GjbKsOkN1Oc.json

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        "text": " Welcome to another Nature of Code video. This one is all about mutual attraction and the",
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        "text": " n-body problem. Now I don't know how you arrived here. This is a follow-up to the previous",
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        "text": " what you expect. This is the example from the end of the last video where every one",
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        "text": " very quickly that formula for gravitational attraction. So in the case here, we have two",
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        "text": " is represented by m1 and m2 in this formula. We can look at the vector that points from",
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        "text": " attracted to that pink dot. But if you go back to, hopefully, when I talked about Newton's",
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        "text": " laws and how forces occur in pairs, the gravitational force that two exerts on one pulling it towards",
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        "text": " it, one exerts on two pulling it towards this. If we have two bodies, like one and two here,",
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        "text": " and there are no other forces in the environment, there's no air resistance, there's no other",
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        "text": " bodies exert on each other and what motion would occur is known as the two-body problem.",
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        "text": " One thing I'd like to mention, if you were about to go and do some research about the",
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        "text": " two-body problem, you would discover that it is a solved problem, that the paths of",
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        "text": " the motion of these bodies can be computed precisely with mathematical equations, starting",
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        "text": " with the formula for gravitational attraction and the equations of motion. To execute those",
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        "text": " solutions requires calculus, and here we are in the nature of code, creative coding, fun",
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        "text": " time landscape, where instead of evaluating the scenario through solving differential",
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        "text": " equations, I'm using this technique that I've described previously as Euler integration,",
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        "text": " into the velocity and the velocity into the position. If we wanted to investigate some",
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        "text": " that improve or change the results that you get from whatever examples I make here? Please",
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        "text": " let me know in the comments. But for creating a sort of first pass at the two-body problem,",
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        "text": " then maybe moving to a three-body problem, and then to an n-body problem, I'm going to",
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        "text": " just stick with my simplistic method known as Euler integration. The other nice thing",
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        "text": " with gravitational attraction. But I think I've shown off my outdoor, it's not really",
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        "text": " outdoor, I'm like slightly inside this garage whiteboard enough, and I can go over and start",
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        "text": " writing some code to step into at least the two-body problem.",
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        "text": " So in looking at this example, while if I wanted to do a two-body simulation, I could",
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        "text": " reduce it to just two separate variables, mover1, mover2. I would like to stick with",
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        "text": " idea of an attractor. There is no distinction now in this example that I want to make between",
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        "text": " mover and attractor. I mean, that's not a real world distinction. That's an artificial",
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        "text": " thing that I created for this idea of attraction simulations and visualizations.",
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        "text": " So now I'm just going to completely get rid of this idea of an attractor. I'm also going",
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        "text": " to take out the reference to the attractor JavaScript file. Now, let me go back and get",
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        "text": " rid of this attractor. And now we have two bodies. Now, they are not exerting any force",
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        "text": " the mover. So now the mover has a function by which it can attract another mover. Now,",
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        "text": " to attract two and mover two to attract one. Of course, they're actually zero and one and",
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        "text": " zero because I'm using an array. In arrays, the index values start at zero. So this is",
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        "text": " now mover zero attracting mover one. And you kind of saw that. So I think what I'd like",
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        "text": " this, mover one is attracted to mover zero. Now, I have mover one attracting mover two.",
