Crash de Washington : les Mathématiques du Désastre

Good evening my friends. Today we're going to talk about the mathematics behind mid air collisions. Behind these accidents were two new aircraft collide mid air. To try to think about this mathematics, I ask for help. At the instigation of some commenters on my channel, I asked Chat GPT for help with the paid version. ChatGPT on his mathematics. Not only did he get it wrong once, but he got it wrong twice and he even apologized. So to tell you that he was completely useless. This answer is for those who say, will the plane of the future be piloted by artificial intelligence? You would have a lot of worries if it were. So today we're going to start by talking about a little update on the accident in Washington last week. Not even a few days ago.

A.

Passenger plane that came in to land at Ronald Reagan Airport in Washington, not far from the White House collided with a military helicopter that was flying over the Potomac. Their trajectory converged and they apparently didn't see each other until the last second. Colliding with each other. 67 people died. The worst accident in the United States in a decade. So we know more. We know, for example, that the air traffic controller had asked the helicopter a few seconds before the impact had asked, do you see the plane? The helicopter pilot replied in the affirmative. However, he didn't see it and he crashed into it. There is perhaps a small detail that I don't think has been sufficiently exploited by the various contributors to this subject. Let me tell you about the plane. If we look at his communication with the air traffic controller, he was doing a visual approach, Right. Very often in the United States, he asked the pilots, you see the facilities, you see the Runway? Yes. So continue visually and maintain visual separation. That means he's passing the responsibility to him. You continue your navigation visually, you're going to land. And also, well, you're separating yourself from the others visually avoid them. The air traffic controller continues to monitor his radar and when he sees a converging trajectory, he'll inform the pilots. Pay attention to what you see, such and such aircraft and so on. What information was not used. Take a good look at the aircraft.

What.

What was its intention? The first intention was to land on Runway 01. But where did it want to land at the time of impact? Runway 33. When did the change occur? Relatively late, shortly before landing. By this time, if the controller had not intervened to make him change runways, this accident would never have happened. Shortly before landing, the controller asked him, can you give me Runway 33? The other one says, yeah, okay, go ahead. On Runway 33.

Okay.

