper-J-I there. And I might mechanically write them down and send them to you, and I would say, well, what happened here? Could you simplify that? And instead of looking at the picture and simplifying the topology, you could look at my algebra and say, well, let's see, this is summing on I, and these are inverse matrices, so this is just a delta.

37:30 And here's another sum on K, and these are inverse matrices. That's just another delta. And your two deltas multiplied to a final delta, which is just delta AB. So you could tell me without looking at the picture that this was canceling, right? So that's how the matrix algebra and topology interact with one another, like that. Of course, if the matrices were not inverses of one another, then we would just get their product. Now, why am I telling you this? Well, let's make a little transition. First of all, there are other ways of making the products. I guess I just might as well do this as long as it's on my slide. Here's a circle, just the bare circle. And in that case, you see the two matrices that are tied together, the lower one and the upper one, and you're summing on both of these because that's what you do, you're summing on tied lines. So this is the sum of the products of the corresponding entries. And that's the value of the loop itself. And any loop I draw, like this more complicated one, will evaluate exactly the same as this one, you see, because of the fact that the matrices cancel each other out. So any Jordan curve that I drew would evaluate like that. So that's another thing you could do if you're playing algebra versus geometry. You draw something and turn it into algebra and hand it to the computer, and the computer gives you a value that's different from this, then you know that whatever it was, it wasn't just a Jordan curve that would transform itself. And there's an analogy here with our bras and kets, which I've drawn. Maybe I should go this way. You see, you could think of one of these as a ket and a bra, and the other one as a ket. And in fact, I quite intend to do that. I want to think of this as a preparation, because I'm thinking of things going upward. And I want to think of this one as a measurement. And I want to think of the two together as a combination of preparation and measurement, brah and kett. And if we do that, and we combine it with our earlier notion of associating a matrix with a two-cubit state, then we're going to be in the position of having a nice diagram for thinking about teleportation. We'll see. So remember the previous idea. same way. I'm going to associate a matrix with a two-cubit state. I'm thinking of the matrix as

40:00 diagram like this, just like we did for the topology, and the matrix is exactly the one that I would associate to the two-cubit state in the usual way. The 0, 0, I'd be corresponding to the 0, 0 state and so on. So that means that this preparation, you can think of it as a little Feynman diagram which says that in the course of time we created this situation, and the situation is a superposition of these two-qubit possibilities with these coefficients, right? That's the creation. And a measurement is a dual vector like this one. I put some upper indices on it, and it's also associated to a matrix in the same way. And then we want to figure out the following quantum information scenario. I want to figure out what happens if I start with a state here, one-cubit state. I prepare this two-cubit state. I measure here in the first two tensor factors by some two-cubit measurement, and I want to know what the result is. There may be some transmission of classical information from here to here, and this is intended to be exactly the kind of thing that happens in the teleportation scenario, only it's got a little topological picture so you can by now actually see into it what's going to happen without doing the calculation. But the calculation comes on the next slide. So in that teleportation scenario, here's Alice over here. Alice can make this measurement. She has control in the first two tensor factors to make the measurement. This is the EPR state. You'll see a diagram of this in a moment again. But Alice can contact this end of the EPR state. The other end is very far away. So maybe this diagram really should look like this. Alice, Bob. Okay. Just to indicate that the EPR state has gotten its two pieces separated from one another in space. Alice has access to making a measurement over here. So that's what we want to think about. Now think about it before I do any more. We know that if we were to just multiply these matrices

42:30 according to the topology scenario, and this was one matrix and this was the other, this would be their product. It might be the identity. If it were the identity, then you would expect that what happened here would end up being equal to what happened here. That would be direct transmission of the state. If it isn't the identity, then you'll get some matrix applied to this. And what I'm asserting, and you can almost see it, I think, is that the matrix multiplication that I'm doing in the topology is actually exactly the same as the scenario prepare this state by using its matrix and measure these two factors using that matrix. it's exactly the same as what I was doing in the topology where I wanted to find out the resultant of concatenating those two it's exactly the same if you're used to these things you've seen it already by now and if you're not let's take a look it's fun to take a look anyway so we're going to do it according to the quantum formula so here we are non-informalism. This is the preparation, and we know that for indices j and s, you will get the coefficient of that state, that's represented by n. And for indices i and j, but j is set equal to r here, if these are separate, but j and r are the same, right here, you're going to have the co-indices for n. I keep putting that arrow because somebody has to tell Bob what happened. And so here it is. Here's the tensor product of psi with the preparation n. And here's n evaluating in the first two tensor factors. So we'll just write it out in the usual way. And you'll see if you get the same kind of thing. Here's psi. Psi is psi k k. Those are the coefficients. And I'm using Einstein's summation convention so you don't have to look at summation signs. And N is M-R-S-R-S. And N is N-I-J-I-J. I-J evaluates into K-R, right, the first two factors. S is left alone. Now, if you go back to the diagram, that makes sense. You see, this is evaluating into the first two factors, and the last factor is left alone, right? and then we

45:00 we have i is equal to k and j is equal to r because that's the only case that anything can happen here and that forces k and r to be i and j and then you see that you have n i and j, n j s, that's the product of the two matrices and it's been applied to psi on the left so there's a transpose there and so you see that you get the product of the two matrices side. And that's not unexpected after all the geometry we were doing with the diagrams because where underlies this is just a little wiggle and the way those two matrices in the wiggle compose. But this tells us a whole lot, right? It tells us that I think I have the diagram on the next page so you can just get the moral from the next page. Could you make the J-I transpose on the diagram? Oh, I could have. I could have done the whole thing with a diagram. I thought I'd better go back to algebra and make it clear about measuring tensor factors. So here we are. Alice is over here, and Alice has the ability to measure into that part of the EPR pair. This is Bob's part of the EPR pair. And then the simplest successful teleportation would be psi prime equals sine. Now let's think about that for a moment. That would mean that M times M would be the identity, right? Because the resultant here is M times M applied to psi. That means that M has to be invertible. And now you see the utter consistency of this whole thing. M has to be invertible, which means it has to have an on-zero determinant. There was no way to take an unentangled state and do this procedure. It can't possibly happen unless the state is entangled. Then you could do it, and that's just a matrix fact. Then the simplest thing would be when M is the inverse of M. So let's set up, let's see what else is on the slide. So Alice and Bob are sharing the EPR pair, and this would be an example of a successful measurement when these two are the identity. I'm caught, so you were standing in front of the star. Oh, I'm standing in front, right. So that's just summarizing what I said. Okay. so the simplest teleportation would be when the end state is equal to the beginning state after a successful measurement and that could happen if the measuring state that succeeded