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        "text": " And I have mover one attracting mover zero. And you can see this little dance that they",
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        "text": " do with each other. So just to try to understand and see what's going on a bit more, let's",
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        "text": " add back some trails. And let's see if we can give these circles more precise locations",
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        "text": " and with specific initial velocities to see if there's any kind of more expected patterns",
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        "text": " to come out of this. So now I have the mover start on opposite horizontal ends of the canvas.",
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        "text": " Let's see what happens if I start one mover maybe moving directly up and the other one",
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        "text": " start moving directly down. So I'm going to add two additional arguments to the constructor,",
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        "text": " which gives me the x and y components of the velocity vector. Well, I love this. They're",
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        "text": " velocity. And let's increase the strength of the gravitational force. And we get this",
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        "text": " nice, beautiful, sort of like repeating circular dance. Now again, I'm not solving for the",
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        "text": " precise equations of motion. I'm not using the proper units of measurement that map to",
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        "text": " actual celestial bodies in terms of their distance and mass, the universal gravitational",
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        "text": " constant. But I do see this. It makes a lot of sense in terms of what we are seeing here.",
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        "text": " I almost don't want to keep going because I want to stop here and just like play with",
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        "text": " the parameters and variables of the system ad infinitum. But let's add a third body.",
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        "text": " Now I mentioned that the two body problem is a solved problem. The three body problem",
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        "text": " is not a solved one. We could approximate the motion of three bodies using sophisticated",
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        "text": " methods of integration. Let's see how close we get just with my examples right here. I'm",
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        "text": " going to add a third body just in the center. And I'm going to give it no velocity just",
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        "text": " for the moment. It's not participating in this at all. Now what I need to do is I need",
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        "text": " to say, OK, 0 attracts 1. 0 attracts 2. 1 attracts 0 and 2. And 2 attracts 0 and 1.",
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        "text": " Amazingly, the body in the center is not moving because its attraction forces are perfectly",
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        "text": " balanced between those two particles on the outside. That's kind of unbelievable. That's",
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        "text": " really cool. Let's just push it slightly to the left. So this is not super sustainable.",
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        "text": " If I want to have four movers, five movers, I have to have four movers. I have to have",
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        "text": " four movers. If I want to have four movers, five movers, I can't keep individually assigning",
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        "text": " which ones attract the other ones. But I can write a loop that handles this. Now I have",
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        "text": " a loop down here to update and show all the different movers. I want to keep the force",
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        "text": " calculations in a separate loop. But I need a nested one. This loop now, it's not behaving",
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        "text": " the same way. And I'll get to why in a second. But this nested loop is doing exactly what",
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        "text": " this says. Start with mover 0 and have it attract 1 and 2. Then go to mover 1, have",
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        "text": " it attract 0, 2. Then 2, 0, and 1. But there is a bit of a problem because every mover",
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        "text": " attracting every other mover means that the mover will also attract itself because it",
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        "text": " is another mover in the array. So one way that I could approach this is by just saying",
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        "text": " as long as mover is not equal to other, and this is checking if two objects you can check",
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        "text": " same object, then this should now be exactly what I had before, only it's not because I",
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        "text": " have it twice. So I could delete this now. And we should now be seeing exactly the pattern",
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        "text": " of motion of these. There's probably a lot of things you could try doing in terms of",
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        "text": " how you visualize this. One idea would be to connect them with lines. So I'm no longer",