So the plane left the centerline at the last second and did a sort of loop to land on the other Runway. This potentially surprised the helicopter pilot. Why? Because he was flying in the area for training. He saw the planes lined up on the centerline for Runway 01, and that reassured him. He had this impression that they were all coming down that path. And suddenly one of them left the centerline, went to the other side, and then came back to the left and bam. Do you see what happened? There was this deviation. So let me tell you something. Many international airlines, large airlines, I'm not talking about regional airlines, many airlines of large airlines would have refused such an instruction. Why? Because they have what are called approach stabilization criteria. There comes a time when the approach must be stable. That means we are aligned on the Runway. We are on the axis, we are on the glide path. We have the right speed, we have the right configuration of the aircraft. We are perfect. We just have to make small adjustments, small corrections, until we land. Well, if we ask at the last minute to leave the axis and go to land on another one, it is not possible because the stabilization criteria will no longer be respected. How do we change runways in this case? If we have to, we put the power back on, we take off, we do a big loop, we come back to present ourselves 20 minutes later on the Runway, on the other Runway that was given. But in no case, we just change the Runway like that, improvised at the last second. This is the case for many airlines, just to tell you. But in any case, in my opinion, the helicopter pilot could have been surprised by this situation. Something else the pilots were also criticized for. The helicopter is not allowed to fly at an altitude above 200ft because the pilots have to stay below 200ft. That's extremely low. It's very hard to fly very low, especially at night. Even during the day, it's hard. There's the risk of getting hit by an electric pole, a wall, a tower, even if there aren't many of them. You know, Washington is a swamp. It's built on swamp, the famous swamp that we're talking about. That's why Washington is not New York. There are no skyscrapers in Washington because the ground is very soft. But the fact remains that when you have to fly low, it's hard because there's always the risk of hitting something. So the altitude meter can't help, because the altitude meter indicates either zero or even negative values sometimes. So the only way to add a little bit of personal comfort, it's trying to fly Slightly higher. So probably we'll see later why there are black boxes and all that. But we can assume that, for example, the helicopter pilot, to improve his comfort, says to come on instead of flying at 100ft, 120ft, which is very, very low, which requires very increased vigilance so as not to crash into an obstacle. I'm going to fly a little higher, let's say 250ft, 300ft. I'm higher than all of Washington there. I'm not at risk anymore there. There are no more structures. There's almost nothing that can hit me. And so I relax a little. And that's where a plane crashes. So we're going to talk a little bit about mathematics and I'm going to show you a little bit where the GPT cat crashed. So, first of all, so that we can see an object approaching us. And this is valid in aviation, but it also applies to road safety. There are two conditions that must be met. The first condition is that the apparent size of the object must be large enough for it to appear in our field of vision and be prominent. If its apparent size, its apparent angle is extremely small, we risk not noticing it. Secondly, what is the other thing? The other thing is movement. The more an object moves in our field of vision, the more likely we are to spot it, to see it. If it moves like this, for example, if I take lateral vision, which is very. I can't read an object here. I can't read a page if I put it here. On the other hand, if something moves there, I will see it. Our peripheral vision is very good at detecting moving objects, except that when airplanes land, well, sometimes we don't see them. They don't move at all. Of course, they are flying, they are moving, and they are even moving very quickly compared to a fixed frame of reference. But sometimes I'll show you an image. Sometimes it doesn't move at all in the frame of reference of another plane. About frames of reference, there are a lot of people who said, well, I don't understand how this helicopter and this plane didn't see each other. Because I look at the images, it's pitch black. They both stand out very, very well in the night. I don't see why they can't be seen. Yes, but this vision that we have, this vision of surveillance cameras or people filming from the ground, this vision does not exist for pilots because we are on the ground, the camera is on the ground, and we see these planes silhouetted against a black sky. That is not the vision of Pilots, These pilots are above and they see a brilliant ultra bright ground below. Taking the plane for example, that is descending, what does it see? Its window is full of all of Washington. Lights, lights, lights everywhere. Runway lights, approach lights, highways, cars, all that, all, all that lighting. And in there there is a helicopter. Yes, there is a helicopter navigation light there, which is black, by the way. And on top of that, it is night. So as you can imagine, these are not optimal conditions for us to be able to see an aircraft. It is not, it is not easy. Now, compared to the plane that doesn't move, I'm going to show you a video. Notice this plane that remains at a constant angle. When we see a plane at a constant angle, we say it has a constant bearing, right? That means we're going to hit it intuitively. It can be reassuring to say, hey, it's there, it's not moving, it's at an angle, it's not moving. We might have the impression that it's okay, it'll pass. Well, actually we would have preferred it to move. We would have preferred it to pass by. Either it passes in this direction or it passes in the other direction. But an object that remains fixed, if you arrive at a crossroads and you see a vehicle, a motorcyclist or something else that is fixed at your angle in your line of sight, it's guaranteed to hit you. If after a few seconds it doesn't move, you're going to hit it. You see, I'm going to show you the math behind it. So remember, you all know my notebook, right? I found it, I had to look for it, right? It was used for drawing, for painting. Mind you, there's a really nice piece of art on it, right? So I had to find another one that I'm going to use. So in the video that I showed you, right, we have a plane that is in constant bearing. What does that mean? Geometrically? What does it mean? It gives similar triangles. I'll show you. We're just trying to show you. I'm going to try to do this. Look. Now we have a first plane that's here, okay? Do you see it? First plane that's there. Second plane that's there. So this pilot who's here sees the other one. He sees this pilot going, lets say like this, for example, ok? And this pilot here is going like this. It doesn't have to be an angle, right?

Huh?

I didn't draw a right angle. I wanted to make an angle that was maybe a little bit more open, but that's not important. It's not very open, but that's not important. So he sees it from this angle.

Okay?

He sees it from this weapon. A little further on, a few seconds later, 10, 15 seconds later, the two planes have moved. They don't need to have the same speed. The two planes have moved. Notice what happens. That's the new situation. See that? So here what we have. That's a very good situation.

Okay?

So what we have, we have a constant angle that doesn't move geometrically. We're talking about similar triangles. These are triangles that in my case here, have the same angles, and they actually have the same proportions too.

Okay?

So these are similar triangles that start with a large triangle and then we're going to have smaller and smaller triangles, but that have the same angles until we reach the three points of the triangle. So the first plane, the second plane, and here is the supposed location of the tragedy of the accident that is there. And all these three points will suddenly collapse. Suddenly, these three points will merge with each other. That means the two planes and this point will all be in the same place. And at that moment, it is the impact.

Okay?

