Okay, good afternoon. So if you remember, at the end of the last lecture, we introduced this new concept called sensitive dependence on initial condition. So let me remind you, this is a general notion for maps of metric spaces. So F exhibits sensitive conditions. If there exists epsilon, let me make sure I write it correctly here, such that for all x in x and for all delta greater than 0, there exists y in x with distance between x and y is delta and n greater than 0, such distance between n and n. So why is this an interesting notion? It's to do with predictability. I will make now a little, just a couple of minutes, a little parenthesis, a little bit philosophical parenthesis about dynamical systems and the history of dynamical systems. So dynamical system in some sense is motivated by applications by the study of systems that evolve the change in time. These systems can be mechanical systems where you want to study the effect of something moving under some forces. That's a dynamical system because it's something changing in time. Or it can be just trying to model a population, the different generations where there is some dynamics that jumps from one size of population to another. There's a million ways in which a dynamical system can be used as a model to model some natural phenomena. Now it turns out that for some natural phenomena, it's much more successful than others if you think of this dynamical system as existing to be able to predict what happens. So all of you know that one of the real success stories of mathematics and physics in some sense is the application, the study of astronomy. So the study of the motion of the planets is one of the first fields in human civilizations where people really that motivated a lot of mathematics and a lot of study. And amazingly, accurate predictions about astronomical phenomena have been made even hundreds, even thousands of years ago. It's possible to predict exactly when the moon, there will be certain eclipses or when certain planets will be in certain positions. And this is by modeling the system, the planetary system, with some equations that depend on your knowledge of the physics of the situation of Newtonian physics. But there's other phenomena which are much more difficult to predict. For example, the most typical one is the weather. You all know the weather is also a physical system. It obeys physical laws. The development of the weather depends on what the temperature is and what the pressure is and the humidity. And all these things and the geographical landscape determine how the weather develops. And physicists know basically all this stuff. And a little bit we can predict the weather, right? So a little bit usually you can say tomorrow probably will be like this or will be like that. But to say next week will be like this or to say that next month on a certain day at a certain hour, the weather will be in a certain, will be is impossible. So what is the difference between the planets as a physical system and the weather as a physical system? And why is it so, why is one so predictable and why is another so unpredictable? It's not because for a long time people thought, well, the weather is just too complicated. The weather has too many factors, too many variables, too many degrees of freedom. This in part is true, but it's not the essential reason for which the weather is unpredictable. There are many simple systems which are quite unpredictable. For example, if I take a coin and I toss a coin, this is often thought of as the most typical random system. You toss a coin. But of course there's absolutely nothing random about tossing a coin. You have an object that has a certain mass, a certain shape. You apply a force to it. It flips a certain amount of time. The number of times it flips depends exactly on how you apply the force and where you apply the force. And the outcome depends exactly. It's completely Newtonian classical mechanics that determines the evolution of the coin. So this is not a random system. This is another system that we can model. We understand the physics of it. We can define a dynamical system to model it. But the fact that we can model it does not make it so that we can predict the system, the evolution of the system. And the weather is actually very similar to the effect of the coin. And what is the reason then that the coin is so unpredictable? What would you say? Why is tossing the coin unpredictable? Why? Why can I not tell if it's going to be heads or tails and I cheat and I always win when I toss the coin with my friends? Even though we know, I think the essential problem is that the mechanism by which we toss the coin is not sufficiently precise to allow us to control that force with sufficient precision. So because we need to kind of flip it in this way, it's just too delicate. And if you think that if you flip, it flips the coin before it falls, it flips maybe 10 times. If you apply 10% more or less force, it will flip nine times or 11 times instead of 10 times. And so it will come out tails instead of heads and so on. So the margin of precision, if you were really flipping something big that only flipped one or two times, then you would be able to control the force to a sufficient level of precision because the difference in force between flipping just once or just twice would be sufficient. But because it's flipping many times, just a small change in the force will make it flip one more time. And we really cannot control the number of times, okay? So how is this related to this? If you think of this as dynamical system, I'm not going to try to give a precise model of this coin flipping, but just as an idea, if you think of dynamical system that models the physical of the system, then the evolution of the system depends on the initial condition. And if you assume that everything else is fixed the position of the coin, the size of the coin and so on, then the only initial condition has to do with how much force you apply to it. That's the initial condition that determines how the system is going to evolve, right? So the fact that if we just apply a very small difference in initial condition, we get a completely different outcome, this is intuitively sensitive dependence on initial condition, okay? When you think of it in practical terms. So sensitive dependence initial condition means that even though you take two points that are really close together, arbitrarily close together, right? For all delta, at some point, they will be very different. So I'm using this epsilon that can be a little bit misleading maybe because generally epsilon is used to think of as a small number, okay? But the point is that epsilon is fixed. So in relation to delta, we think of epsilon as a large difference. If two situations are epsilon different, it could be like heads or tails, sunny weather or cold snowstorm, yeah? It could be just two completely different things. And we are looking at two initial conditions which are very close to each other. And nevertheless, if we wait long enough, the outcomes of the initial conditions are completely different because they're separated. So if this is related to the modeling of a physical system and the real system, the real initial condition is x, but the measurement you make of the initial condition contains a small mistake, okay? And really what you measure is the initial condition y. And then you run the model, your prediction based on your measurement y will give you one thing after a certain amount of time. But the reality will give you something that is far apart from that, right? So you're trying to predict the weather. Our measurements are not perfectly accurate and that problem with the measurement means that we cannot predict because the weather in one week's time really depends on level of precision of the measurement which we just cannot achieve, right? This is the essence of this. And that is why this has become so in the, this is something that actually was known by mathematicians and physicists even two, 300 years ago. It was understood that even though if you have a differential equation, one initial condition has a uniquely determined evolution, nearby initial conditions perhaps might have different evolutions. But it did not really become really understood how dramatic and how important it is until people started doing simulation with computers, essentially in the 1960s and the 1970s and so on. And this notion of sensitive dependence to initial conditions was coined by Lawrence, Edward Lawrence in the 1960s as anyone heard of the Lawrence Attactor. So Lawrence was studying, it was actually meteorologist. He was actually studying the weather, right? He was trying to study the weather and how to predict the weather and there were lots of mathematical models for the weather. And he said, well, I'm going to study a much, much simplified model of the weather because the models of the weather were extremely complicated. So he said, I'm going to study a very simplified model which is a system of differential equations just in three dimensions, okay? Called the Lawrence Attactor. Now it's called the Lawrence Attactor. And he said, well, if we can't understand the complicated models, at least let's try to understand the simple models first. And he had this computer, you know, the computers in the 60s were not like the computers we have today, right? So he had computer that was probably in some air conditioned room as big as this room and he had to, he did not have a monitor that would just plot the graph of this equation. He would just have to put in cards and get out numbers. And it was very difficult to study these equations. But he was trying to study them and he was looking for some periodic, fixed points or periodic solutions because as you see also, our first approach in dynamical systems is to look for fixed points and periodic points and simple behavior like that. And he could not find it, he could not find it. And he was plugging an initial condition and trying to integrate the equation and trying to find out what happened and he could never find the system. And then what happened one day, he says this in this paper that he published in the 1960s, right? So what he was getting was this list of numbers that he then had to plot into a graph and try to find some periodic solution. And one day he had this list of numbers and when he started the next day, he started the program instead of just starting from the last number and plugging it into the computer and continuing the program, he took some number up here and he plugged this into the computer and he waited for the computer for the program to evolve and after a very short time the numbers were very different from the ones he had the previous day, right? So the previous day starting from this initial condition which was, so the previous day he started with some initial condition x0 and then he had got up to some xn, you know, a whole bunch of points of the system. And then the next day he come in and instead of plugging xn into the computer and continuing he plugged in some intermediate value, xi, right? He called it yi and he plugged into the computer