Square Orbits Part 2: Moons Upon Moons

[@B0Boman]

I've heard a moon of a moon called a "second order moon" before, so the final moon here could be called a "hundredth order moon". Although I prefer the terms Twoon, Throon, Foon, Fivoon, Soon, Sevoon, Eioon, Noon... all the way up to Hundroon!

[@videogamesarecool9280]

should have gotton up to eightoon so that you could have called 17 suboptimal rather than seventoon

[@deltamico]

Tbf then the Moon should be Onoon, or base it from mono -> Doon, Troon, Quatoon...

[@gaopinghu7332]

Soon?

[@deltamico]

@@gaopinghu7332 soon da Soon and Noon will be in noon

[@nos9784]

@@videogamesarecool9280 you meant to say "suboptimoon", right? :D (For anyone wondering- if you even exist: KZbinr Jan misali's channel has great videos on different bases for number systems. Base 17 is supoptimal.) I think calling 17 subobtimal is misleading. 18 isn't optimal. Its _three_ times six.

[@themakuta]

4:35 A pretty strange looking trajectory, indeed...

[@olivergarrett7973]

I was looking for this exactly

[@stefanalecu9532]

That's a nice sideways hat, what do you mean? :^)

[@kaminachos5129]

thank you for reassuring me that I saw what I saw.

[@buidinhnguyenngoc4326]

looking very sus

[@legofreak5769]

its a rocket ship

[@aaronspeedy7780]

This is a great way to teach Fourier series!

[@AllThingsPhysicsYouTube]

Thanks! I thought so too! :)

[@lucythebrazen]

@@AllThingsPhysicsKZbin While I absolutely agree, doing it 1 dimensional seems a bit more approachable? Maybe I'm biased, because that is the way I learned about it, in the context of approximating 1D functions using sin/cos functions. Expanding that to a second, complex dimension afterwards seems trivial. In the end it doesn't make a difference, because if you understand one, you'll understand the other, either as a simplification, or an expansion on what you've learned. I guess that could just boil down to the slightly different approaches people with an engineering background compared to people with a more theoretical background take...

[@tamelo]

I thought about Fourier after the first video. I'm glad I still remember Calculus 15 years after graduating.

[@taproot0619]

When I saw the title of the 1st video, I immediately thought about fourier series. I was pleasantly surprised when he was able to achieve something very squarish with just a three body system. I am glad he did go into fourier series eventually though.

[@wissahdahmastah]

I am a mathematician, but the physics approach to Fourier series is really nice and intuitive, keep up the good work!

[@silmarian]

Agreed! I'm, at best, good at arithmetic and interested in mathematics. Typical middle-aged mom stuff ;). This (and the previous video) really clarified my understanding of how the Fourier series/transformation works.

[@AllThingsPhysicsYouTube]

Thanks, will do!

[@AllThingsPhysicsYouTube]

So glad to hear that!

[@matteoinventore7496]

He's like the physics's Bob Ross 🥲

[@1234567890Pirlo]

nicely said, surely agree, and thats quite a compliment

[@rpyrat]

Or the physic's 3Blue1Brown : )

[@silmarian]

Happy little orbits

[@superscatboy]

If Bob Ross, 3Blue1Brown and Weird Al had a baby

[@tristenarctician6910]

4:25 so artistic

[@iveharzing]

I have learned about Fourier Series and Fourier Transform in university, so I kind of understood how they worked, but when you said: "make the vector stop rotating" it REALLY clicked for me! Thank you for this amazing video, I can't wait to see what you will make next!

[@AllThingsPhysicsYouTube]

Glad to hear the video helped! Feel free to share with others who might be interested!

[@josiahtaylor8714]

I can see this channel blowing up to 100k pretty quick, and I’m glad to be here for the early days and proud to be one of the first 5000 people to subscribe

[@AllThingsPhysicsYouTube]

I really hope you're right. I'm really new to this and don't really know how to get the channel to truly blow up, so please feel free to share the video with others you think would like it!

[@frogz]

ditto, im here at 7.5k subs after 2 weeks from this post, it is going to grow big :)

[@AllThingsPhysicsYouTube]

@@frogz Once again, I hope you're right!!

[@MatesMike]

What an amazing video, congrats 😊

[@AllThingsPhysicsYouTube]

Glad you enjoyed it!

[@samicalvo4560]

Grandeee Mike. Este canal tiene todos los números de ser el 1B1B de la fisica 😂

[@AllThingsPhysicsYouTube]

@@samicalvo4560 Thank you for the kind words (Gracias por las amables palabras).

[@cara-seyun]

OMG!! Thank u!! I’ve been trying to make a square orbit for my planet system for the longest time. Very helpful, will recommend to any others who want square moon orbits.