The danger with this is that I told you that it can be reassuring because we see a constant angle, so we are not afraid. Another thing, another thing is that. Imagine that we have, for example, a structural element of the plane or a screen or something that hides this device. It always arrives at a constant angle and it is hidden. So if we sit like this, we look, we do not see it. It is there. It has been arriving for a long time, but it remains at a constant angle. Rule, Rule. It remains at a constant angle until we pick it up, until it suddenly grows and then, bam, it explodes, you see? So that's also the danger. That's why sometimes when you're driving a car and you're holding the car upright, sometimes instinctively you end up doing that. For example, when you're on a bend, you end up going backwards and then going forwards to cover more angles to try to see if there's something hiding behind the clock. Now, the size. I talked about movement and I told you now that you can have a plane that is stationary and in fact is coming towards you stationary in the direction of constant bearing. Now, something else. The apparent size. What is the apparent size? I'm going to show you where the GPT cat is collected. In fact, the apparent size follows a very particular function. It's a function called arctangent. It's a function that means that for a very long time, to make simple, for A very long time. The apparent size of an object approaching us remains relatively small, varying very little with distance. This means the object is getting closer. But while it was at 10,000 meters, it is now at 5,000. The relative size has hardly changed. Then it goes from 5,000 to 3,000. The relative size has hardly changed. This means the angle from which we see it, the apparent angle remains ultra small, changes very little until we are very close. And then suddenly the size increases very sharply. It fills all the windows. And then it is too late, like in the case of Washington. They must have come together. They must have seen each other maybe a second and a half before impact, or maybe not at all. It was too late. Impact and that's it. So this unfortunately creates a trap, this visual trap. It's like when you get caught in front of a municipal field by train, watching people play football. A spectator like that, eating without popcorn, and suddenly a ball hits you in the face. How does the ball see it? We didn't see it. And when we're on the ground, what do we see? We only have one or two images of the ball. We have the image maybe three or four meters away, which is already like this. And then the next image, it's already completely black. That means the ball was small. And just before the last seconds, the last half before we hit it, its apparent size increased until it completely filled our view. But now it's too late. You see? So the function is called arctangent. I'll write it down. So this is the function. It means alpha. Alpha is the apparent angle, right? It's equal to two times. And so this is our arctangent function of the wingspan of the plane divided by two times the distance. This is the function that allows us to explain this relationship between alpha and distance. The span is a constant, so the span is not important here because it remains a constant value. Now, where did the GPT cat end up? Let me tell you. I told you that despite the fact that I studied mathematics at the Polytechnique, I practically stopped at primary school. That means that frankly, I don't know how to do it. My brain only knows how to do division, addition, multiplication, subtraction. That's what my six year old daughter already does. I haven't learned anything else in reality. So when I see a function called arctangent, I don't have the arctangent function in my head.

Ok?

Yes. I'm no better than anyone else. You tell me arctangent of zero. I tell you zero. Yes, but I know that because I Know it by heart because I've used it, that's all. I can find it in tables, specific values where you can use a calculator. But mentally I don't have a cross. I'm not equipped to tell you what the arctangent is. So when I hear the word arctangent and a bunch of other functions, uh, I'm going to explain them differently. So how else? I'll tell you When I see arctangent, I think like this. For me, the arctangent of this. So alpha is roughly equal to sigma. It's a sum.

Okay.

We'Re going to start from N to N infinity. I'll explain more or less what that means in a second. Minus 1 Caret 0 by x Caret 2 n + 1/2 n + 1. That seems like too much. It seems terrible. It's very simple. Stay, my friends. Don't. Don't run away. Stay. This is elementary school level math. It works better in a human's head than arc tangent. Ar tangent doesn't mean anything. It means. I'm telling you, I'm telling you and we'll talk about it. Look, in my case here, this is a sum. And each time I take an N, it's going to create a new element of this sum, a new member of this sum. So now we're going to start doing the sum here to show you where GPT messed up on something really simple. You'll see. It's really simple. So we're going to start with n equals 0 to 1 carat ISS 0. What's that one? Okay, multiply by one. It's like we didn't do anything. There you have it. X 2 xn 1, that's x carat 1, that's x and that's divided by 1. So that's x. Okay, let's continue. Now, 1 carat ISS 1 is minus 1. OK, so it's minus minus.

What.

If we have 1 that's Xcarat 3 divided by 3 plus x caret ISS 5 fifths Xcarat 7 7. Can you guess the sequence? Well, you can guess it. It's x caret plus x caret 9/9. The 2N 2XN1 is simply creating odd numbers for me. And the minus 1 to the power of N, it's doing this sometimes it gives me plus once minus once plus and once minus. It alternates plus minus plus minus plus. Okay, so that's what's called a Tyler sequence. Quite simply, it's transforming something complicated, arctangent, a complicated function, into a Simple function.