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        "text": " drawing the dots, only the lines. This doesn't look like much yet, but I have a feeling once",
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        "text": " the three start to move a bit more chaotically in relation to each other, we'll see what",
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        "text": " we get here. Curious just what happens if I do four movers and kind of put them in a",
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        "text": " like a bit of a diamond pattern. I don't know what I expected to get, but I'm quite pleased",
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        "text": " with these interesting results, and I think there's a lot for you to sort of try and continue",
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        "text": " playing with. Just out of curiosity, let me go back to making them random. So we can see",
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        "text": " with them random, I'm getting quite an unstable chaotic system. But I think what I'm trying",
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        "text": " to emphasize here is that by thinking about the initial starting conditions of the system,",
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        "text": " the results that you get might have some symmetry or sort of character to them that you could",
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        "text": " take advantage of for a particular kind of design. And there it's off and running. Hi,",
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        "text": " sorry, I'm a few days in the future finishing up this video. I didn't get a chance to finish",
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        "text": " it before the other outfit I was wearing, so excuse the awkward transition. But I'm",
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        "text": " going to look at now what kind of initial positions and velocities could I give an array",
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        "text": " of mover objects in an N body simulation. So let's dive right back in. And since I'm",
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        "text": " them as circles. Here's my P5 canvas. I'd like to set up all the mover objects inside",
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        "text": " of this shaded area. Now, ultimately, if you happen to skip ahead and watch a bunch of",
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        "text": " videos in chapter 3, you're going to learn all about polar coordinates and various trigonometric",
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        "text": " functions that could make doing this in a spiral pattern and all sorts of fun possibilities",
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        "text": " happen for you. But right now, I'm going to make use of the P5 vector function random",
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        "text": " 2D. What this is going to allow me to do is create a vector that points in any random",
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        "text": " direction, like these five here. Then maybe the inner circle here, if I said that was",
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        "text": " then I can set those random vectors' magnitude to a random length between 100 and 150. Offsetting",
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        "text": " function just so I can see. Oh, they all are there. Huh, that doesn't seem right. Oh, I",
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        "text": " vector itself, but another way I could deal with this is just by adding translate into",
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        "text": " draw. So now I have five mover objects anywhere inside this donut. If I were to say 500, we",
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        "text": " can see clearly they're all starting just like I mapped out on the whiteboard. Let's",
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        "text": " see what happens if I let it just run. Explosive total chaos. Let's consider one example mover.",
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        "text": " It just happens to be placed here with this vector mark noting its location. What if I",
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        "text": " gave it an initial velocity that is perpendicular to this vector? So no matter where it starts,",
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        "text": " tangent to this circle. Now I'm starting to use a lot of terms like perpendicular. Its",
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        "text": " angle is 90 degrees. It's tangent. Again, follow into chapter 3 and all will be revealed.",
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        "text": " But I think right now as a quick way of doing this, I can use the rotate function that's",
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        "text": " and give it 90 degrees or pi divided by 2 radians, then I will have taken a vector that",
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        "text": " looks like this and turned it into a vector that looks like this. And again, not to be",
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        "text": " a broken record, if you're wondering about radians and pi, don't worry. The next video",
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        "text": " is going right into that. But stick with me here. This is going to get us something,",
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        "text": " I think. So here v is the velocity vector. Let me name it vel to be a little bit more",
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        "text": " clear. I'm also going to start it as a copy of the position. So now that position vector",