[@harriehausenman8623]

You have a planet system? Cool! Any chance you have a left-over moon or something? Doesn't have to be big, athmosphere not necessary (I bring my own), but shouldn't be too close to asteroid belts. I'm looking for non-earth based real estate. 🙂

[@harriehausenman8623]

Forgot to mention: and should be AI friendly ;-)

[@ivanpalomo1586]

I imediately thought of 3blue1brown when watching your videos but with a more physics aproach. cool video and I really liked it

[@harriehausenman8623]

WOW! That's what I call a proper compliment 😍

[@harriehausenman8623]

And Manim always shows.

[@AllThingsPhysicsYouTube]

There's a minor issue in this video from 10:26-10:29. I don't know exactly what happened here; I typically triple check everything before uploading, but somehow I must have uploaded the not-quite-final version. Fortunately, it looks like this the only problem. Sigh. Unfortunately, there's no way to fix an uploaded KZbin video without deleting it, so I guess it's going to stay this way. Apologies for the oversight.

[@ishanagarwal766]

your channel is a real gem with the quality of videos. hope it blows up soon

[@AllThingsPhysicsYouTube]

You and me both! Please feel free to share with others who you think might be interested!

[@ishanagarwal766]

@@AllThingsPhysicsKZbin i already did share it.

[@AllThingsPhysicsYouTube]

@@ishanagarwal766 Great! Thanks so much!

[@harriehausenman8623]

Great Video! I like the style :-) Good editing, fine choice of background music, excellent visuals, keep them coming!

[@AllThingsPhysicsYouTube]

Awesome, thank you!

[@harjutapa]

This is one of the best channels on KZbin. I hope you get more subs and views.

[@AllThingsPhysicsYouTube]

Wow, thank you! I hope so too. Please feel free to share the video with others who might like it!

[@nexerkarigum4031]

I understand almost nothing but it's absurdly relaxing to listening to it

[@BillySugger1965]

Yes I too watched Grant’s video and as an electrical engineer I really enjoyed his treatment of the Fourier series. But I also have a fascination for orbital mechanics and was especially taken with your version of it. A wonderful journey, thank you very much!

[@AllThingsPhysicsYouTube]

Cool. Thanks!

[@ZomB1986]

I'm so happy that you addressed the glaring shortcoming of 3B1B's video: Varying the speed along the curve to obtain different (and possibly better) coefficients!

[@AllThingsPhysicsYouTube]

Well, I'm not sure I'd call this a "glaring shortcoming," but I must admit that I found this pretty surprising initially.

[@ZomB1986]

​@@AllThingsPhysicsKZbin I encountered this problem during an uni course for reverse-3d-model-searching, where my team used FFTs of shadow boundaries of 3D models in 12 dodecahedral orientations.

[@ProgressiveEconomicsSupporter]

Grant of 3Blue1Brown is great! Thank you for this deep dive homage to him 😎🙏🇩🇪

[@cionnar]

I'm amazed you managed to go a video and a half before finally naming the Fourier series. Very nice presentation.

[@AllThingsPhysicsYouTube]

We’ll, in the first video, it wasn’t really a Fourier series yet!

[@spiveeforever7093]

This is amazing. My brother in high school sent me a video explaining Fourier series from scratch, and you actually did it line by line. Great work.

[@AllThingsPhysicsYouTube]

Thanks. Please consider subscribing and forward the video to anyone else you think might like it!

[@X3MgamePlays]

This whole video was wholesome. First I thought, does he know about the Fourier series? And there it evolved. Then I thought, does he know about the accelaration in the corners? And you simply mention it like 5 minutes later. Lastly I thought, should I share a link to a particular video that draws pictures with the Fourier series? And you simply mention the channel. 19:40 And here is where the beauty of the system begins. 22:38 Yes, I, Am! The only thing missing... what happens if you have the objects gravitiy influence each other as well? What would happen over time? You need mass (density times size, so the size can be made small if needed) and the distance between the objects. And of course the movement speeds with a direction at time = 0. Maybe you can plot the systems in universe sandbox to see what happens.

[@adissentingopinion848]

You have done exactly what I had dreamed of, to present much like 3blue1brown a mathematical concept so plainly to see, even with intense mathematical processes. I'm more tempted than ever now...

[@AllThingsPhysicsYouTube]

Cool! Good luck!

[@gabedarrett1301]

_This_ is how math (or anything else complicated) should be taught in school! It should give the simple big picture explanation, be intuitive, and explain the reasoning behind each step as you build towards a conclusion (this avoids hand-wavy derivations). By doing these, we answer the question of "how would I derive this on my own?". You've also managed to explain the importance of the topic so we know when to use these tools. Excellent work! Subscribed

[@AllThingsPhysicsYouTube]

Thanks so much! Please feel free to share with others you think might be interested!