and very soon this y did not correspond. When he got to yn he had something completely different from xn and so he was very worried and he was didn't understand what was going on. Maybe the computer is making some mistakes which would invalidate all his studies and so on. Can anyone explain? Does anyone have some idea what was going on? So why is that? So he still put in the same initial condition. So even though you have sensitive dependence it's always true that if you have a dynamical system one point just has a well-defined trajectory, right? In general. But the point is that he did not insert exactly the same initial condition because it turns out that the computer memory was working with six decimal places. But the computer was printing out these numbers only to three decimal places, right? So when he looked at this number and then the next day he plugged this number back into the computer he plugged in a number with three decimal places which therefore was different from the number that the computer was using in here which had six decimal places. And so the two numbers were indeed different although he thought they were the same number. And because people did not, they knew. He knew that the computer memory was working with six decimal places. But he just did not really realize that even fourth decimal place which is when you're thinking really of practical measurements, it's like measuring things in meters and worrying about the millimeters or fourth decimal place is a small number. And he took him a while to realize that even that small, very small difference in the fourth decimal place could very quickly make the two trajectories be completely different. So this is just a little bit of the history of the modern theory of dynamical system. This is the beginning in some sense of the theory of chaotic dynamical system. So dynamical systems which exhibit sensitive dependence on initial conditions. Which is a class of dynamical systems. And what I want to show you is for example, the shift map and the maps we've just been studying they exhibit sensitive dependence on initial conditions. So sensitive dependence on initial conditions is nothing mysterious. It's something that occurs very much. But I just gave you this little parenthesis to make it a little bit more, less just technical and dry. This notion of sensitive dependence on initial conditions. Okay, so let's check that the shift map satisfies this lemma. So shift map sigma from sigma L plus sigma L plus conditions with epsilon equal to one. So the proof is trivial, right? How do you prove this? So the result is quite remarkable, right? It's a very simple map. But what we're saying is that arbitrarily close initial conditions eventually are far away of order one, right? So proof, yeah? Take any element, any two elements, right? Yeah, so take any two elements arbitrarily close and how do we know that they move apart to distance one? Yes, yes, if they're close, yes. So wait, let X difference from Y in sigma plus M. So we need to show that at some point some two images of this separate the distance between some images. So the distance between sigma N of X and sigma N of Y is greater than or equal to one. Sigma N of X and sigma N of Y. That's right, yes. Okay, it's the same thing actually. If you think about it a little bit, all I'm saying, so I formulated it in this way, but here we have even a slightly stronger version. So the way what we can say here is that any two points, any two distinct points after some number of iterations, the images have distance greater than or equal to one, right? Which implies this, it's slightly stronger. So this is not just for every delta there exists a Y, but in our case, we will have for any X and Y, however close at the existing N such that this is greater than or equal to one. Okay, and why is that? You were saying? Yes, exactly, exactly. So let me say this. So the fact that these are different means that they must exist. So X different from Y implies that there exists some N bigger than or equal to zero such that if X is equal to X zero, X one, X two and so on and Y is equal to Y zero, Y one, Y two and so on. X N is different from Y N. Because if they're not, they exist such an N, then they would have to be the same, if they were the same sequence, right? And therefore clearly, then the distance between sigma N of X and sigma N of Y must be greater than or equal to one because they have the first term of their sequences now will be different, okay? Very simple proof. It's just intrinsic to the shift map that this holds. What about the lambda, the tent map? So if we have our tent map that we had, we have been studying zero, one-third, two-thirds, one, I one, we have our usual lambda. So lemma, F restricted to lambda exhibits sensitive dependence on initial conditions with epsilon equals one-third. And why is that? So this is a very simple proof. You don't even need any of all the construction that we did before for the conjugates. It's just almost immediate from the map. Why is that so? Take any two points arbitrarily close X and Y in lambda. In lambda, so let X different from Y in lambda. I have some iterate, the distance will be greater than one-third, greater than or equal to one-third. Why is that? Well, if they're already on opposite sides of I zero and I one, then this is obvious because this length is one-third. So the distance is one-third. So all we need to show is that at some point they land on opposite sides of this. One is an I zero and one is an I one. That's right. So now you're using, you're basically using the conjugacy that we have here. Yes. You can know that's correct. So using the conjugacy, basically it's for the same reason here, the two points are different. They have different combinatorics because we have proved that. And so it works. You can also argue very simply, again by the mean value theorem that if you look at the interval between X and Y, then this interval, the derivative is three here. So the size of this interval is growing all the time as long as this interval stays inside I zero or I one, it's growing. And so sooner or later it has to grow bigger than one-third and they have to be on opposite sides. Okay. So whichever way you want to look at it, there's various proofs, but using the construction. So it's simple. So there exists some n greater than zero, such that fn of X is in I zero and fn of Y is in I one or vice versa. Okay. And so the distance between fn of X and fn of Y is greater than or equal to one-third. Okay. So finally, before we finish this chapter, let me make a comment, something for you to think about, about how this conjugates. So when we conjugate this lambda to the symbolic dynamics, there is a map that goes from the ship space into the interval, right? For each sequence, you get a certain combinatorics. Now, if you remember in the last lecture, I gave some generalizations and I said the whole construction can be easily generalized for an arbitrary finite number of closed disjoint intervals and you can do the same thing. And you don't really need to worry even about what happens outside, right? All you need is for the map to be well-defined inside these intervals and you can do the same construction. Now what I just want to, and also if you notice in the pictures that I do, it's, of course, for this conjugacy to hold, this picture does not have to be necessarily like this. It can also be, for example, like this. So in one case, the map can be like this. So where the derivative, these are always straight lines, but also they don't need to be straight lines, right? But just to simplify, even though they're straight lines, here the derivative is positive and here is negative and all you can also have the situation where you have both, in both cases, the derivative is positive. So in both cases, the whole construction was perfectly well, of course, as you, if you look at the construction, you see that nowhere did we worry about the orientation, right? For the future, we'll talk about similar situations, so we can say that in this case, we have positive orientation and in this case, we have negative orientation. We never worried about it. So in both cases, we have the conjugacy to the shift space. So in both cases, we have this, let me call this H1 and H2. But the way, so because this conjugacy is a bijection from the space of sequences to lambda, which is contained in the interval, you can think of it as a kind of embedding of this shift space inside the interval, right? What every sequence corresponds to a unique point somewhere in this interval. So it is a natural question to ask, okay, how does this embedding work, right? What does this embedding look like? And if you remember as part of the construction, we saw when we actually construct this embedding, we studied a little bit how this happens because what does it mean to say that this is I0? So this is I0 and this is I1. What this means is that every point in here will have a sequence that starts with zero and every point in here will have a sequence that starts with one, okay? So if you think of the embedding of this space in here, you can separate this into two subsets. One is all the sequences that starts with zero and the other one is all the sequences starts with one. All the sequences that start with zero are mapped somewhere inside this region of the interval and all the sequences that start with one are mapped inside this region of the interval, okay? So that's the first observation about how this is embedded in the interval. Then you can look a little bit more closer. So you see that there's a finer structure, right? So this structure, I0 maps to the whole interval like this, this is the interval zero one, this is the interval zero one, which means that here there is a smaller interval, yeah, so this is I0, this is I1. Here there's a smaller interval which I call I00, right? Which is all the points that belong to I0 and that image also belongs to I0 because this is exactly, if you remember, so this is I0 and this is I1. So this interval here is exactly the interval gets mapped by the graph into I0, right? And here there's an interval which I call I01 because it is exactly the interval that gets mapped into I1 and I do something similar here. So here I have an interval which is called what? I11. Because look, it maps to I1, right? So I11 and here is an interval which is called I10, okay? What does this mean in terms of the embedding? It means that we can divide sigma two plus into four disjoint subsets. The union is all of sigma two plus and you look at all the sequences start with zero zero and all of these sequences will be mapped somewhere in here and so on, okay? You look at all the sequences start with one zero. So the embedding, the way this is embedded has a certain structure that you can understand by defining these finite time intervals that we studied in some detail, okay? Now what is the situation here? How different is the situation here? So here we also have an embedding, okay? But here, and here we have I0, we can also call this I01, I0 and I1. So on the first level, we have the same embedding. So if you look