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        "text": " I can take velocity and rotate it by pi divided by 2. That's 90 degrees. And let's see what",
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        "text": " to a constant. Let me lower the value of g quite a bit. And then let me also give",
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        "text": " them a higher starting speed. That was kind of cool. Let's make more of them. I'm going",
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        "text": " to lower that force of gravity even more. And this is what I was hoping to see, this",
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        "text": " almost undulating, spiraling galaxy-like formation. Let's give myself more space. Spread those",
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        "text": " movers a little further out from the center. Give them an even higher initial velocity.",
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        "text": " Let's try 250. I'm quite enjoying this. And let's create a little bit more variation now",
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        "text": " in the mass and initial velocity. I wonder how many of these I can draw before it just",
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        "text": " runs too slow. Already at 350, you can see the frame rate is choppy. Now, why is the",
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        "text": " frame rate choppy? It is not because I am drawing 350 circles. You can see the animation",
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        "text": " is quite smooth if I comment out mover.attractother. It is the fact that I have to check every",
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        "text": " mover against every other mover. This is an n squared problem. 350 times 350 is 122,500",
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        "text": " computational cycles. Let's go back down to 200. And you can see that's probably about",
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        "text": " the limit that I can get just with the way that I've written the code. Now, there are",
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        "text": " many efficiencies that you could add to a system like this to be able to increase the",
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        "text": " number of mover objects that you can draw and the amount of forces that you could calculate.",
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        "text": " Probably the most well-known technique for optimizing an n-body simulation like this",
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        "text": " is the Barnes-Hutt simulation. I've covered something quite close to this in a previous",
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        "text": " coding challenge where I looked at something called a quadtree. So you could go back and",
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        "text": " look at the quadtree coding challenge and bring the quadtree code into this example.",
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        "text": " In fact, I will make sure to do that and include a quadtree Barnes-Hutt version of this n-body",
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        "text": " to show you here, but here you're seeing the result of that. One thing I'm curious to try",
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        "text": " canvas. So I'll make a variable called sun. Sun will be a new mover at 0, 0 with a velocity",
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        "text": " if I gave it a really huge mass? And I don't even need to draw it. Just out of curiosity,",
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        "text": " what I get. Let me go back and add some more trails. And here it is, my n-body sort of",
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        "text": " galaxy-like simulation thing. It's very hard to stop tweaking all the parameters,",
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        "text": " but I'm going to settle on this as kind of the example that I'm going to release. I reduced it",
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        "text": " stable system. A wonderful reference that I would encourage you to take a look at is the paper,",
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        "text": " Classification of Symmetry Groups for Planar N-Body Choreographies by James Multaldi and",
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        "text": " Katrina Steckles. In the paper, they demonstrate a set of initial conditions for n-body systems",
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        "text": " interactive demonstration of these choreographies that was created by Dan Griese that you can play",
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        "text": " around with. Figure 8, super 8, I can change the speed, I can hide the trajectories. I could really",
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        "text": " do this all day. So my proposal to you, my exercise for the viewer, if you're interested,",
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        "text": " is take a look at that paper, take a look at this. What kinds of initial conditions can you",
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        "text": " create in the system that I've set up, and how might you visualize those simulations with color,",
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        "text": " simulation like I'm doing here. And ultimately, my physics simulation, as I talked about earlier",
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        "text": " Nature of Code. Maybe I'll come back and explore some of these ideas further. Let me know in the",