[@mrwillard95]

I'm glad I stumbled upon this channel, such good videos to watch👍

[@AllThingsPhysicsYouTube]

Glad you like them!

[@sinecurve9999]

Very nice explanation of how to compute Fourier coefficients without mentioning inner products.

[@AllThingsPhysicsYouTube]

Glad you liked it

[@MouseGoat]

This what? your fifth video? and im already fully on board XD It is easy to see how 3Blue1Brown inspired you, but you humor is also great. Also I really love circles, they like the core of everything, I would even say the symbol of everything.

[@AllThingsPhysicsYouTube]

Thanks! Feel free to share with others who might be interested.

[@rubixtheslime]

I'd argue that 𝕌 is moreso the symbol of everything

[@MouseGoat]

@@rubixtheslime okay but thats just a sign we humans have made up to describe something. circles are fundamental mathematical principles that show up everywhere in everything of our univers

[@kenthedawg6383]

Your videos are so well done! Excited for the next one :)

[@AllThingsPhysicsYouTube]

Thank you so much!! I've got some really cool things planned, but they take a lot of time.

[@emma5068]

Using a Fourier series to figure this out is really clever. I didn't think of that in the last video. But it makes total sense now that you described how it applies here.

[@AllThingsPhysicsYouTube]

Glad you approve!

[@nwunder]

I absolutely love that you addressed the three-body problem! that's the first thing i thought of I figured you were going the fourier route, but assumed that would have to disregard multi-body interactions, really cool that you were able to show that it still works! I also found myself wondering if 3+ bodies could allow for similar results. I cant help but think now, what if instead of a fourier-like system you had a solar system more similar to our own, and heavily tweaked the mass and radius of each orbit. if lagrange points are able to exist, surely multi-body dynamics would allow for potentially intricate orbit shapes as well. I suppose I'll have to try it myself!

[@realroadrunnr]

Although I'm barely understanding any of this, I find it very fascinating and interesting to watch, just like your previous videos. So, keep on making videos :)

[@AllThingsPhysicsYouTube]

That's my plan! Glad you enjoyed it!

[@lorenzoaste9173]

This has been the best introduction to Fourier series I’ve seen yet

[@AllThingsPhysicsYouTube]

Thank you so much! Please feel free to share the video with anyone else you think might be interested.

[@PhantomEye11]

What would it feel like to stand on the moon with a square orbit? Would there be any significant effect when going around the corner of the square orbit? Edit: forgot a part in my question

[@NoNameAtAll2]

you would still be pulled to the ground it's the celestial mechanics (trajectories on the sky) that will be strange and cool

[@mrphlip]

The gravitational forces on you, and the gravitational forces on the ground you're standing on, would be essentially the same, since you're in the same place, so you'd be accelerating together at the same speed, you wouldn't really notice it. This is the same effect that causes weightlessness in orbit - both the astronauts and the orbiter are falling at the same rate. One big exception to this could be in the first version of the square orbit where it takes a sharp turn around the corner at speed... in order to achieve this, the combined gravitational force on the moon at that moment must be massive, and so I'd expect the combined tidal force to be quite large as well, which could cause some severe effects. Tides are essentially the part of the gravity formulae that accounts for the fact that... while we were approximating that you and the ground you're standing on are in the same place, that's not exactly true, and if the gravity changes steeply enough that can matter.

[@NoNameAtAll2]

@@mrphlip tidal effects are about _difference_ of forces in the moon force being big at the corner doesn't make change of force through the moon be big

[@PhantomEye11]

@@mrphlip i forgot a trivial part of my question. I was wondering what it would be like going through the corner of the square. If there would be any effect because of the sudden change of direction

[@mrphlip]

@@PhantomEye11 You probably wouldn't even feel it. Compare it to sitting in a car going around a corner. The parts of you that are in contact with the car... your skin, your clothes... get dragged around the corner by friction. But your gooey internals, and the fluids in your ears, they want to keep going in a straight line, until pressures build up inside to transfer that momentum and keep you all in one piece. That internal pressure is what you sense and recognise as the feeling of acceleration. But gravity doesn't work like that. Gravity pulls on every part of you, all at once, inside and out. So all of you is accelerating together around the corner. No internal pressures needed to keep all the parts of you going the same way, nothing to feel. Hence, the comparison to weightlessness.