at the first two big subsets of sigma two, the sequences start with zero and the sequences start with one, they also get mapped to the two intervals I0 and I1. Now how do the smaller subsets get mapped in here? Well, we have a similar thing. Here we have, what? I00, here we have I01, here we have I10 because this belongs to I1 but its image belongs to I0 and this is I11. So you see that there is a difference here. There's a difference here. These are swapped. Yeah, when you think about it, it's natural. I mean, it's clear why this is the case, okay? But this continues. So as an exercise, you can think at home about how this continues. So the fact that we have an orientation reverse, this is in some sense the more natural ordering if you want on the line, right? Whereas here, these are because this orientation reversing is swapped and this orientation reversing means that things continue to be swapped at deeper and deeper levels. So if you now look, for example, at this element here, how this is broken up, this, remember that each one of these at the next stage gets broken up into two intervals, right? And so you need to say, okay, one of these is going to be 010 and one is going to be 011. Which one is which? The left hand is what? So let's see, I have to think about that. So this is zero. This gets mapped to I1, right? And then from I1, this gets mapped like this, right? So the left-hand side gets mapped to the left-hand side of I1, which means it then gets mapped to I1. So this will be 011 on the left-hand side. And the left-hand and the right-hand side will be 010, okay? Because if you look at the way this gets mapped in an orientation-preserving way into I1, but then here's orientation reversing, so it gets flipped. So this map, so the left-hand side of this will then be mapped to I1 instead of I0 and it flips. Okay? So it's a little bit subtle, okay? And I'm just giving it this to you because I think you should spend a little bit of time at home looking at these various possibilities. There are other possibilities. You also have the third possibilities in which both branches are orientation reversing, right? You could also have this possibility. And you can study the different embeddings in each of these cases. So the way the topological conjugacy, because these are all topologically conjugate to each other, will come back to this. So this is a fairly subtle point. What we have here is that how can we have a topological conjugacy between lambda here and lambda here? If this topological conjugacy has to map this interval to this interval and this interval to this interval, how can we find a homomorphism that is going to preserve the combinatorics because remember the conjugacy, everything is conjugated to the same here, right? And so the way you construct the conjugacy between lambda here and lambda here is just by composing these two conjugacies. So the topological conjugacy between the lambda here and the lambda here is the conjugacy that maps all the points that have the same combinatorics, two points that have the same combinatorics, right? The conjugacy exactly goes via the combinatorics here, which means that the topological conjugacy between lambda here and lambda here has to map this to this and this to this, so it must switch these over. Where's it here? It maps this to this and this to this. So tell me, how can we get a homomorphism that maps these two like this, but then switches these over? How is it easy? It has to be a homomorphism. What homomorphism? No, no, no. A conjugacy has to be one homomorphism. Do you understand the problem? You don't understand the problem. Do you understand the fact that these two, that when I define our mindsets lambda here and lambda here, lambda prime here, I have two counter sets, right? Both of which are topologically conjugate to the shift map. We agree with that? Yes? Okay. You agree that I'm saying that the way this shift map is, what this conjugacy does is it maps each sequence to the corresponding point that has that sequence, that has that communitlex. So using both the fact that both of these are conjugate to this, we get that the two sets lambda, F is sticked to lambda here and G is sticked to lambda prime here, are topologically conjugate using that. And this conjugacy maps points with certain sequences, to the corresponding point with the same sequence, right? This conjugacy is exactly the conjugacy is the map that preserves the combinatorics because that's how it's defined, right? Because this is exactly how both of these are defined here. But my problem is that this conjugacy is supposed to be homomorphism and here, as you can see from the combinatorics, they are all twisted, not just at this level, but inside. Inside here, this will be the opposite as here because here it's just orientation preserving. So at every level, they continue to be twisted. Well, homomorphism is just a homomorphism, you can. Excuse me, say that again. Was that a solution or was that a question? You said that your strategy doesn't work. Do you understand the problem now? Sorry? So you're going in the right direction here. Identity between counter-sets, what does that mean? That's right. So on what set is the conjugacy defined? No, it's not identity. Counter-set is just point. No, that is going in the right direction. So I will come clean. I'm trying to cheat. I'm trying to fool you, okay? Because that is exactly the right observation and this is what I'm trying to draw your attention to is that this conjugacy is not, does not need to be a conjugacy, a homomorphism from the interval to the interval. Indeed, it is not a homomorphism on the interval to the end. It is a homomorphism from the counter-set here to the counter-set there. And homomorphism between counter-set, counter-sets have a different topology from the interval so there's no problem in defining its homomorphism in such a way because the counter-sets has lots of open and closed sets inside and so there's no problem to define the homomorphism in such a way that it does not extend to a homomorphism of the interval. The embedding, the way these counter-sets, this counter-set is embedded in here is a way that is not, the two counter-sets are homomorphic but the homomorphism does not extend to a homomorphism of the whole interval. They're not homomorphic as embedded counter-sets with that structure, okay? The homomorphism is defined only on the counter-set and so it's not the identity. It's not right to think of it as the identity on the counter-set but it's going in the right direction. The point is that you are just conjugated dynamics on this counter-set with the dynamics on this counter-set and that's where the map is defined, okay? Yes, yes, exactly. So you can do that because you do not need this homomorphism to extend to a homomorphism of the whole interval. It cannot do that. This is a homomorphism. It's a map, this composition's map that is only defined on the counter-set here and it maps to the counter-set there and you cannot extend it to a homomorphism from the whole interval to the whole interval. It does not because it's all twisted. So look at some examples and think about it and think about this issue, okay? Because it's a level that is a subtlety that is important to comprehend because sometimes the proof is just so fairly simple in some sense that you forget that it is hiding some fairly subtle issues in here, okay? You say, okay, yeah, they're all topologically conjugated and so on. It seems simple. Any more questions on this? Okay, so this topic is not completely closed but we'll open in some sense a new chapter which is very closely related to this but as I said last time, we're going to consider. So let's think again of the family of tent maps. Let's record the family of tent maps because they're shaped like a tent and we saw that for lambda greater than two, they look like this. This is one, this is one half. So one half is the turning point between the negative and the positive slope and as long as lambda is strictly bigger than two then this implies that f lambda of one half is strictly bigger than one and we have this picture here and as long as this is strictly bigger than one we always have a small interval, open interval here which is our delta and we know that all the points outside zero one they just map to minus infinity all the points in delta map to plus infinity and here we have two intervals i zero and i one the derivative is lambda, sorry, this, yeah. So the derivative is bigger than two here, okay? Particularly it's bigger than one. So this is expanding and therefore we can do the whole construction that we did before, okay? You realize that that this is exactly the same situation. The only difference is that lambda is slightly different is not exactly the case lambda equals three but we have i zero, i one and it falls exactly into the general generalization that we gave before in which we have two closed this joint intervals that map to everything else and even in this case we have the conjugacy we have a canto set here and we have the conjugacy to the full shift everything is exactly the same. Now the question we will be interested in now is what happens when lambda equals two, yeah? So when lambda equals two, the picture is this it's somehow almost exactly the same but not quite. So now what we have, for points that are outside the interval zero one we still have exactly the same situation but if you define now, first of all if you define lambda equals the set of points x such that fn of x belongs to i zero or i one, well, we can still define i zero one so we can still define i zero union i one or even just the whole interval i for all n greater than equal to zero and what is this set? This is the whole interval simply, right? Because the interval gets mapped inside itself so every point in the interval zero one gets mapped back into zero one so the whole interval is invariant. So we no longer have that this set is a canto set. We have that this set is the whole interval. In particular, we will not be able to have topological conjugacy from this to lambda. Why can we not have a topological conjugacy from sigma two plus to lambda? Exactly, it's a canto set. This is an interval. You cannot have a homomorphism between a canto set and an interval, okay? But of course the dynamics is very, very similar to this. You imagine, okay? What has changed? We still have i zero i one, it looks like, I mean it's not like all those periodic points and all those dense orbits can have just disappeared, right? Until up to anything lambda greater than two you still have this canto set with all these periodic points and all these dense orbits and all the sensitive dependence on initial conditions and so on. And what has happened of course when lambda is very close to two this gap is very close. So also all the pre-images are very small. In some sense this canto set is thicker if you want, okay? Although not in a precise way. The gaps are all much smaller. And what we've done here in the limit we have closed this gap and the moment you close this gap you close all the gaps of the pre-images so you're taking this canto set