47:30 had its matrix the inverse of the EPR pairs matrix now the simplest example of that would be if the EPR matrix was the identity that's a good one, that's 1100 that's entangled, so we could do that that would mean that the best success would be measuring in the identity matrix. Oh, I see I'm repeating myself a bit here. You're running out of time, too. Oh, that's okay. I'm almost done. So here we could take M to be 0, 0, and 1, 1. That's a high CPR pair. And it's the identity matrix. And then the best possible measurement would be the identity matrix. and then everything is sent through. And the remark about the determinant is the same, so let's quickly go on to the next slide and figure out what we need to do in general. So in general, we're going to get N multiplying sine, where N is the measurement. So now what we need is a good basis of measurements. And now we come to the point where if you're reading a book on the teleportation procedure, they will tell you basis or something, and then you check that everything works. But I'm setting it up so we can invent it. Because we know that these other guys should be rotating, they're going to be rotating the original state. And so we need some nice matrices in order to rotate. So you take your favorite poly matrices and maybe modify them slightly, and make sure that they're linearly independent as four vectors. And then take the corresponding states. One of them is the identity and the others are known to you and each one has associated with it a rotation, a unitary map whose inverse will undo this setup. And then you can have the agreement between Alice and Bob about a little code that tells Bob I got this one or I got this one or I got that one and then Bob knows what unitary to apply over at the other end and rotate it back and get the original state, and that's the teleportation procedure. But you see from this point of view, you could almost have invented it yourself. I was looking for a way to tell myself that I could have invented this thing because it was such a clever looking thing, and it turned out that the topology was a way for me to understand it. Now that of

50:00 course is not saying that topology is fundamental to the teleportation procedure, it's just showing how somebody who's used to topological language might think this through. And yet, on the other hand, there's something kind of interesting about the fact that lying in back of the formalism is this little wiggle, which is showing how things are connected up. And since I'm out of time, I won't speculate further, but I will point out that we can generalize this. There's a one on paper by some other fellows, I'm sorry, I didn't check my references, that points out that you can you can teleport entire transformations and indeed you can teleport entire transformations because this whole procedure was about transporting a transformation right a matrix transformation so if we wanted to if we if our aim was to have a higher dimensional vector and to apply a unitary transformation to it that we had selected then we could also play the same game look for the successful situation that will apply the unitary basis and get the other rotations that will do the compensations and thereby create essentially a teleportation quantum computer or process maker which will via teleportation cause certain unitary transformations to be applied to vectors. So you can make quantum computers entirely out out of measurements and certain classical bit transformations like that, if you want to. And the little topological wiggle informs that. Now, there's more that one could try to do, and we intend to. You see, there are other procedures which have patterns that are topological, like this is the procedure for what people call, this is the picture behind what people call entanglement swapping. measurements, and you can make up all sorts of complicated, interlocked preparations and measurements, and then analyze them quite quickly by thinking this way. So, I'll stop there. A couple questions. The main thing is crypto-transmitted security, because in a sense, what you're really interested

52:30 keeping secret is the thing you teleported. That's right. It's very important that we're not measuring that thing we're teleporting, right? That's why you have to go through all these machinations, because as soon as we measure it, it goes away. And I want to keep it as a quantum state and maybe do another transformation on it later, or take it somewhere else, move it around. So that's why that's the reason for this. And people who are thinking about abstract quantum information design, to regard teleportation as a fundamental element because it's going to be very hard to know how to get a quantum state as quantum state from here to there in any other way than by teleporting it and you're going to have to move them around inside this possibly distributed machine. That's why these guys get lots of grand one. Oh, okay. Thanks, Lou. I should mention that, of course, I'm really interested in seeing some really fundamental relationship between topology and entanglement, or non-locality. And these things are certainly interesting, but they're very much on the surface. Has anybody made any of this yet? Has I implemented it? How much did it cost? As a machine. Yeah, people have, but just with very few cubits. I mean, the problem is the scaling up is quantum computation, basically. Quantum computation is certainly a nice metaphor for talking about fundamentals of quantum theory. And it may turn into something real. I'm not in a position to. The last I heard from you, which is Newton's at least six months old, is that we built a quantum computer with four bits and showed that 15 factors into 3 times 5. The Shor's algorithm is good at factors, and if we've managed to factor 15, you can do that too. Yes, this thing can do it massively quickly. Well, the thing is, you give it a big number, and then you start to notice the difference because a quantum computation gives you an exponential speed up, and that cancels out the difficulty of the factorization see the whole the whole factorization thing comes down to you produce a key an encryption key