Okay.

Almost all functions can be approximated by a Tyler sequence. The precision depends on what? Well, the precision depends on how many n's we're going to take. If I take N equals two or three, right? That means if I add two or three orders here, well, I would have average precision, but it's up to me to add 50 if I want, or 10 or 15 or 100. You know what I mean? So that allows us to approximate things in a pretty impressive way. By the way, when I say in a pretty impressive way, because there are people who hear the word approximation or approximation, you shouldn't think about it in the pejorative sense of the term. Look, if you're machining something, you're manufacturing something in your instrument, your tool is capable of working to the tenth of a millimeter. Are you going to do a calculation for this tool to the billionth of a millimeter? No, it's useless. If you manage to do a calculation that has the same level of precision that your tool is more than enough, anything beyond that will be useless. It's a waste. It's wasted energy. These are calculations that you're going to do, but that won't be of any use to you. You see, it's like, I don't know, you wanted to paint this room and you're going to measure the surface area of the walls and you're going to start measuring them to 100 decimal places to buy your paint. It doesn't make sense. It doesn't make sense. If you can figure out the surface area or put it square, that's more than enough. You see, every application needs a level of precision. And when we say approximation, it means that we're going to match. We're going to go with our desired level of precision and we're going to make the necessary approximation. And our approximation is well within the level of precision that we have with others. So I was saying I did that right, And I thought, okay, instead of calculating the Tyler sequence of arctangent 1x myself, I'll ask Chad. GPT. I told him, go ahead, do it for me. I said, okay. There are so many people who come to the comments to advertise chat. GPT.

Okay, go ahead.

We'll give in. We'll ask him. He made a mistake. He made a mistake. I told him, but what you did is wrong. It's not possible. What you're saying isn't true because visually it didn't work. And then I said, yes, you're right, I'll correct all that. He corrected it.

Okay.

A little later, I asked him about the variation of the function. It's linear. It's not possible. You're talking nonsense, my friend. Oh, yes. No, it's not linear. Yes. So to tell you, my friends, this is not about criticizing Chad. I farted here. I'd like to tell you something. I'm old school. Never do a calculation without knowing roughly what the order of magnitude is that you're going to get. Because that's how you make mistakes. That's how you make mistakes. Never trust an instrument, never trust a computer, a calculator, a program, because they make mistakes, especially when your life depends on it. I know people. I knew someone who trusted them, and they died. You see? You shouldn't trust them. Always have an idea. I'm not asking you to do the math in your head before doing a calculation with a calculator or a program. No, that's not it. I'm not telling you to go and do it with a sheet of paper or a pen to compare.

No.

Just always have an order of magnitude of what you're looking for. Also, if you're looking for a function, what kind of function it is, how many degrees it is, how it varies and so on. Those kinds of things that allow you to have a good nose for errors. And when there's an aberration, it sounds like a wrong note. You don't necessarily know the right note, but you know that this note is wrong. There you go. That's the idea. So I was telling you that variation is a variation that is hyperbolic in nature. There too, you're wrong. Luckily, the old school is there. Honestly, I had a really lonely moment. I won't hide it from you because I said to myself, we can't let go for a second. We can't let go for a second. We can't trust anything. We're forced to be alone. We're forced to. I said to myself, what are we going to do when the old school is no longer there? The Earth will make up for everything, right?

Okay, let's go.

Go on a negotiation trip, like the reports, but you understand me. So, one thing. The variation, the variation in the apparent size of the apparent angle of an object approaching you follow a law called hyperbolic. There you go. That's the idea. It's that when you look, you can approximate. I'll make it short. You need a big table. Actually, if I want to do more math, you need a big table. You have to approximate this to 1 over D. That's it. I Messed up. I told you it's linear. It's not linear.

It's not.