that is embedded in the line and you're closing all the gaps of the canto set. And identifying the points of the boundaries of these gaps and you're getting an in-tool, right? So we have to try to understand exactly what that means for the dynamics and how we can describe precisely this process in this situation. So it turns out that it's not really that difficult. We just have to understand closely what these identifications mean, okay? But so what I'm going to do rather than what I did in the previous case is we studied this particular example in detail and then I gave you these generalizations. I think in this case it makes sense just to write out the general formulation right from the beginning, right? So we want to generalize this picture here and we want to have a similar picture in which we have zero one and we have some finite number of intervals but this time with no gaps, I zero, I one, I two, I three. And then we will still want this kind of picture so we will still want each interval to be mapped onto the whole original interval. If we want the map to be continuous we have to have the orientations positive and negative but that also will not be really a necessary part so to make it even more general we will just assume that we have, for example, this case is negative orientation, here we got positive orientation and maybe here we got negative orientation. For example, something like this, okay? So you can see that the, you can also see this as the limit of a situation in which you start with four closed intervals which are disjoint so they have gaps, okay, which would be similar to this situation here and then we can apply our general theorems that we had before, when you have gaps we just get a counter set but what we're doing now is we are closing so the bound, so these intervals are no longer disjoint because they intersect just at the boundary and then we have a picture like this and then we have to study what happens here. So let me write down the formal definition. So a map F from a unit interval or an interval I to I is called, is a full branch piecewise expanding, is let's say full branch piecewise expanding. If, so there are various ways to formalize the way these intervals are so I will do it in one way but you could say it in different ways so if there exists disjoint open intervals, I zero and minus one, lambda greater than one, such that for all I equals zero and minus one, we have these three conditions. First of all, the closure of I is equal to I to the derivative X is greater than equal to lambda I so these are basically exactly the same conditions we had before and then the crucial thing is that we don't have any gaps so the way we write it is that I is just equal to the closure of the union. I think this takes care of it. Another way would be to write that you have a union of closed intervals that intersect only at the boundaries, there's various ways of saying it but I think this is one simplest way. Okay, you recognize the picture in this definition here, okay? So this says there's no gaps and this says that each interval is mapped to everything in a uniformly expanding way because the derivative is bigger than lambda. In particular, it means it's a bijection, right? Because the derivative, I'm assuming here that so because the derivative is bigger than lambda, it means it has to be monotone on each branch. Yeah, we can also include C1, the two EI. Well, it follows from the fact that the derivative is bigger than lambda really, that but we can say C1 differ if you want. It's really not so important. C1 differ, let's assume that it's a C1 differ. Because the derivative is bigger than lambda, it cannot fold back on itself, so. So, what is the theorem? The theorem, F full branch piecewise expanding. There exists continuous surjective map H from sigma L plus to I such that H composed with sigma equals F composed with H. H fails to be injective on a countable set at star in I on which it is 2 to 1. So, remember, I think we discussed some examples before. So, this is not quite a conjugacy. What is this called? Semicondugacy, exactly, right? Because it's not necessarily invertible, but it's a continuous map that satisfies this. So, the fact that it's surjective is important. Why is it important that it is surjective? Sorry? The definition of full branch? Oh, what is piecewise expanding? Ah, okay, that just is because usually it's just a terminology. Like usually, if you say expanding, you are implicitly saying that it's C1 everywhere because expanding, usually you use to say that it's C, you know, the derivative is bigger than lambda everywhere and here there are some points where the derivative is not defined. So, piecewise, it's just a way to draw your attention to the fact that it might not be continuous. The map might not be continuous or might not be differentiable everywhere, okay? It's just not a, it's just a terminology that has become standard. I could just call it full branch expanding map. It's, yeah. When you see, whenever you see in a book, this is piecewise C1 or piecewise expanding. What it generally means is that you have some partition of your neighborhood and on each domain, it satisfies that property, but there might be some boundaries in which it's not C1 or expanding or continuous and so on. So, surjective is important of course because if you just, well without this sentence here, you know, otherwise it's trivial, right? Forget about this injectivity. Suppose you don't know anything about injectivity yet. Why is it trivial that there exists a continuous map satisfying this if I don't include surjective? No, not just what we want, but why is it trivial that this is true? I can find such a semi-conjugacy if I don't include surjective in here. Yes? The intervals are disjoint, and so? Where is lambda? Lambda in this case? No, no, but now we are looking at this kind of, this map here, right?  My question is the following. Suppose my statement was just, then there exists a continuous map H satisfying this. Why is this trivial if I do not include surjective? Yes? Well, no, you cannot restrict yourself to one interval because this interval is mapped outside it.  I don't know if you knew it or not, but which are the points that stay inside one interval forever, that stay inside the same interval forever? How many points is that? Which points do you think? Are there any points that stay inside I zero forever? There's a fixed point here. Each branch actually has a fixed point, okay? So this point here will stay inside I zero forever, but how is this related to my question? So my question is, let me, because I think it's important to think about, let me, suppose my theorem was the following. F, full branch piecewise expanding, okay? Then there exists continuous map H from sigma L plus to I, such that H composed with sigma equals F composed with H. I claim that the proof of this is trivial. What is the proof of this? And what's the difference between these two statements? Forgetting about this part, just a little bit, okay? Between these two statements. Forgetting about this part, just the first part. Give me a continuous map, such that this is satisfied. So you need to map the sequences to some points here in I. Suppose it's not subjective, which means it maps to some subset. So you are going in the right direction, saying the canto set and the one interval and so on, but give me just one easy example. So what if I map, so what if I call this fixed point P? So let P be a fixed point. Let P fixed point for F and let H of X bar equals P for all X. Does it satisfy these conditions? It's a constant, it's constant. So there's a big difference that you add subjective. It's crucial to get just a semi-conjugacy, a continuous semi-conjugacy, semi-conjugacy. If you don't have any additional information, it can be a trivial statement. Because every time you have a fixed point, you can take any other system and you can just map everything to the fixed point. And it's automatically a semi-conjugacy, because it satisfies this condition, okay? So just a semi-conjugacy, that's why conjugacy in general is much better than a semi-conjugacy. Although in the semi-conjugacy if we have some additional information, so we know that is surjective, means it cannot be this kind of trivial situation. It is mapping to everything. Not only that, but we're also making some statement about injectivity, is that it only fails to be injective on a small set. And on this set it's only two to one, right? So you cannot have the whole set going to one. Because also a map can fail to be injective only at one point of the image, but it's the only point of the image and everything maps to that point, right? Then it fails to be injective only at one point of the image, but it's infinite to one, everything maps there. So what do you think that this set is going to be when it fails to be injective, the set in the image, right? What we're doing is we're going to construct, so this is our map. In this case we would have sigma 4, sigma 4 plus continuous surjective map. When we say here that it fails to be injective on a countable set, of course, it fails to be injective. So we're looking at the image. There's a countable set in the image, right? That has more than one pre-image upstairs. So what is the obvious candidate for this set? Certainly these points, and what? The orbit or the pre-orbits of these points? The images or the pre-images of these points? We'll see, okay? But it's obviously related to the set because this is the set. If we want to do the coding, this is precisely the set in which there will be some ambiguity. When you fall in here, you don't know if you're an I0 and I1. So it's likely to have some two sequences maybe going to the same point in that. But what is important is that it's only on a countable set. So in fact, for most points on the interval, this is actually a conjugacy and this is one to one. So I won't have time to do the full proof, but let me start by, let me just in the last 10 minutes say how we start the construction. So in fact, we will do very, very similar. We will really try to generalize the argument that we had before. We'll try to construct those sets. So first of all, so to construct, we first of all define first a multi-valued map, F bar. And we define it in this way. F bar is equal to, sorry, the first I needed to define. So first of all, for all j equals 0 to n minus 1, let Fj equals F restricted to ij. So we write F as a kind of union of L different maps on each ij. And remember the way I defined it was that these ij's were open, right? But because they are c1 different morphisms from ij onto the image or the cloche, it's a c1 map and a c1 extends to a c1 different. I remember now why I hadn't written c1 different, because it's just a technical issue about exactly what you mean. It's a c, because it's defined on an open set, technically speaking, it's a c1 different morphism onto the open interval. Onto the open unit interval, right? But because we assume that the image of the closed interval is the full interval, then it extends essentially to a c1 different morphism from the closed interval to