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    "transcription": " Welcome to another Nature of Code video. This one is all about mutual attraction and the n-body problem. Now I don't know how you arrived here. This is a follow-up to the previous video about gravitational attraction, but I'm actually making it almost two years later. I just watched the end of that video, so I think I know, if you're watching this in sequence, what you expect. This is the example from the end of the last video where every one of these orbiting circles is attracted to that pink circle in the center. Let's review very quickly that formula for gravitational attraction. So in the case here, we have two bodies. I'm going to call them body one and body two. They each have a mass. That is what is represented by m1 and m2 in this formula. We can look at the vector that points from one body to the other. We can call that vector r. The magnitude of that vector squared is the distance between the two objects squared. And then this here is the unit vector to describe the direction of that gravitational force. So in my code example, this is the pink dot in the center, and this is the other circle that is orbiting around that that's being attracted to that pink dot. But if you go back to, hopefully, when I talked about Newton's laws and how forces occur in pairs, the gravitational force that two exerts on one pulling it towards it, one exerts on two pulling it towards this. If we have two bodies, like one and two here, and there are no other forces in the environment, there's no air resistance, there's no other things, other bodies, the pure essence of just the gravitational forces that these two bodies exert on each other and what motion would occur is known as the two-body problem. One thing I'd like to mention, if you were about to go and do some research about the two-body problem, you would discover that it is a solved problem, that the paths of the motion of these bodies can be computed precisely with mathematical equations, starting with the formula for gravitational attraction and the equations of motion. To execute those solutions requires calculus, and here we are in the nature of code, creative coding, fun time landscape, where instead of evaluating the scenario through solving differential equations, I'm using this technique that I've described previously as Euler integration, where we just use large time steps, the frames of our animation, and we add acceleration into the velocity and the velocity into the position. If we wanted to investigate some more sophisticated methods for doing, performing this kind of integration, we could look at something called leapfrog integration, we could look at Verlet integration, we could look at Runge-Kutta integration, and I would love to hear from you if you try to execute a version of whatever I make in this video with one of these styles of integration. Does that improve or change the results that you get from whatever examples I make here? Please let me know in the comments. But for creating a sort of first pass at the two-body problem, then maybe moving to a three-body problem, and then to an n-body problem, I'm going to just stick with my simplistic method known as Euler integration. The other nice thing about what we're doing here is I'm foreshadowing some things that are going to come later in the nature of code, in particular in the autonomous agents chapter, because one of the things I want to look at are scenarios like flocking simulations, where every agent moving around a system interacts with every other agent. That's what we are beginning to do here, but with gravitational attraction. But I think I've shown off my outdoor, it's not really outdoor, I'm like slightly inside this garage whiteboard enough, and I can go over and start writing some code to step into at least the two-body problem. So in looking at this example, while if I wanted to do a two-body simulation, I could reduce it to just two separate variables, mover1, mover2. I would like to stick with the array because that's going to help me later. So what I'm going to do is just change the array to having just two elements in it, and I'm also going to get rid of this idea of an attractor. There is no distinction now in this example that I want to make between mover and attractor. I mean, that's not a real world distinction. That's an artificial thing that I created for this idea of attraction simulations and visualizations. So now I'm just going to completely get rid of this idea of an attractor. I'm also going to take out the reference to the attractor JavaScript file. Now, let me go back and get rid of this attractor. And now we have two bodies. Now, they are not exerting any force on each other whatsoever. They're just two bodies with an initial random velocity. I'm going to take the exact attract function I put in the attractor and move it over into the mover. So now the mover has a function by which it can attract another mover. Now, because I have only two mover objects, I can write the explicit functions for mover one to attract two and mover two to attract one. Of course, they're actually zero and one and zero because I'm using an array. In arrays, the index values start at zero. So this is now mover zero attracting mover one. And you kind of saw that. So I think what I'd like to do is just give them very little starting velocity. And we can see every time I run this, mover one is attracted to mover zero. Now, I have mover one attracting mover two. And I have mover one attracting mover zero. And you can see this little dance that they do with each other. So just to try to understand and see what's going on a bit more, let's add back some trails. And let's see if we can give these circles more precise locations and with specific initial velocities to see if there's any kind of more expected patterns to come out of this. So now I have the mover start on opposite horizontal ends of the canvas. Let's see what happens if I start one mover maybe moving directly up and the other one start moving directly down. So I'm going to add two additional arguments to the constructor, which gives me the x and y components of the velocity vector. Well, I love this. They're moving quite slowly. So a couple of things I could do is let's give them a higher