55:00 which is the product of two very large primes this public key the public key encryption then you can publish the product and you encode it in a clever way yourself and then you know it's you you as the yes who wants to send a message send a message to Bob, she uses the public key, which is the product of these two big primes. And anybody can then send a secure message to Bob. But in order to crack the message, you have to factor this really big number into two very big primes. And that's a computationally difficult algorithm. When's the interest in cracking things? Well, if you have a sufficiently elaborate quantum computer, you can do it in the same And then once you, so you get an exponential speed up by using the quantum computation because you have all the states that sit in position. And so that allows you then a chance of cracking these very large numbers. Now the next step is quantum primes or something. But the telecorp can be used to send messages in that part. Oh, yes. And you can also tell somebody saw one of the bits. Teleport is good, because if anybody does steal the bit, it's just the message, the key rather, it's destroyed. Yeah, and you can tell if somebody's peaking, you can tell if they're peaking. It gives a very curious, very expensive communication system. Paul? As far as the speed issue, there's a whole passion idea of an analog computer that solves differential equations. You can put Schrodinger's equation on an analog computer and solve it. Is that faster, slower? The same. Sounds a lot cheaper. It's very much hotter. I used to do things like that, and they'd put me in the computer in a very small room at the top of the building, and I nearly died of it. No, the problem is the stability. The stability of the analog components means that, in practice, you can't do anything very complicated. and you can't factor large numbers it's a question of precision precision and stability as soon as the thing runs it gives you so many different answers whether it's hot or cold whether the windows open or hot and all that kind of thing because then it's very hard you have the infinite precision of the real line

57:30 at the same time you want to know exactly which point on the line you got and you're out of luck okay well let's stop it there the next speaker is who is going to tell us about further developments in her analysis of the If you want to take a three-minute break, this is a good turn to do it. Thank you. Thank you. Thank you. Thank you. I don't know how to explain. Just the practice. When you're seeing something, I'm very scared about something. Thank you. I'm not worried about that. Thank you.

1:00:00 Diane? Where is Diane? I have two papers here that are on the general theme of starting over in special learning theory. How long do you expect this to take to start? It's different Can anyone know how, Diane? To see how it might change up in the upcoming theory. Do you know how to get, oh, we have to. It might provide a different, uh, basic. Basic mechanical adjustment. Science mechanics. I just thought it might be, um... It's amazing what difference between the theory. This is a kind of thing I'll call about that. Yes, it's slightly rotated. That's it. You can slightly rotate it. And then slightly find the way. I'm going to move as quickly as I can. who heard me speak with Luke Hoffman here, believe it or not, it was six years ago. And so a lot has happened in the last six years. Dick Schaub, who I know wishes he could be here, calls me the cosmic energizer Benny, because I refuse to stop, and with no obvious means of just kind of keeping going. I want to say several things before I start. The first one is that this is going to be an inside-out presentation, and I won't explain that any further except to say that it may not...I'll just leave it at that. As most of you know, I don't speak your mathematical or your scientific languages. All I and is statements that I'm going to make are subjective.

1:02:30 Okay? Very important. I'll leave these on the board so these guys can read it coming in. The meaning, reason, and understanding that I draw are from the singular geometric matrix in which all are contained. So there's a context with a capital C. please consider this an opportunity to consider a radical new perspective of what you believe or what you know is let me see if I can get this to come up a little higher can we maybe just look into this is that possible? I have a book some books in here, one, this might get that another one yeah, alright copy of the Anthropocene got that help? well it doesn't get it all on there it says it's an invitation to engage your intellect and imagination because the unity of intuition and cognition is what leads to dialogue from my point of view, okay? So, for those of you who don't know, what I'm going to show you started 17 very long years ago, when I had what's classically called a near-death experience. I've been clinically dead three times. The most recent time was in March of 2001, at which time it's never a religious experience. for me, it's always been like going to Geometry University. And so I've been able to, thanks to Luke Hoffman and Bob Gray, map what it is that I've been seeing. Now I want to give you a very look at what I see starting from the very beginning. Imagine that there's a world. And the world blows itself up. Okay, there's got to be a beginning point. And out of that world blowing it up, two fundamental things, aspects, particles, whatever you want to call it, blow out of it. One is rolling one way, one is going the other way. I would like to have you consider that one is water, what I consider to be salt water.

1:05:00 The other is what I consider to be a diamond. They're geometric structures. You'll see them in much more detail in a few moments. And then what you have is absolute nothingness. And that's why I call this something, S-U-M, something from not thing, which is a play on the word not, the trafoil. So, all of a sudden, out of the nothingness, you see a point in the line. They're fairly far away. As they come closer to you, all of a sudden, your perspective changes, over a point. If you change your perspective again, if you are the observer and observing, all of a sudden you see a line under the point. Do you mean that the line turns into a point and the point turns into a line? No. You, as the observer, are changing your perspective. It's like if I point my finger to you, you're seeing a point over a line. Okay? But if Peter looks at my finger, he sees a point under a line. That you see a point over a line. That's what I meant. The finger looks like a point. That's right. Exactly. But again, if we change our perspective, if someone else is looking from the bottom of the floor, from the ceiling, they see two lines over a line. But in fact, what I want you to consider is that we're so far away that we can't see what they are and they're actually spiraling geometries. And I'm not going to go into the details of those spiraling geometries except to say that one is significantly larger than the other. And the water, which is, I would posit as the gold, is much larger than the red, which I believe is a diamond. Now, now we're going to watch how these two come together and interact. And what they're going to do is form embedded spheroids. What you just saw was the water closing at the south. The diamond, you can see, is inside and is still forming and is about to move into the center. So you're saying it would look like a wiggle from the distance turns into a polyhedra? It turns into polyhedra.