the closed interval. So we let Fj equals F restricted to ij. And then we let F bar j equals the continuous extension, Fj to the closure of ij. This is just a little technical point that we have to deal with because of the way this is defined. So we have that F hat of the closure of ij is equal to the original interval i. Sorry, Fj hat. Is that clear what I'm doing? I mean, this is a annoying little technical point, and it's really just to formalize something that is intuitively obvious, but to formalize it, we need to write it in these ways, in this way, because F is strictly speaking just defined on the open interval. But each Fj is now defined on the continuous extension. The reason we need to be a little bit careful here is because strictly speaking for the original map F, the original piecewise expanding F, the image of this point in which the two intervals meet, it can only have one image. The map F is single-valued because maps are single-valued. Otherwise, it's not a map. And so the image of this point has to be either here or here, depending on how the map is actually defined. In this case, there's no ambiguity. So in this case, it doesn't matter because on both sides, the image is here. But on a situation like this, it's either here or here. But we also want to give a meaning to the fact that even if the original map F is defined and taking the value zero here, we also want to be able to define what it means to apply the map to the closed interval here. And so we want, in some sense, the image of this point. If you think of this point as belonging to the element I want as being up here. Okay? And all of this is solved by making exactly this definition here. Topology in action. So what we've done now is we've defined a union of disjoint maps, if you want, on these closed intervals, thinking of them for the moment as disjoint closed intervals. Because each of the boundary points has two images, depending on whether you think of this boundary point or possibly two images, one or two images. Depending on whether you think of it as a boundary point of the left-hand interval of the right-hand interval. Okay? And we use exactly this fact to define this multi-valued map. So do we define this multi-valued map as Fj of x. If x belongs to ij, these are open intervals. So the ij, the way I've defined them, so there's no ambiguity there. And as F bar i of x, union F bar j of x, if x belongs to the intersection of the closures of these two intervals. For some ij in zero. So this is where it becomes a multi-valued map. Defining this map F bar for some points has two possible values. And then now with this definition I can actually, this definition is a little bit technical, a bit complicated but it's very useful because with this definition everything else becomes almost exactly as in the previous situation. So what we can do now is we can define. So now for all a bar in sigma n plus, we can let equal by definition the set of point x such that F hat i of x. Rather than writing that it belongs to i ai because this can be, this can have two values because it's a multi-valued map. I would like that this intersects closure of i ai. So notice that if we happen to do all this stuff in the situation where those intervals are actually disjoint, then this gives you exactly the previous definition line. In the case in which we have a union of disjoint closed intervals, then this intersection never occurs because the closure of these intervals is disjoint. So we have that F bar of x is just equal to Fj of x is single valued everywhere. And therefore these are closed to begin with. So the fact that the section is non-empty just means that Fi of x belongs to i ai and we have exactly the same definition as in the previous case. So this is exactly a generalization of the previous case because it coincides with that in the case in which these intervals are disjoint. Okay, so the rest I will do, I will do the next lecture, but it follows in very similar way. We will show that this is a non-empty and consists of a single point and then study the lack of it. So this allows us to define the map H because to each A bar, we can define a single point here and so on and everything else will be fairly straightforward actually after we have fixed these definitions. So try before the next lecture to think a little bit about this case. So the fact that the notation is a little bit complicated to become familiar with here, okay, to deal with these intersections. Okay, thank you very much.


which is fans. This is one of the biggest fears in football. Not only a player being injured, but a player being injured in the off season. We haven't even sniffed it down the football yet, but Eagles guard Brandon Brooks is already done for the year. And that is a big blow to the Eagles. He has somebody, he's somebody that's established himself as one of the better guards in the NFL, is one of the best guards in the NFL. He's made pro bowls. He came in right away with the Eagles. He has been injured recently quite a few times, but he came in right away with the Eagles and filled the role. Cause he was with the Texans before some years back. Oftentimes I forget that myself, but he came in and he did what he had to do and he did a really good job of it too. But now the Eagles, they lose that for the year. And he didn't do it playing basketball. He didn't do it playing soccer. He didn't do it playing on the steps with his dog and he think he did it running on the practice field. He was on the practice field. And I stand by this, I say this all the time, this is the worst part about football, man. It's injuries because they can happen out of nowhere and nobody is safe from them. Nobody is immune to injuries. Nobody can avoid injury. If an injury happens, it happens and you gotta deal with it and move forward.  But that's a pretty big role to fill. Now hopefully for the Eagles sake, this creates an opportunity for somebody behind Brandon Brooks to where they find a diamond in the rough because that's how a lot of people's careers end up getting started.  They make the most of it. So hopefully that ends up being the case. But we'll see what goes down. But I feel for Eagles, man. Feel for Eagles, I feel for Eagles fans because these things hurt, man. They hurt. And I read that as soon as he went down, he knew what it was because he's dealt with this before. He tore his Achilles in both legs, both legs.  You think his teammates are hurting. You think his coaching staff is hurting. Imagine how he's feeling right now. And he knew it right away when it happened. The second he went down, he knew what went down. He knew it because he been through that before. That's gotta be crushing, man. But again, now it's next man up. You gotta hope that he can come back stronger because the Eagles still got him under contract for like three or four more years. Yeah, this season, yeah, till 2024. So three more seasons because this season runs through 21. So unless, I mean, we ain't even gonna talk about them cutting bait or anything like that. I'm sure that'll be talked about down the road, but they got some decisions to make. But anyway, I feel for y'all Eagles fans, man, straight up. So team keep it clean. I love y'all. I appreciate y'all. And unfortunately, like Brandon Brooks is for the season, I'm out.


provides here. So I'm joined for those who don't know me, I've been found for a while I'm running the Twitter, by the way, I live for Graphics Media, so if you have any complaints about that, it shouldn't be. But I want to talk about a project I started like a few months ago following something that you might know of which is this project called Fort Awesome. For those who don't know, it's an icon font, right? It's been quite popular, not Fort Awesome, font awesome, that's the one I'm already jumping. So you guys have to help me. Sorry, who does the sheet? Fort Awesome. No, I wanted to start with Fort Awesome actually. So it's part of the confusion. So Fort Awesome creates this project called Font Awesome, which has been used by millions of websites. You probably know the version 4.7, which is all, you can only find it from here, by the way, and the appetizer for the new version. But basically, it's this icon font that's been used on millions of websites. You've seen them everywhere. Who doesn't know about this project? I'm curious. One, two, three. Okay, so if you design websites and you don't want a web app or whatever you call it right now, it's like if you want a quick set of icons, you just import this icon set and then you're pretty much covered for a lot of cases. But what happens is that they advertise themselves as an open source project on GitHub. Let's find their GitHub page. And I've been actually annoying them because I made, basically, file issues and make pull requests. And especially since last year, because I'm a big fan of a project called Master Long, I asked last year in April 2017 to add the Master Long logo to the icon set because I think people will find that useful to ask to add the icon next to their handle name and use that on their website. But as you can see, the issue is still open a year later. Even if the main developer of Master Long joined in, he came with a new logo, showed the files, and in the end I made myself, I did the pull request for it, which was denied. And what's denied for good reason is that since 2016, they've stopped accepting any pull request to add new icons and they started a Kickstarter campaign, of which they received a million dollars for it. They were asking for $30,000. And they switched to a different model where now they're releasing the file in three versions or two versions, a free one still with Creative Commons licensing, but they're promoting for a pro version which you pay for. And since my pull request wasn't accepted, and I was, there have been so many pull requests, unaccepted for people who wanted to add new icons, I decided to fork it and I forked it. I first asked, actually the console of two people that are here, Nathan and Brandon, to see if it was a good idea to fork and if it was legitimate, you know, forking is still a big deal in the free software movement. So, yeah, I decided to fork it to a project that I called Fork Awesome. And that's what we've been doing for the last couple months. What we started doing is adding a lot of new icons, which you can see if you go here. So, we've added a lot of icons, such as for massive amount of forms, but new social hubs like that, and even Scuttlebot for those who know the social network. We added icons for open source projects such as Charlie or any other like non-open source project in the mix. And so this has been, my point of doing this is that I was frustrated that what they were calling themselves an open source project, an open source design project if you wish, were not accepting pull requests for design improvements or adding to the sets. And also they were like keep focusing on basically the main author of the project, which is Dave Gandhi, which did most of the work of the following self. But I thought that it wasn't really beneficial to the community, and especially with the