initial velocity. And let's increase the strength of the gravitational force. And we get this nice, beautiful, sort of like repeating circular dance. Now again, I'm not solving for the precise equations of motion. I'm not using the proper units of measurement that map to actual celestial bodies in terms of their distance and mass, the universal gravitational constant. But I do see this. It makes a lot of sense in terms of what we are seeing here. I almost don't want to keep going because I want to stop here and just like play with the parameters and variables of the system ad infinitum. But let's add a third body. Now I mentioned that the two body problem is a solved problem. The three body problem is not a solved one. We could approximate the motion of three bodies using sophisticated methods of integration. Let's see how close we get just with my examples right here. I'm going to add a third body just in the center. And I'm going to give it no velocity just for the moment. It's not participating in this at all. Now what I need to do is I need to say, OK, 0 attracts 1. 0 attracts 2. 1 attracts 0 and 2. And 2 attracts 0 and 1. Amazingly, the body in the center is not moving because its attraction forces are perfectly balanced between those two particles on the outside. That's kind of unbelievable. That's really cool. Let's just push it slightly to the left. So this is not super sustainable. If I want to have four movers, five movers, I have to have four movers. I have to have four movers. If I want to have four movers, five movers, I can't keep individually assigning which ones attract the other ones. But I can write a loop that handles this. Now I have a loop down here to update and show all the different movers. I want to keep the force calculations in a separate loop. But I need a nested one. This loop now, it's not behaving the same way. And I'll get to why in a second. But this nested loop is doing exactly what this says. Start with mover 0 and have it attract 1 and 2. Then go to mover 1, have it attract 0, 2. Then 2, 0, and 1. But there is a bit of a problem because every mover attracting every other mover means that the mover will also attract itself because it is another mover in the array. So one way that I could approach this is by just saying as long as mover is not equal to other, and this is checking if two objects you can check if they're equal, meaning they're the same data in memory of the computer, that's the same object, then this should now be exactly what I had before, only it's not because I have it twice. So I could delete this now. And we should now be seeing exactly the pattern of motion of these. There's probably a lot of things you could try doing in terms of how you visualize this. One idea would be to connect them with lines. So I'm no longer drawing the dots, only the lines. This doesn't look like much yet, but I have a feeling once the three start to move a bit more chaotically in relation to each other, we'll see what we get here. Curious just what happens if I do four movers and kind of put them in a like a bit of a diamond pattern. I don't know what I expected to get, but I'm quite pleased with these interesting results, and I think there's a lot for you to sort of try and continue playing with. Just out of curiosity, let me go back to making them random. So we can see with them random, I'm getting quite an unstable chaotic system. But I think what I'm trying to emphasize here is that by thinking about the initial starting conditions of the system, the results that you get might have some symmetry or sort of character to them that you could take advantage of for a particular kind of design. And there it's off and running. Hi, sorry, I'm a few days in the future finishing up this video. I didn't get a chance to finish it before the other outfit I was wearing, so excuse the awkward transition. But I'm going to look at now what kind of initial positions and velocities could I give an array of mover objects in an N body simulation. So let's dive right back in. And since I'm going to create an array of many, many mover objects, I've switched it back to just drawing them as circles. Here's my P5 canvas. I'd like to set up all the mover objects inside of what I might consider a donut. I want to place the mover objects anywhere inside of this shaded area. Now, ultimately, if you happen to skip ahead and watch a bunch of videos in chapter 3, you're going to learn all about polar coordinates and various trigonometric functions that could make doing this in a spiral pattern and all sorts of fun possibilities happen for you. But right now, I'm going to make use of the P5 vector function random 2D. What this is going to allow me to do is create a vector that points in any random direction, like these five here. Then maybe the inner circle here, if I said that was 100 pixels from the center, and the outer circle here was 150 pixels from the center, then I can set those random vectors' magnitude to a random length between 100 and 150. Offsetting from the center of the window, then I have mover objects that are placed randomly around here. Let's try just that first. So instead of setting the x, y as random numbers individually, let's create a random vector. Set the magnitude to between 100 and 150. And then I'm going to give the mover that pos x and pos y position to start. I want to comment out the update function just so I can see. Oh, they all are there. Huh, that doesn't seem right. Oh, I forgot to offset it from the center. So I could adjust the x, y properties of the pos vector itself, but another way I could deal with this is just by adding translate into draw. So now I have five mover objects anywhere inside this donut. If I were to say 500, we can see clearly they're all starting just like I mapped out on the whiteboard. Let's see what happens if I let it just run. Explosive total chaos. Let's consider one example mover. It just happens to be placed here with this vector mark noting its location. What if I gave it an initial velocity that is perpendicular to this vector? So no matter where it starts, its initial velocity is always sending it in the direction which