1:07:30 It's snakes. It's polyhedral snakes. it becomes spheroidal geometry, it becomes sphere-like. We've demonstrated this, we can prove this, and I'm just showing you the fast way through this. Now, earlier today, we can demonstrate it, we can unfold it into a planar geometry and fold it up to get this... Or there's a natural way to start? There's a natural way, yes, yes, we've got it. Now, this morning, Dirk was presenting his membrane theory, and I'd like to ask you to consider this to be a membrane, because it's like a bubble. Okay? And where Dirk was having the masculine penetrating, he was using an almost tantric example of a penetration. What I'm showing is on the inside. And here what we have is the feminine, not gender but a principle and I'm going to show you why it's feminine is because it's breathing it's taking energy in it's utilizing the energy and it's putting it out this is a masculine principle as you're going to see and you're going to see why so now that we have and I want to show you with this, this is 120 faces, this is 144 faces. Both are unknown. Both are unknown polyhedrons. It took Bob Gray three months to write the computer software just to do the 144 in the center. Can I ask you to hold your questions so we can get through this? Because it's really, I'm moving fast and I know and I'll be here the rest of the week Here's the 144. Again, it's unknown. When the 144 comes together in the middle of that structure, the hands are a perfect model for it. There's three edges that have to come together in that geometry. If you watch my hands, it goes one, two, three. Where does the energy go? because it's like a diamond, and it's a spark, a spark, and a spark. It rolls in. It implodes. It's an imploding energy, and it goes into the center, but it can't contain it. And when Dirk said this morning, like little volcanoes,

1:10:00 it is going to erupt that energy out into the system. And here we see how it sends out 48, because that's how many vertices there are in that 144-faceted polyhedron. It sends out 48 rays of light. They're clustered very unusually. Bob Gray, when I showed him, I said, you've got to look at this 4 and 2. He said, well, that's not logical. And I said, well, that's the way it is. I want you to know it's the only way it works in this geometry. Once that light escapes, the 144 is done. It's like the masculine principle. It's like Tantra. You have the orgasm. The man just sits back and has a cigarette, just spins, and waits for the feminine to do her thing. Okay? Too often. Okay? So the light goes. So we have another polyhedra here. In March, we have just mapped what that polyhedra is. I've always called it the focusing sphere, because it's going to now show how that energy, how what I believe is light, escapes that focusing sphere. And I mapped for Bob, I said it clusters, and it's going to dock into that mother structure, that bubble, that 120-fasted bubble, There's going to be a band, an equator that's free. Changing your perspective. Changing your perspective, the same thing. Now, let's look and see if it fits inside the 120. We take all facets off that 120-phase polyhedron, and we have an icosahedron, which we know water clusters and icosahedral clusters. Now let's put the facets back on, exact perfect fit. Now, whoops, got the wrong button. But the light can't stay at this point. You can see what's happening. You've got six rays of light. They're clustering and they're going to become one. So the one where those 48 clusters are going to turn into eight bulbs of light spinning around inside that context of the 120.

1:12:30 When that light comes out, it says, it hits water. If it's something warm and it hits cold water or something that's cooler, it says, what the hell? I don't want to be here. I'm going back. I want to go back to that, which is in here, the masculine principle that's spinning where it first came from. But guess what? it hits that focusing sphere, and it ricochets. And so the light ricochets around, and here we've got a curve happening. This is the first curvature that's happening. And so what it does is the light ricochets around, hits the 12 vertices of the icosahedron, and finally, after doing this trajectory, all eight of them, They come back to themselves, and they form. Remember we had a line beside the point and a point beside the line? Now we've got an inside-out. One's on the inside and one's on the outside. And if we turn it 90 degrees, you see a vasocopisis. When I saw this, Bob then, and we put it into the straight line, up. At 19.5 degrees, it opens from this, which is flat, to this, which is beginning to define the sphere, which by the way, that's a trifoil knot. Oops, sorry. And we open it up to this, and some of you will remember this from the cover of my book, which came out in 1997, called a pattern. It's opened at 19.5 degrees, which is the tetrahedral angle. Now, we put it back inside of the sphere, inside of the 120 polyhedron, and we cut it in half. We said, let's look at what's going on in the middle, and we cut it in half to look inside. How many of you know this picture? The hourglass nebula. You're looking at a black hole. Well, we thought it was pretty interesting that if you superimpose them, one on top of the other, they're virtually identical. The geometry. Now, when James drew his drawing

1:15:00 of the planets he drew, he put up this map the other day, and he showed a flat perspective of the Earth with the Milky Way. I thought, well, that's very interesting it's a repeating pattern of that yin-yang going on if you take it in into that realm. So here again is the 120. This 120 polyhedron is breathing, and I'm going to show you movies in a minute. It's expanding and it's contracting, it's convex, it's concave. Through 10 of the vertices, which are pentagonal, energy is emitted when it breathes. So 10 come out. There's a stabilizing. That's a five-fold. It stabilizes over a four-fold. At this point, it bursts a new bubble, and the remnant falls back in to regenerate the whole system again and again. Here's where the energy comes out. I can show you movies that proves that this actually happens in the dynamics of the geometry. Of course, Bob Gray said, no way is that going to happen. shocked when it did. It stabilizes over the fourth hole, and it re-enters over the three-fold. So, in 1998, this is what Lou said. This is one of the many things that Lou has said along the years. I'll just let you know. I don't think so. Okay. Now, let me show you what I think is very Okay, this is the 120 colohedra, I'm sorry, it's a little kind of skewed there, let me see if I can move this, it might help a little bit, this is a picture of the 120, if we change little bit different. I don't think I can advance this. Oh, I can do that. Good. This is the structure of that 120. It's a very unusual triangle. When I first drew it for Bob, he said, that's not going to make anything. And I said, wonderful. Prove it doesn't make anything and I'll go back and try to

1:17:30 be normal. Okay. And we actually have this down to, I think, 20 decimal points. The buckyball is in So that's some of the detail about the one that is the triangle. That is the triangle. 120 of these triangles. Are they all identical? Yes, they're all identical. It's basically 79 degrees, 37 degrees, and 64 degrees. Now I want to show you the dynamics of what this this geometry is doing. How this This is the breathing motion of that structure. And as it's going through the dynamics, all of those triangles are spinning. They're all rotating as well as the whole system is spinning. and the 144 and the 120 are spinning in opposite directions which you will see in a few minutes but this is the dance that 120 is doing between an icosahedron and a dodecahedron it's actually in the phase shift it shifts between 120 phases and as it's moving towards the dodecahedron it adds an additional 60 phases which is what I saw when I died in March of 2001 I came back and I said, Bob, it's not just a 120 When it spins, it adds 60 new faces. It goes up an octave and it goes to 180. And he said, are you sure? And I said yes. And so this is the movie that he did to prove that that is actually what happens in the spinning and the rotational spin of every single triangle. And as you can see, it's obvious, it becomes obvious how the system is opening and closing. and i appreciate that i'm moving very fast on this let me just take you through some of the slides