latest move they made, which, I mean, I'm very glad for them that they managed to take that much money and turn it into a profitable project. But somehow it was kind of for me betraying what I think open source design could be. So what we've been doing with this project right now is I've onboarded in the project a few contributors. I've also, in the website itself, mentioned and did the work of going through all the contributions that have been done in the past to actually reference them and acknowledge the contribution of every people that have participated in the fund awesome, but also since then, since this moment, all the people that have participated in the full awesome project. I'm giving also access to the repo to some people and we've been tackling issues and discussing design and improving on the project. As you can see, there's already like 35 issue closed. The price itself has received, I can't, or you can see that, but yeah, 300 stars and people are starting to use this in their own projects as a replacement because it's basically for people who want to replace for awesome with for awesome. It's really like using either CDN files that we're using right now and you can see it's getting quite some hits or downloading the files and just replacing. We basically haven't changed anything. We're keeping the code as it existed. We're just adding new features and new icons. And so if you find this interesting, I welcome you to talk to me or to join us on GitHub and participate. The goal is to really turn this into a community project and move out of the single person who decides everything and owns everything in the end. That's it. That's my presentation. Actually, in my slides, I wanted to show you something. We've been accepting full requests for icons, especially where they said there's one for XMPP. Basically, the issue was open since 2012 on the front awesome. So we finally closed that issue in 2018 with Fork Awesome. So that's the kind of things that I'm proud of for this project. I invite you to join. Thank you. This is a very much improvised talk. So the slides are actually in German and I hope I can take from the context what this is about. And there also is a chance that I already did a talk on this project at the library graphics meeting. This is something that I'm being kind of obsessed now again, although the project itself is something that is with me since at least 2006. The goal is to have a MIDI to SVG project, basically trying to use MIDI files and convert them to something you can plot out on a laser cutter to play music from. So the talk I gave at the make up on a few weeks ago, I gave together was Gjörgen Land. He is a musician and also writer for mostly Orion. And let's just skip to the story. Basically, we met at a summer music festival, which is like a world music festival somewhere in the middle of Germany. And yeah, basically it's quite a colorful event and there were a lot of discussions going on. And they were discussing, I've witnessed people discussing a present for the organizer of this event. And they were talking about having a custom music box playing a song from the organizer. And yeah, I told them actually I did something like this. So, I'm sorry. So this is actually the start of my part of the story there. This kind of music boxes we are talking about here is something I know for a very long time already because my father had one of these things. I remember playing with this thing as a child. In 2006 I bought my own and the idea is basically I can just show how these things look like. There's some. So very nice little things and the music is encoded as holds in the paper strip. And so the first thing you do is actually you get paper strip with black dots printed on it.  And yeah, this is something that gets annoying fast. So my first goal was to actually build a robot to punch holes. And I actually started and bought products which were supposed at some point to end up as this kind of robot. And I did a design and a meaning play and I found what kind of works to do before to find the right things to punch holes in the paper. And that's what my mechanism meant whatever. Checked out, I'm not a mechanical engineer. So this is something basically it's in the same state today this project as it was. But then I'm used to projects which take quite a long time. And then in 2011 I attended the case communication camp. And it turns out there was a favelet truck from the Netherlands. The favelet truck and it looks actually like this. Which this is actually not a photo from this event that the truck is the same. And it's just a favelet on wheels. There is a laser cutter in it. There are 3D printers in it. And so at that point it made like click and I thought okay I can do that because I know how to deal with graphics. I know how to deal with media. I already had written software to pass media files. And extract the relevant information. So all I had to do was to create graphics that I could then print out on the laser cutter. Yeah and basically this is in 2012 then the meeting with Jürgen happened. And this actually was a totally different view on my project because he also has bigger plans. Let's just skip to this a little bit to the technique things. I know of 4 different variants of these kinds of music boxes. They basically look like this. This is a 50 note C major scale scale. So this is a bigger variant. It has 30 notes. It actually has some chromatic notes in the middle. And it plus some bass notes. And this is actually mechanically or from production quality the nicest one. But it is limited to 20 notes and a major scale as well. So how do these things work? This is a sample version. You can see there are the metal strips that produce the sound by getting plucked. They get plucked by these kind of wheels. And the paper strip basically goes through like this. And the holes enable the pins on the wheel to enter the hole. And then get pulled forward, plucking the metal strip down here. There are 4 of these pins around the wheels. You can see that here. This is from the bottom. So basically, yeah, this is about it. You have 2, 2... No idea. Paper is transported between 2 rows, thank you. And the holes allow the wheels to grip into the paper strip. And, well, the next pin then flux the metal strip. What I find kind of interesting is that there is a whole range of things you can actually do. You have these kind of objects you are more or less obsessed with. You can start by just printing out something which you then clip by hand. You can try to find a way to do actually laser. Use a laser cutter to do this. You can do it with a whole punching robot. And you can try to automate the playing as much as possible. And you could also be happy with just sampling the music box and just using your regular media software to play this stuff. Or you can even go a step further. This is not a project for me. This has talked up in the last few weeks. It's actually actuating the pin wheels with individual servo motors. So you have like 13 servo motors here. And it actually is possible to do live music playing with this thing. It's quite an impressive project. And, yeah, so you have this big kind of spectrum. And you have to find a way a project is. And my project actually stops for now at the laser cutting of the whole strips. And, yeah, that's it. I'm not sure into how much of a detail I should go. I could tell a lot about browsing these files. I could tell you a lot about creating the graphics using, in this case, Tyro. But I guess I'll stop for now. And if you're interested, feel free to contact me. Thank you. How about using Vox with you? Unfortunately, I didn't think of it. Sorry. No, it won't show again. I can show you generated PDFs on my laptop. It's basically just an outline with lots of circles. And, yeah, there is a little bit of strategy involved. Because the laser cutter in our FabLab has some mechanical issues. Regarding the positions, all I had to do was slide back around. But this is very much nitpicking. Thank you. So, I think something you may be familiar with this project that I've been working on. I presented it a few years ago at another LGM in Leipzig. And it's still going. It's fits and squirts, but the project is still kind of active. And there's still ongoing experiments. So I wanted to, I guess I should give some people some, if you're not familiar with it, some background. So the idea of this greenless office is to come up with, I call it an artistic operating system. Not because I think it's somehow pretentious, somehow superior to non-artistic operating systems. But because by calling it an artistic operating system, I can get away with not being efficient or working in a way that would be expected of a proper operating system. So I decided, I mean, I guess one of my problems is that I feel like the screen is, I mean it's obviously very powerful interface, but it's kind of obvious in some ways. It gets us into certain habits of ways we work and maybe some of these aren't good or maybe they can be kind of boring. So if we're going to do something that's experimental, why don't you do the most difficult and get rid of the most obvious aspect of the whole thing so we get rid of the screen. So things become really difficult, but then maybe there's some interesting ideas that come up. So what I sort of started with the first functional thing was I thought, okay, how do I get through my day? I'm still using the internet, but how can I not use a screen? What do I need? So the first thing was, all right, so I can avoid sitting at the breakfast table with my laptop or some other device looking at that. And so I started making, basically, it's an RSS reader that this is what I print out every morning. So I have... So you reinvented newspapers? I've reinvented newspapers, but it's up to the minute. It's really up to date, you know, it's not like hours old, like one of these old newspapers. This is fresh. Yeah, sure. Yeah, and so basically then the way it works is it... Yeah, I have RSS feeds. It grabs the RSS feeds, then in order...  