It's hyperbolic. Why is it hyperbolic? Because if I have that, it means that alpha XD1. That's hyperbolic. Why? I'll tell you a story. It's just between us, we like these things. So we're talking about, uh. Look, my friends, if I tell you a set of points, a set of. Listen carefully to this one. A set of points that are equidistant from a point. There's a fixed point. There's a set of points that are equidistant from this point. It's. What is the shape? It's a. It's a circle. Yes, it's a circle. It's a circle, isn't it? There you go. I have a point here, and I have a set of points that are equal distances apart. Well, yes, I just formed a circle. Now, if I tell you a set of points in which the sum of the distances from two points is constant, what is that? So I have a point here, a point here, and I have a set of points in which the sum of the distances from these two points is constant. Many people will tell me it's an oval. Yes. That means if I take any point in this oval, this one, for example, this distance a one plus a two, it is equal to all the other points you take any point you take. This one, a one plus a two always equals always. All these points there have this characteristic, by the way. That's why, you see, this one is close to this point. It moves away from that one, for example, you see, because the sum must remain constant. It's simply geometry or mathematics explained in French. Quite simply, if I tell you the set of points that are equidistant from a point and a line, what is that? Equal distance from a point and a line. It's a parabola. Now, if I tell you that's why these definitions are important, you need to know them because they help later on in reasoning to discover errors like here and error in artificial intelligence. What is the set of points in the sum or in the product? Not the sum. The product of the coordinates is a constant. Let's take a constant equal to one, for example, or equal to a span in the coordinates that interest us. And taking points in the coordinates, xy XY will always be equal to E. Always constant. What do we call that? We call it a hyperbola. Do you see the thing? All the shapes that surround us that we know have a mathematical geometric Definition, but also a definition in good English. This is the geometric definition that I am giving you. It finally gives a cross to all of this. It gives a cross. It gives what? I am going to show you Arctangent of what I told you. Arc alpha equals times arctangent of the span divided by 2 times the distance. If I make it simple on a curve, it gives you this. This is our alpha. This is our distance. Alpha apparent angle, distance, distance of the observer, right?

So.

So it gives me this. There are people who are watching, okay? To people who are always complaining about the scissors. So look at this. Look at the explosive side. Widows and orphans are explained by this curve, my friends. Look, let's take this first zone, this one. What is happening in this first zone, my friends? The delta D distance varies enormously. Look at this distance as it varies. Did you see the distance? Big distance, right? If I look at alpha, that's delta alpha. Compare, compare Delta alpha. Compare it right to Delta D. Great distance, Great variation in distance, right? Great distance. Very little variation in angle apart. And yes, that's the hyperbola, you see? Now, when the object approaches you, when the object is close, be careful. Things are going to change. Here we. We enter the explosive vertical part of the hyperbola. Here the slightest variation in distance will cause a large variation in angle until it completely fills your window. But it's too late to react. That's why it kills. Look, here we have, right? And I'm going to make a small variation in distance right here. And we'll see what that gives.

Okay?

So here what you're going to see is a small variation in distance. Look at this delta D. And look at what it corresponds to in delta alpha. Look, small delta D here, large delta alpha, large variation in apparent angle. That's the problem, my friends. That's the problem.

That.

That's what kills. Because towards the end of the story, towards the end, when the drama is finally unraveling, when we arrive at the crime scene at that moment, what do we have? A small variation in distance in the large variation in apparent angle. That means, in other words, the plane, the helicopter, the object, the microlight, whatever grows very quickly. The biker, who for one thing, grows very quickly in your window, in your sky. Too late to react.

Okay?

That's it. That's how it works. That's why people die. Because as I've been telling you since the beginning of this video, we have two problems. The first problem is the constant angle, the constant bearing. It doesn't move. It's supposed to move, but it doesn't. And if it's hidden behind a screen, behind an upright, behind a stain, behind anything, we won't see it. It'll stay there until it completely fills the space. Secondly, the fact that it remains very small means that the variation of the apparent angle as a function of distance follows a hyperbolic function. Roughly speaking, we're on an arc tangent. Well, what's going to happen? For a long time, it's going to stay flat, it's going to barely move. Then suddenly, in the last second, it fills up. Bam. Okay, that's the big problem. These things aren't linear, okay? And the third problem is that you should never trust trust. Because when you trust trust, you become trusting. In fact, the problem is that you should never trust artificial intelligence, computers, calculators, anything, all these things. You always have an idea of what you are constantly calculating. You always have a preconceived idea like that. If there is ever something wrong, you say to yourself, okay, let's look at this. There is a little surprise. There is something that doesn't add up. Let's look at it.

Okay?

Thank you, my friends, for watching this video so far. I will explain it to you one day. Another day. If this video interests you, I'm not even sure if I should publish it. Too much math, I think. I don't know, it scares people away. Even if it's inclusive math. I'm doing simple things, if you've noticed. I'm simplifying things to the extreme. But the word mathematics still scares people. There you go, my friends. See you in the next video. And until then, take care of yourselves.