1:20:00 this is what you see at one of the perspectives in that movie it's very interesting that it's a symbol for nuclear energy right there in the middle of it here we're looking at the system Now we're going to watch it slowly opening. Bob's got the arrows giving you the directionality. Now you're looking into the system, and it's closing back up. That's about mom. Remember the focusing sphere. we have just discovered what that middle sphere is and the discovery of what that is the geometry and how I was able to see it and Bob was finally able to match it is a cluster of icosahedrons it's like water clusters and icosahedral clusters so here we're seeing the 144 in the center which I'll show you in another picture in a moment and this is actually a cluster of five icosahedrons so that's modeling of the middle sphere, and in a few minutes you'll see a movie of that. And then the 144, again, I'm sorry this picture's a little bit small, again, it's an unknown structure. It took Bob a long time to map this. Here we're looking just at different perspectives of the same structure, the same thing. Here we're looking at what fits into that structure. We see that it's how spheres pack, closest packing to spheres. We're looking at a cube octahedron fitting exactly inside the system. And here we're looking at how this is going to fit into a cubic face centered lattice of nine and how it's an exact fit exactly into that structure. This is a small model that Bob built. It's about this big. It's a real jewel.

1:22:30 And inside of it are tetrahedrons, cubes, rhombic dodecahedron, and an octahedron. And that's all it fits in, this model. And there you see that you can see straight through the 144. for. Now I'll show you what for me is very exciting. This is a transparent model of a 120 polychaedron. And what we wanted to find out is what fits into this 120-faced polychaedron. What makes it what it is? And so we went looking for the geometric structures. So this is, of course, a tetrahedron. And the tetrahedron fits exactly into the intake energy, the vertices where energy enters into the system. So the question was, how many tetrahedrons fit into the system? So we see 2, 3, 4, 5, 6, 7, 8, 9, 10. Ten tetrahedrons embed within the system. And now let's watch them move. It's interesting that the tetrahedron is the ancient symbol for fire. Okay, and so here we have 10 tetrahedrons moving, and what you may notice is that there are pairs. One spins clockwise, one spins counterclockwise, so there's five pairs. Two of the pairs, it's actually, we forgot to animate one pair here, but one of the pairs is static. it sits still it doesn't rotate as well because the others are spinning at 44.4 degrees around the faces of this of the single one that's a fixed fixed in the

1:25:00 system and what's interesting for me as well I will come out of this and just you one picture. What you notice here is how all of the tetrahedrons combined form an embedded five-point star. So, next in filling out the volume of the system, coming up from the tetrahedron, we have the cube. And the cube is the ancient symbol for Earth. And so, how many cubes fit into the system and where do they dock into the system? Will they also fit into an incoming energy vertex. So there's two, three, four, five. And I want to point out to you something philosophically that's very important. If you look here, you see a five-pointed star. Okay? If you look at the vertex of every one of the five-pointed stars, there's a six-pointed star. In every ancient tradition around the planet, wisdom tradition, when the five-pointed star and the six-pointed star come together again, there'll be 10,000 hearings of peace on Earth. Let's look at what those cubes are doing when they move. We were talking earlier about the Rubik's Cube. This will give you a new thought on a Rubik's Cube. So this is what the cubes are doing. And so what the system is actually doing is that this vertex here, it's blowing baby dodecahedrons. Okay? So if you freeze frame it, at any point, you don't have a five-pointed star and you don't have a six-pointed star. And that five and six, that reunion you might say, that five and six, happens at every level in this system that I'm going to show you. Now, when we went to model the icosahedron, this was what was so exciting to me.

1:27:30 This is an icosahedron inside the system, and what we've discovered is there's scaling of icosahedron in the system. that inside of the system, because of the dance between the cube and the octahedron, which we haven't seen yet, an icosahedron occurs. And so we said, how many are there? And what we discovered, two, three, four, five. and there's actually there's coincidence there and there's actually 10 but we don't count them because they're hidden sorry about that and so this is the icosahedron moving in the center and defining that focusing sphere and it's very much like a pulse it looks very much like tides moving up and down in the system. And on the original storyboard that we did six years ago in when Bob was working just to map the geometry, my knowing, and again, this is a subjective knowing, is that this is very much related somehow to gravity. And again, I don't know your language, but I feel this very strongly. Okay? So let's show you the systems moving together. It's a little hard to tell, but they're spinning one direction and one the opposite direction. So there's three embedded spheres. And I will show you in a few minutes what they're doing as they do their very complex dance. Next comes the octahedron. which is, the octahedron is the ancient symbol for air. It's an eight-sided. There's two, three, four, five. And the minute we add the fifth one, again, we get the pentagon, the pentagonal form.

1:30:00 And interestingly enough, it's this geometry that I told Bob, I said, this is what's stabilizing the system. How many of you made little origami cups that did this when you were kids? Did the boyettes ever do that stuff? You know? Everybody's saying yes. I'm going, no. Well, it's doing that exact same thing because this one stationary one keeps any energy. It stabilizes somehow. It's the throughput in the system. And so it's exactly that same motion that it's doing. So, so far, we've only looked at the regular polyhedron, the platonic solids, and here we have the first irregular, semi-regular platonic structure. It's called the rhombic dodecahedron. It has a rhombic face, 12 sides, and where the others all have a single task inside that bubble inside the 120 polyhedron the tetrahedron is energy in the cube is energy in the water is beginning to eicosahedron is beginning to move energy out the octahedron is stabilizing this one is multicasting it's taking energy in it's dealing with energy and you might say and it's It also has a stabilizing effect, and there are two, three, four, five. There are five of them inside the system. And again, I want to point out the five-fold symmetry in the system. And we'll look at how they move. I think you can all appreciate why I appreciate Bob Gray, who's done all this modeling. And again, we see how all five of those are working together. And again, you have the same five- and six-ness. If you notice, you'll see a six-pointed star forming here in the corners while the five-pointed star is right above it.