So I go and I get the open graph headers from the HTML and pull out some images maybe. And then if there's a particularly interesting article, then... So these are just the summaries, then I take out my barcode scanner and I scan the barcode, which then prints out the web page. So then the web pages look like, like here's a few examples like this. You know, it's a pretty kind of boring layout, but... If anyone has design critiques, I'm open for it. We can talk about whatever. And here, if anyone's interested, I'm actually using the browser. So I'm using Firefox and rendering it to PDF. And the trick I do is when I pre-process it, I strip out, I use some... I think, yeah, basically it's a Python implementation which strips out most of the junk. And that's also one of the things that I like, that it's sort of like taking the control back with the interface, that if you're browsing the web, but you're no longer... There's no pop-ups, there's no annoying thing that's hovering over it and saying, please turn off your hand blocker or this kind of stuff, that it's all stripped out and I've just got the parts of it that I want. I mean, there's a lot of sites that don't quite render properly, so then occasionally I do have to open up my laptop, but I'm okay with it. And then, yeah, the main sort of operative thing that goes on is I scan through all the links and then I make a number, a footnote, number where the link would be. So then if you're reading the text and you see a little footnote, then you can go to the end of the document and then instead of clicking the link, you scan the barcode and then that prints out the next one. So that's gotten through a certain amount of my day with that. I also have... the layout is awful, but I can... And I also have my inbox, so I have some simple iMap stuff that grabs my inbox, tells me everything there. And in some ways it's kind of nice because I have one that's like spam and delete, so I can really quickly just... they're really nice stuff. But I can also print out pages and then what I have in terms of... I don't think I should show you this as well. So then Twitter, of course. So this is... So it will print something. Yeah, so I have a receipt printer. I don't have my A4 printer, but I also have a receipt printer. I recently implemented, so I can even do the debugging. So now if there's a stack trace, it actually prints out. Which is kind of nice. Usually in screens you do that kind of stuff, so that's a little bit of... without that. So, yeah, so I've been working on that for a few years. Different people have helped me. And the other thing I have is, in terms of input, I built a document camera. So the software right now, it's running on this sort of... It's actually like one of these Raspberry Pi-esque things from this Bulgarian company called Olimax. It's not very fast, it can do some optimization, I mean, the whole point of this is to do a little slow computing anyway. So it works out. But, yeah, so I use this document camera. So if I want to post something to Twitter, I have this... I should have brought an image. I don't have one here. But I have a sort of A3-sized black blotter and I just put something down on it. You know, it could be one of these things, or I could write something on a piece of paper and there's a switch so that I can set the size of the actual image that I want to post, somewhere between A6 and A3. And then I can set the orientation if it's landscape or portrait. And then when I scan here, the... Yeah, exactly. So here's the menu of my software. So when I scan here, tweet, or even I did implement Facebook, although I think it just seems quite the sound. But I occasionally post to Twitter and what I can do is I just scan this barcode and I have an LED light that turns on and it takes a photo of whatever's on this surface and then uploads it as a tweet. Do you bring the photo before tweeting? You see if it's good. No, I just have to hope that it's good.  Actually, no, I have a fixed focus. But yeah, for some reason the composition... I guess I could see that. But yeah. So that's sort of going on. How much paper do you buy per month? Not that much, actually, because that's the thing that I realized is that by... I mean, paper, you see it. Once you start, you realize how much I'm printing out. I printed like 10 pages one morning while I'm drinking my coffee and I think, like, this is maybe enough for now. And then I'll sit there and read it.  I use both sides. I try to be aware of it. But there's this sort of feeling that I mean, come on.  You may have something, if you have a decent-sized screen, that thing probably uses like, maybe, I don't know, 100 watts. 60 or 80 watts or something like that. And you're sitting there for hours going on some deep dive into obscure parts of Wikipedia or something. And you don't have... There's no point where you see this stack of paper that, oh my God, I just wasted two hours researching obscure snakes or something. And it's never... Whereas here, at some point, you're like, this is ridiculous. I just printed 40 pages. I have to stop. You know, this... For me, it's a way of being more reflective about this usage, because I don't agree with the screen. It sucks, you know. That's part of what makes it amazing, but it's also scary. Yeah, so the input stuff is still kind of the main problem. So, a friend of mine is a Dutch hacker and he ran into one of these things. Moleskine puts out this very fancy digital pen, which works with their very fancy notebooks. And these notebooks have this micro-printing on them. And the way the pen works is it has a camera inside of it. And the camera will recognize the X and Y coordinates and which page it's on, which page in the notebook. And so, it saves this as a sort of little stream of data on the pen. And then if you are connected by Bluetooth to your PC, then they have some apps that are up to a smartphone or whatever. It then will send this stream of basically vector data of, you know, pathing it to various apps. Now, if you haven't figured out that they had published some kind of specs to this thing and we went through this back and we realized it wasn't that complicated that we managed to get a sort of rough prototype working. And I thought, alright, this could be cool. This could be a nice way. What if we work it out like basically like a screenless content management system?  You want to update the website, okay, here. There's the pen, the paper. You do it.  The racing wasn't quite good, but that's the same way with pens. Consistent. So I can give you guys a quick demo. So it's kind of primitive right now. The way it works, the synchronization is essentially, so this is an SVG embedded in a minimal bit of HTML. And then what we do is then append the end of the SVG with more path data. We're just using ourselves to push that so that it's not so we don't have to sync the whole thing every time. But this probably should be done with some kind of cool WebSockets whatever thing. If anybody knows about this stuff I'm interested to hear about it. But let me give you a quick demo. A little latency, but it seems alright. Yeah. So what I brought the thing is right now I'm not sure what to do with this thing. I think it's kind of cool and it's fun, but I think this is room full of interesting people. So I have one and this is a cool thing about these pens is they actually publish some PDFs so you can roll your own and print your own whatever formats you want to cook up. So I thought if people are interested and want to try it out maybe we could do something where we try to make a kind of improvised publication I don't know if you can start to comment. I don't know what people want to do. But I thought it would be an interesting format to play with. So I have this book and maybe we can have a beer tonight and people can give me some suggestions and then tomorrow we can maybe do some kind of scheme for passing this around and say if we can produce some interesting little experiment. Yeah, so yeah, that's all I have to say. Is it printed long? Days? No, usually I throw it out. It's really like, you know, just recyclable it and be done with it. A few things I do keep or I like, the other thing I brought into kind of like scrapbooking. So I'll have if I have an interesting image or one of these something here or a tweet that we're like the image or the text then I'll just cut it out and paste it into my notebook. And then there's a lot of other stuff where I mean I think it's good to be able to there's one thing that's also a tendency with digital is that storage is cheap it's getting cheaper every day and everyone becomes like a sort of crazy librarian and we never throw anything away and then you get a lot of crufts and it's kind of maybe good to I mean I like this, I can totally understand this it's great, you know, archive.org but at the same time I think it's good in terms of like I don't want my apartment to be full of crap and being nice with my digital life could also have a certain level of tidiness to and just maybe my weird way of trying to figure that out any other questions? If it's a practice operating system, who's the audience? Who's the audience? Are you? I think I, I mean it's totally egotistical, I'm the only user here. But it's also it is, if there's code and it's actually not too bad, there's only one dependency where I just figured out how to I mean it's complicated to get it going and depending on what hardware you want to use, we may have to I may have to help you out, but it's not so exotic so if anyone is interested I'm totally surprised surprised and I'm very happy to help you like get your own screen this office running. But I think for me it's about, it's also about confronting this idea of universality. Like that's one big problem I have with a lot of interface stuff is it's based on this sort of old modernist idea which is in some ways kind of nice, you know like the Bauhaus and this idea that we should design for people, but it implies this sort of, there's a kind of arrogance, you know, that like this idea that there's a universal standard that is good for these kinds of things and I'm by doing this in a way I'm provoking to say like, well maybe that's not really true, maybe there's like a weird messed up personal way that all of us would like to interact with these things and maybe that's okay to do that, to come up with your own more personally acceptable way of dealing with all this needy because it's not easy to get into magic yeah yeah


Hi, this is Tanesha Johnson and today we will be exploring how to make presentations in Canva on iPads Canva is a digital tool that makes creating presentations fun and easy So we will get started by accessing the Canva app on my iPad So on your home screen you will have needed to download it. It's on my bottom row. It's the Ombre app with a C So click on that. I will suggest that you create a teacher account if you have not done so already Canva gives teachers access to way more content within the application For being a teacher so make sure that you do that But I'm gonna go ahead and dive in to showing you how to create fun presentations you can access All of your templates within Canva in multiple ways But I'm gonna start on this purple toolbar at the top and click on presentations And I'm just gonna tap on this first box That is gonna take me to the like the the the area where you create your designs So within this I can search for templates Canva has Whoo, they have so many that you can use that are already created for you You can just click on the one that that you're interested in and then tweak it to make it what you want So as you can see these were my recently used at the top. These are ones that they're suggesting for me But then as you scroll down, there's educational presentations. I've used this very first one for grad school. Yeah, I I used it to create a presentation for one of my professors to Present my end of the semester content and it made it so fun and engaging So and it didn't take me a lot of time to create it because it was already done for me I just had to plug in my information So I'm gonna show you how to do that So as you see just wanted you to see that the the Possibilities are