would be essentially tangent to this circle. Now I'm starting to use a lot of terms like perpendicular. Its angle is 90 degrees. It's tangent. Again, follow into chapter 3 and all will be revealed. But I think right now as a quick way of doing this, I can use the rotate function that's embedded in the p5 vector class itself. If I take this particular vector and call rotate and give it 90 degrees or pi divided by 2 radians, then I will have taken a vector that looks like this and turned it into a vector that looks like this. And again, not to be a broken record, if you're wondering about radians and pi, don't worry. The next video is going right into that. But stick with me here. This is going to get us something, I think. So here v is the velocity vector. Let me name it vel to be a little bit more clear. I'm also going to start it as a copy of the position. So now that position vector is copied into velocity before I've resized it. And then after I've resized position, I can take velocity and rotate it by pi divided by 2. That's 90 degrees. And let's see what happens. Oh, I named it vel and I didn't rename it here. Well, we've still got quite a jumbled mess of stuff going on. Let's take out the trails. I think the relative scale of the forces is too high. Just to get a handle on things, let me set all of the mass values to a constant. Let me lower the value of g quite a bit. And then let me also give them a higher starting speed. That was kind of cool. Let's make more of them. I'm going to lower that force of gravity even more. And this is what I was hoping to see, this almost undulating, spiraling galaxy-like formation. Let's give myself more space. Spread those movers a little further out from the center. Give them an even higher initial velocity. Let's try 250. I'm quite enjoying this. And let's create a little bit more variation now in the mass and initial velocity. I wonder how many of these I can draw before it just runs too slow. Already at 350, you can see the frame rate is choppy. Now, why is the frame rate choppy? It is not because I am drawing 350 circles. You can see the animation is quite smooth if I comment out mover.attractother. It is the fact that I have to check every mover against every other mover. This is an n squared problem. 350 times 350 is 122,500 computational cycles. Let's go back down to 200. And you can see that's probably about the limit that I can get just with the way that I've written the code. Now, there are many efficiencies that you could add to a system like this to be able to increase the number of mover objects that you can draw and the amount of forces that you could calculate. Probably the most well-known technique for optimizing an n-body simulation like this is the Barnes-Hutt simulation. I've covered something quite close to this in a previous coding challenge where I looked at something called a quadtree. So you could go back and look at the quadtree coding challenge and bring the quadtree code into this example. In fact, I will make sure to do that and include a quadtree Barnes-Hutt version of this n-body simulation linked from the video's description. So right now, I don't know what I'm going to show you here, but here you're seeing the result of that. One thing I'm curious to try is what happens if I bring back, somewhat artificially, this idea of an attractor, a stationary object that's maybe with a larger mass positioned right in the center of the canvas. So I'll make a variable called sun. Sun will be a new mover at 0, 0 with a velocity of 0, 0 and a mass of 100. Let me say sun.show. There it is right there. But now, I want the sun to attract every mover. It's hard for me to tell if something really different is going on here, but I do feel that the system itself is not kind of wandering off. But also, you can see that the power of that object in the center is somewhat overwhelming. What if I gave it a really huge mass? And I don't even need to draw it. Just out of curiosity, if I lower the gravitational forces quite a bit, this is kind of interesting to see what I get. Let me go back and add some more trails. And here it is, my n-body sort of galaxy-like simulation thing. It's very hard to stop tweaking all the parameters, but I'm going to settle on this as kind of the example that I'm going to release. I reduced it just down to 10, and I constrain the ranges of masses and velocities to just create this stable system. A wonderful reference that I would encourage you to take a look at is the paper, Classification of Symmetry Groups for Planar N-Body Choreographies by James Multaldi and Katrina Steckles. In the paper, they demonstrate a set of initial conditions for n-body systems that create these symmetrical and stable choreographies. There's also a beautiful interactive demonstration of these choreographies that was created by Dan Griese that you can play around with. Figure 8, super 8, I can change the speed, I can hide the trajectories. I could really do this all day. So my proposal to you, my exercise for the viewer, if you're interested, is take a look at that paper, take a look at this. What kinds of initial conditions can you create in the system that I've set up, and how might you visualize those simulations with color, with trails, with shapes? So many possibilities. I should mention that this particular interactive simulation, these paths are drawn with precise mathematical equations, not with a physics simulation like I'm doing here. And ultimately, my physics simulation, as I talked about earlier in the video, is a very crude one with these large time steps. So this wraps up chapter 2 of Nature of Code. Maybe I'll come back and explore some of these ideas further. Let me know in the comments if there's particular things that I proposed or talked about here that you'd really like to see in a video. But now, it's really up to you. Do you want to go on to chapter 3? I hope so. There's lots of fun stuff in there. All sorts of things about angular motion, rotation, polar coordinates, pendulums, springs. Have fun with that, and I will see you next time on The Coding Train.",
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