1:32:30 The dodecahedron was the ancient symbol for the universe. In the Pythagorean school, the punishment for saying dodecahedron out loud was death. right? It was the big secret. It was the ancient secret. There's only one that fits into that 120 collohedron, and it docks in at, this is a vertex, an input vertex, where energy is going into the system, so there's only one. And then scaling, as I said before, we have this one large icosahedron in the system. I'm sorry that's not very good graphic, but we'll look at how it's moving and you can see how it seems to change the direction that's how we just rotate things that just rotate and steady yeah it's only going one direction yes it's rotating counterclockwise a little skip saw and told him to put in. The little skip that it does. No, that's just there. That's just there. That's a necessary aspect of That is what's happening. Of this rotation somehow. Of the rotation somehow. Okay, and there's only one at that level. And then last but not least is the other semi-regular platonic solid which is called the rhombic triakonzo-hedron. It has 30 faces. It has no known meaning in the ancient traditions that we know of. We call it gold because this rhombic face is a golden ratio. This is 1, and this is 0.618. And so in our system, we call it, we say it's gold. Now, after we did all of this, I said to Bob, okay, now we have to put it into motion. One, two, now I want you to watch, boom.

1:35:00 Watch for the moment of clarity. If you look away, you will miss it. I'm going to minimize this window and I'm going to put up another film right next to it let me have it put it in still motion for you so you can get a little more of an appreciation for what it's doing That's the entire system moving as one. But now what is it doing when it's doing it? Let me show you what it's doing. No doubt you all recognize this. Yes? The Taurus. All of that geometry is generating a knot. If you take the vertex of a tetrahedron and you put it into that motion and you put it into that context, when it's moving, the tetrahedron, the vertex is handing off to the corners, to the vertices of a cube. There's a handoff. It's like a relay race. And then the cube hands off to the octahedron. And it's just all being handed off. And what it's doing is it's tying a knot. So it's like on the outside, out here, we think of geometry, especially if you call it a platonic solid. We think of it as something. Well, this is nothing. This is a knot thing. These are lines of force moving inside of a context, a container, the 120. And as it's moving, it's generating this knot. But it's actually not generating a single force, it's generating a double force. Now let me show you the knots.

1:37:30 One knot, and again, it's hard to tell. The next film that I put up, it'll be a little easier to tell. The knots, and I have pictures of the geometry. If any of you want to see the details on how the knot happens, I have it. But the knots, there are two knots. One is spinning clockwise, one is spinning counterclockwise. One is a right-handed knot, and one is a left-handed knot. They're not the same. They don't interfere with each other. They never intersect. except they dock, they share polarity, nor have sound, in the system. Let me show you, let's look down on the system and see what we see. And so what you see here is again, this figure eight, a yin-yang, just like what James showed in his picture of the universe but you have not just one but you have two. And again this icosahedron in the center is what's keeping the knots from collapsing as well as the 144 in the center. Lou's done quite a bit of work on this particular knot. Have you read Jana Ledin's How the Universe Got Its Spot? Just finished it. Just finished it. Oh, right. In anticipation of this meeting. I don't need to say any more then. It's a fabulous book, and I loved it when I got to the page where her kind of picture of the universe is very much a spherical dodecahedron. So here we're going to look at, some of you may recognize this from my book. This was the cover of my book. At that time, I didn't have the details of the geometry. I was advised not to put the whole, not everything on the cover, so we left the 120 off the cover. I had a child, six months after the book came out, a 10-year-old child came up and snarled at me, but Lynn Clare, where's the bubble? She knew there was something missing on the outside. So here we're just changing perspective. Again, here's a little clearer shot of that knot,

1:40:00 of that yin-yang perspective. This was actually the cover of my book. I came back from this near-death experience saying, I saw the strand of the tapestry that wove all creation. I said, it's space, it's music. To me, this is resonance. It's a wave that folds back on itself. knot, it's very, can I write on this? It's a waveform that folds back on itself and creates this knot. And if you turn that knot 90 degrees, this is what you get. That's just a 90 degree rotation of that. It's exactly the same. These are not different pictures. Nothing is changing except your point of view. Right, so you say you've seen some of these patterns in a near-death experience. I would point out, although I believe is true, that people who are subject a long time to sensory deprivation see very similar things, and it is thought that ancient cave painters disappear into the cave, into the blackness, for a few days at a time, until they were seeing things like this, and then they painted them. Well, the $64,000 question is, I mean, I've been doing this for 17 years, because I somehow think it's important. I think it's very important. It's like, Bob Gray says, well, I hope we ask a lot of questions. Maybe I should put some of those questions up, too. Well, actually, let me leave this up there and see if I can articulate them. First of all, the system, as you're standing, is pure gold. The golden ratio is very, very important to this. Inside the system, as we know, outside the system, the golden ratio is 1.618, what is it, 0.833 on forever. It has no, it's irrational, right? Inside the system, we have what I believe are two rational golden ratios that tie the knot.