endless with Canva as to what you can access and use so I'm just gonna pick one And as you see I picked one the child board of mathematic class Educational presentation it tells you at the top that it has 17 pages So I can choose to click on all 17 pages or I can just only pick the pages that I want But I'm gonna go ahead and apply all 17 So now you can see at the bottom here They are and I can scroll through and pick the ones that I am interested in So to change this presentation and make it what you want. All you have to do is just click on the boxes So as you see it brings it up. I can double-click and now my toolbar comes up so I can change the wording I can change the font so Drop that down for you as you see there's tons of different fonts that it can give me I like this right there. So I'm gonna change it up. There we go. So I Want it to look a little bolder. So there you go. I don't know about that one, but It was kind of cool looking So I changed my font Now I can change how big or small it is up here See how it grows it gets smaller so you can change the font size I can also change the color. So say I don't like that yellow. I can make it purple. I can make it green I can make it blue. I can make it purple. I can make it whatever color. I want so Then you have these three dots. I can change the position. I can animate it have effects underline it change You may get centered left right. I can add bullets. It's the possibilities are endless with what you can do So whenever you are in here creating a presentation I would just recommend that you just take time playing with all these different little Options that they give you and explore how to use them so I Changed my font. I can change the words and so say I wanted to add some pictures because This is math. I don't see anything that has to do with the numbers up here So what I want to do is click this black toolbar on the side and now I clicked elements So I want to add numbers So you can search for anything. So I'm gonna add some numbers There we go Now it's gonna pop up Numbers for me and I can scroll through and and add whatever numbers I want to add on here So I'm just gonna be basic and go with a one Two and I'm just gonna stop there for sake of time, but I've now added them So I'm gonna get rid of that. So here they are. I've got my fun numbers on here I can drag them around move them just using my finger to make them bigger smaller But say I feel like they're white and they're kind of boring I want to make that a different color So you see I've clicked on it and at the top that little white bar There's a black box and a white box. I'm gonna click on the white box and Now it's popped up Different colors. So it's gonna give me different colors that I can use to change it So maybe I want to make it green because there's some other green on the screen now I'm gonna click on my two and I want to make the two purple So you can change the colors so quick and easy Again, you can do follow that toolbar at the top and Animate and do all the different things that you decide you want to do so say I Did want to do some animating in here You know, you can play with it figure out what it does Gives you information allows you to flip things Flip it however you want Crop so you can make it bigger smaller See how I'm playing with that Love cambo. Oh my gosh. So Page animations so you can do the elements where the elements will pop up come and go and you can also do it with Your pages. So say I'm done with page one. I'm gonna go to page two Now again, I can manipulate this in any way shape or form say I don't want this little rolly cart I'm just gonna get rid of it. See I just slid it right off to the side Say I just don't like where all this stuff is positioned I can click on every single thing on this page and I can move it around I can change it. I can make it what I want it to be I don't want the carpet yellow So I've clicked on the yellow box. I want it green. It changed the lines to green I want the yellow to be pink. So I mean I can manipulate any and everything in this presentation and make it Exactly what I want it to be. It is so quick and easy So going to page three again, I can change all of my words I don't like that. This is centered say I decide I want it to be on the side I Can just oh Lord help me. What did I do? Pardon me people. Oh, well here you go This is a good thing that that happened. I messed up, but I'm gonna use it as a teachable moment. I Accidentally got out of my Presentation so now it's taking me to all of my designs. So as you can see, this is a page where everything that I've created I've made my resume in here. I made some end of the year fun Flyers to go home, but this is my design. It was trying to generate it and bring it over But I got bored waiting for it. So I got out of it by accident It was saved right there in all of my design So look I pulled it right back up and it's right where I left off great teachable moment Sorry about that flub it up But it was great for you to learn that in case you didn't know so anyways You can just go through manipulate this in any way shape or form make it what you want If you don't want to use every single template in here, you don't have to so say I've decided that I don't want this second page in here. There's a little trash can at the top. I got rid of it I mean, I brought all 17 pages over, but I've realized that I don't want to use them all So I just click on whichever ones I don't want and I just hit the trash can and it's gone So again, you can make this presentation exactly what you want it to be Okay You've played with it. You've made it what you want. You've changed your colors You've added in your your things again, the elements shows you how to add in any Anything that you want any kind of element then You can go to your uploads. I'm going down the side toolbar for you. So say you can do your uploads. So Right here. See how there's a picture in the middle. I'm going to Add my picture and I want to put that there instead So did you see how I just dragged it over and it automatically filled it in so now my picture is in there And I wanted to say Freckle because that is the freckle app that we use so now my kids know that They're gonna be getting on freckle You can add videos. I Haven't added any myself, but you can add videos. You can add audio. So there's music They also have a Cajillion different Photos already added into cam before you so say you don't have an image that you want to use you can just search in the top and you know Like these are ones I've already looked for so there you go students and teachers say you want to show some kids working together You can click on it and it'll add it right into your presentation. You can change the text. They have some really really really fun Different types of text already created for you With the colors. I mean I couldn't make this stuff myself if I tried so this is why I love canvas so much because It just has all of these fun and amazing things already created for you You just click on it drop it in there But again, I don't like how it looks so I'm gonna change it my toolbar comes up at the top And I can change this and make it anything I want Um, it's so so simple. So let me get back. I got off track went down a rabbit hole So I met text so I found the text that I wanted to add then you could go to your audio So say I don't want that. I'm dragging it off. See how quick and easy that was so say This is my freckle app. I want my friends to work on freckle for five minutes. I can down I can not download but take one of these musics That they've already got uploaded and I can just drag it over and it's gonna go into my presentation I Don't really know what that one says but see it popped up at the top. You can hit play It's gonna put it in the background for you so that maybe you like to play that calming music Why your students work? You can add videos that they've already got created for you. You can change the amount of time Maybe you want this ocean down here as some, you know calming background Waves why your students work? You can also access pictures from your camera roll change the background. So say I love This one up at the top and oh see it changed my entire background, but now that I changed my background I'm gonna have to change my words. So it's quick and easy fun. You can add emojis. You can also download your Your What's that thing called your bit moji and hide that in there? So anyways, I'm done. I've gone through I've shown you all the different ways that you can change this I mean you can just manipulate it and make it what you want you can also Come up here to the top and this purple bar and you can share your designs say you and The you know your team of teachers all are gonna be working on the same design It's just like a Google Doc. You can all go in there and work on it and Make changes to it So that's quick and easy I taught February professional development So the the person I work with we both were able to go into that folder and see everything You can download it to your iPad by hitting the arrow pointing down So that will download it for you or you can go to this arrow pointing up And it shows that you can save it you can assign it you can put it in your Google classroom if that's what you use Save the image. I personally will and you can also just go into Canva and hit present So say you are presenting it to your class You can just have your Canva app up click it and present it You can also hit this share link, which is probably my favorite way to do this And so you can click this top one so you can view the link. So say you needed to make a presentation about Something you're teaching or a concept or a field trip or something that's coming up And you want to share it with your parents you can click this view link copy it and Then you can send that out say in remind or outlook and text it whatever you want to do and Anyone that gets that link will have the ability To see what you've done. They can just view it. They don't have to have a Canva membership They don't have to be at Orikandy schools Employees so anyone can see that then you can share the template down at the bottom So again say you want to share it with you know your coach or your principal or someone And you want to allow them to actually view the template and make changes You could either add them with that first button or you can send them this template link They do have to have a Canva account to do that So I personally use the view link at the top whenever I share these so that anyone can see it and use it so Again today. I was able to show you how to go into Canva and how to make Presentations, I hope that this was useful for you And I hope that you have the option to go in and do this and realize how great it is to use These presentations on your iPad in Canva. You can also do the presentations on your desktop or your laptop But we always have our iPads in our lap. So it's quick and easy So thank you for joining me today and please don't forget to check out our other resources Make sure to like follow and subscribe to all of our dear diss social media pages