1:42:30 And the first one is 1.618-04629, repeating. And the second one is 1.6204166, repeating the same. and that the knot is the movement between the two. And the knot is a very long number. If you divide this number by 90, you get this number. If you divide this number by 9, you get this number. Repeating. You don't really mean divide by 90, though. Yes, I do. If you take this number, you divide it by 90. It's a point. There's a pointer. And you divide this number by 9. But what I believe this number is all about is this. That this is related to the 120. This is related to the 144. And that this 27-digit number on this repeating pattern. This is a closed system inside of an open breathing system. So you have what I would call a continuum. It had a beginning, but it has no beginning or end. Inside of the system. And every time it breathes, because if you put A here, and you go from A to B to C, which is how this is moving, if I put it in my hand, that's what it's doing. If you go from here to here to here, it's energy coming out, stabilizing and going over. That's this motion of this knot. It divides into 11. It's actually 3 cubed. It's 27. But this number is 33 digits long. That's very interesting. So it's 11 times 3. there are 33 platonic solids, regular and semi-regular, that fit in the system. There's five regular and two semi-regular.

1:45:00 All of the Archimedeans are subsets. They're truncations of those seven. So that means that all three-dimensional structures, Platonic and Archimedes are subsets of that 120. Are you saying this is the source of some time? I think so. In fact, you would say it was the source of all. I would take it as the source of all. I would posit that the whole system from the 144 is a radiation powder that we might say is the Big Bend. Okay, but there's no more thing I want to say about this 33 business. Because 33 is an ancient number that has meaning relative to light. I can show you this 120-faced polyhedron has three axes. It has 33 two-dimensional layers. You can shift your poles. So the first one has seven. The second one has 11 floors, and the last one has 15 floors. And so there's 33 two-dimensional layers in the system, all intersected by that same knot, those same two knots. There's 33 digits in this system. If this is time... So I would position this as maybe an inside look before matter, how and how the knot breathes, like I said earlier, in three breaths, it goes from one to more than five million because it self-replicates. But every one has a different perspective relative to the original one. Okay? It's different. There we are. We're off into the universe. Okay? And so it's very much a hologramic model. And it's holistic. And we can lay all kinds of psychology, social architecture, onto this. So, that's what I have to say. Most of your diagrams come up on the screen looking like an elliptical.

1:47:30 They're spherical. It's all spherical. What appears to be elliptical, originally we thought it was elliptical, that it pushed out convex concave. The earlier pictures looked that way because they were. But what we discovered was it was all because of spin. Everything was happening because of the spin of that triangle, that it was opening and closing because of the different edge lengths. This is an artifact. If you look on the screen, it's very much spherical. So if you try to draw a circle on that, it will come out from here. If I draw a circle here, it's a perfect circle. And by the time it gets to that, it's not it. Right. Well, I've seen, of course, most crystals are six-sided, so they're cubic. they have that hexagonal, that six-sided face. Some crystals have four sides. And so, yes, there's definitely, someone showed me a five-sided crystal recently. So this is definitely a crystalline type of lattice. The buckyball is what, 60 faces? 16, yeah. Say, what's that? The buckyball is a subset of this. We can show you why the jitterbug is jittering. we can show you the dynamics of why it's doing what it's doing but I mean it's the funky ball is the highest form of or the most complex form of carbon right but when a quantum holographic system collapses first of all the transverse energy is then released again as longitudinal Well, I don't know, I can't even respond to that, but what I can say is... Well, you said it was holographic. Yes, absolutely, okay. But what I can say is, it's really... I'm just wondering if there's something I've had to do. I think we'll be the answer, Jess, but can I say one thing? Charlie, can I say one thing? And it's really important to you know that I have zero math, geometry, science background at all. None. Zero.

1:50:00 You seem to know quite a lot now. And I've learned a lot. Bob Gray has taught me a lot here. I'm losing look and tell you I've actually learned matrix calculus with Pierre Cartier. No, I had to. He made me. And so, but it's, I've learned, I still, I know what I don't know. And that's why I can say to Peter, I don't know, but can you help me? Because this is an invitation to a dialogue. I believe it is. The golden ratio would be pulled dynamically because it's the principal singularity of a chaotic system. Every structure that is not golden ratio, the minute you put it into this system, guess what? It turns gold. It takes on a golden relationship to the clusters of its siblings, identical siblings. books and to the mother. That's called alchemy. Taking something that's not gold and turning a gold, the golden ratio. Oh, the data on the triangle. Yes, yes, yes, yes. Let me... Oh, this is nothing. I've got thousands of people. Black red squares to dug nuts. Quick at night, he closes his eyes and he watches this on the back of his eyes. Every time he talks, he says, can I take a nap? Can I please take a nap now? I've got to find I have to find where that file is I've got it in here somewhere oh I know where it is, sorry first of all as far as I'm going to show you this this is Bob Gray saying that this is a remarkable set of coordinates

1:52:30 because remember that the 10 tetra 5 cubes, 5 rhombic, all share their vertices. This means that all of their vertex coordinates are a subset of the coordinates given above. What is surprising is that the golden ratio provides a way to organize so many polyhedra within a single polyhedron. The 120 that Linclair experienced and continues to experience. So here's, just so you can see, what the golden ratio, how the golden ratio is in all of these, in the 62 coordinates that are there. So there's this set of coordinates. And here are the angles. That triangle. It's taken out to what? Uh-huh. Eight decimal points? We've been doing some modeling of this in physical models, We've had some very frustrated model makers because they made it. They cut wood, beautiful white birch, to make this big three-dimensional model. And they cut it to five, or ten, they cut it to five, it was to five decimal points of accuracy. And guess what? It didn't work. It didn't fit together. It didn't fit together. They couldn't put it together. And so they had to take it out. They got, Bob gave them decimal points all the way up to 20 decimal points. and when they finally took it to 15 they were able to put it together but they had to use wood filler and glue so they couldn't make it because that's how precise it is and again this is coming from Bob Gray made me draw that triangle in October of 1998 and he said fax it to me and he faxed me back three copies and he says is it A, B or C and I said Bob it's the one I sent you don't test me. The universe doesn't test what you do. He called me on the telephone. He said, if it doesn't work I'm not going to fudge the numbers. And I said I wouldn't want you to. It proves I'm wrong and I'll shut up. How's the right place? The quantum universe is quantum counter. What is that P in there? The golden ratio of P is always related to the golden ratio. So it's 1 minus root 5 or something like that.

1:55:00 Someone else is going to have to read it because I can't tell you what it is. Is the P 1.618 or 1.618? 1.618. The other thing about the system is that Marion, how it forms in the geometry is it forms as two twisting one and a half spindulises that come back and fold back on itself. Now, if the golden ratio was rational, irrational, inside the system, guess what? They wouldn't talk. But, because of this, they do. They link up. Because the Y is that knot has structure. That knot is a Y structure. Yes, well, I'm sure when you say you don't know any of these science, I mean, one of the things about explaining with quantum mechanics is why atoms and things hold together and why they produce these spectral vines when they shift their energy configurations. Basically, you're fitting together numbers. I mean, that's what you're doing. idea was that you fitted waves into various configurations. I don't know whether that could be entirely carried through geometrically but certainly that's what one is doing in wave mechanics. One is fitting a wave to various geometrical constraints. but people like me don't try and visualize it to try and do the math there is a there is a hand whose eigenvalues are the complex of the imaginary the imaginary roots of the Remanzetta now that particular that particular hand

1:57:30 I just want to show you one other thing that's real important to me. And one thing, Sasha Shogun, a chemist, a famous chemist, made me promise that any time I talked about Marion point this out, that it has that it has that it's asymmetrical with perfect symmetry that it has chirality okay? but with perfect symmetry but now I want to put something on here that's really important relative to these two knots, because one is right handed and one is left handed, I want to put the notion of directionality in here, because there's a flow it is moving here, to here, to here, to here, to here, to here. And it would appear that there are one, two, three, four, five, six, seven intersections crossing over, when in fact there are not. This one, this one, and this one are not real. They're artifacts of two dimensions, dried in two dimensions. Have you got a photographic memory? I support you. That's my accident. You were writing those numbers down. Well, what's really interesting is that's not always been true. In my accident, I lost 25% of my brain. or talk normally again. You now know I don't talk normally. You know, I had a stroke. I had to learn to walk and talk again. They said I never would. And I said, but you don't understand. I saw the strand of the tapestry. I have to walk and talk again. And it's not an option. My EEG, if you look at my EEG today, it says abnormal, abnormal, abnormal. The doctors say, where's your medication? When was your last seizure? I mean, the other thing that's interesting about it is the center of it is to be discrete in the geometric sense, and the other, the rest of it is continuous, so perhaps it's the interface between, the holographic interface between the discrete and the continuous. That's what I keep saying. There's no beginning and no end, but what I'm always interested in is these transition points. Are you hungry, Leigh-Claire? Who, me? Yeah. No, I can get a bite to eat.

2:00:00 I would really welcome feedback. Yes, Dr. Crosby. It's like a question. How many of those were probably going to be? At the end, when it's taking energy in, 120. When it's putting energy out, 108. It opens up. It opens and it closes. It goes from convex to concave. And when it opens, the opening is a replica of that same triangle. It's changing geometry. It's changing from... No, it's a convex carcade. It's open. That's right, you add 60 more triangles when it opens. No, it's not 144. It's 120 to 180. The 144 is that inner core structure. That's 140. That stays the same. That never does anything except spin. Yes, exactly. If you're still sitting here, I'll show you how to hold here. I'll show you. Here's where the vortex happens. From one perspective, looking at the pentagon, the pentagonal face. As the whole system is moving, I described it to Bob. I said, if you look at where the energy is coming out, camera lens opening and closing and he said well that can't be that can't work and it works it's exactly what it's doing it's a vortex does it generate energy yes are you saying in the words of the uh the yin yang that it's the internal the eternal pen the creation the creative act which can never be exhausted i believe that the 120 that turns into the 180 is exactly that and that inside of that, which is a breathing dynamic vacuum that's filled with infinite potential, is there.

2:02:30 I believe that this 120 is what fills the sea, the ocean between you and me, right now. That it's what's spinning inside of ourselves, and it's what's spinning here. That that hidden water, because I do believe it's hidden water, because I think we're looking, you want to talk about energy, I think we're talking about somaluminescence, how sound, how frequencies, birth light. And that when you get it right, you're also going to get a drinking fountain because you're going to compress the gravity and there's a medium, there's a water medium between that 144, which you don't want to rupture it to the point where you say, goodbye, name to God. And we wave goodbye and we start it all over again. But that we compress it What will happen, and this is, I haven't been wrong in mapping the geometry. So Bob Gray says, he always says, don't bet against her. So there's a physics called sonoluminescence. Sound birthing light. And I think the problem with sonoluminescence is they're thinking, fission, trying to get fusion. They're blasting sound waves into each other. I agree, that's my question. The other triangle. Look, it just so happens that I made it, for instance, that I made a physical model of this, what you're showing here, without the green bits, way back in 1996. Now, I made it from, I think you'd like to try it. At the end of the day, I had 20 of them. And I got one from the end of the day. I can't pull it down like this. When it counter-rotated and it went big, it started with like a three-place. Well, a truncate isn't a lot of, where a hole is. This doesn't truncate. what this does not truncate well in a okay this this is made of octahedrons inside inside those

2:05:00 are five octahedrons spinning inside of a dodecahedron well they're three-dimensional They're three-dimensional. Yeah, those are three-dimensional entities. I'm sorry, I've got this. Yeah, no, those are five octahedrons spinning inside of a dodeck. What I'm going to do is get back to the other thing. I'm going to say the lead system. We would see all of that actually have faith in the back. Right, the 120. Well, that's what I meant. That's the 120. Okay. It contains all of it. All of that is actually generated, the 120. That's a 120. Is every page the same? Right. And it's this magic triangle with unequal size. And they did it. Right. Okay. What about the 144? The 144 was an unknown. I've got all that detail. I can pull that up for you. I can definitely... I mean, that's going to be so important.