[@robindude8187]

I'm glad to see my intuition was right. When you, in the previous video, asked it an orbit could be square, my reaction was 'no', followed by 'well, not _actually_ square, but maybe squarish with some severely rounded corners'. Turns out, that intuition is correct. When I was thinking of the problem, I wasn't worried about the _mathematics_ of it, I was worried about the _physicality_ of it. An _actual, physical_ system couldn't do it (I intuited), my reasoning being that even if you changed the rotation and got things lined up, the 'corners' would _have to_ curve. And then you asked about sub-moons, and I got to wondering for a moment, but I nearly instantly saw the problem (at the start of this video). ... Those moons of moons and so on were going to go through each other's Hill zones (which, I admit, at the time of the start of the _first_ video, I was just thinking of as 'too close so they'd be pulled into each other or apart). I am left wondering, however, how many objects _could_ be in such sub-orbits and have it remain stable in physical reality. _Part_ of the issue is that the initial mass doesn't seem to have an upper limit _other than_ 'the mass of the universe, minus whatever mass is used for the other objects', as far as I'm aware (that is, I don't see any reason our central object couldn't be an utterly never before seen giganto-maximal black hole orbited by smaller black holes until reaching the size of stars). Still, my intuition here suggests that the number of such objects will ultimately depend upon the mass of that initial object (because smaller and smaller objects will get tiny fast), and so if we used the mass of the whole universe (we'll stick with the proposed mass of the _observable_ universe since, let's face it, what else could we use), and make the 'smallest' object a neutron, my gut (combined with a trivial bit of research and calculation) says you might actually get about 40 of them. The reasoning for this is that each object that orbits another object would have to be about 1% the mass of the thing it's orbiting to be able to orbit it (what is that based on? My gut, nothing more). To get from the mass of the observable universe (10^54 kg) down to a neutron (10^-27 kg) is about 81 orders of magnitude, and each 1% will decrease the order of magnitude by 2, so 40 objects. Of course, the number of objects possible in the universe we _have_ will depend on the mass of the most massive object in space (more research, it's the black hole at the center of the quasar TON 618, 66 billion solar masses), which would make for the potential of 33 objects (... seriously? Only 7 shy of the whole universe doing that? ... Dang). Buuuuut.... all that's almost _entirely_ based on a gut reaction.

[@AllThingsPhysicsYouTube]

I’d say your intuition is pretty good overall, and I like your estimate of how many moons would actually be possible.!

[@COTU9]

You make some great videos. Great and clear. Thanks for making a responding video.

[@AllThingsPhysicsYouTube]

Glad you like them!

[@symmetria5192]

great explanations with amazing visuals! was sad to see that your channel doesn’t have more videos, but now i’m just even more excited to see what you’ll come up with in the future :)

[@AllThingsPhysicsYouTube]

Yeah, I’m pretty new at this. But don’t worry, I’ve got a LOT of ideas for videos, and I think I will get a little faster at making them. This last one took a lot of time.

[@beaverbuoy3011]

I'm lucky youtube recommended this to me.

[@AllThingsPhysicsYouTube]

Feel free to share with other you think might like it!

[@hoodio]

linking fourier series to orbital dynamics in my mind was really helpful

[@AllThingsPhysicsYouTube]

Glad to hear it!

[@robokaos69]

This channel will certainly grow very large with content of this quality :)

[@AllThingsPhysicsYouTube]

I hope you’re right. Please feel free to share with others you think might be interested!

[@pluspiping]

The switch from tracing unusual orbits in space (could be real) all the way to Fourier series with handfuls or dozens of orders of moons (which can't trace out shapes reliably irl but are real in your computer all the time) was smooth and fascinating, and feels like it serves as humor at the end, that moons upon moons can't actually do this in real life in real gravity. Now that was a trip.

[@pauljackson3491]

Yes, I am impressed. When I saw his video I wondered what the pictures would look like with different number of "moons". Such as it's a circle with just a sun and planet. An egg-shaped with a sun and planet and 1 moon, etc.

[@henrikfredriksen4735]

i love this, thanks a lot for the video :) I just cant wait till the next one!!

[@AllThingsPhysicsYouTube]

Glad you enjoyed it! I've got a long list of videos to make, but I'm new at this and they take time.

[@Predator-do6kd]

After having just finished a multivariable and complex calculus course.. the feeling i got after realizing the integral from 0 to 1 of z(t)e^2*pi*i*t being just the parameterization of a complex contour integral, which is always zero as long as z(t) is analytic/entire by cauchy's theorem, was astounding. being able to apply my own learning and experience and seeing how they work out in other fields is phenomenal.

[@AllThingsPhysicsYouTube]

Glad you liked it! Please feel free to share with others who might be interested.

[@Miss_Trillium]

It's really wild to me that I found this channel at ~4k subs, it seems like something that could get as big as 3blue1brown, but here's to hoping. I may not fully understand all of this, but it sure is mesmerizing. I'd love to know then if it's possible with physics in mind--what would the size of each moon have to be, etc

[@AllThingsPhysicsYouTube]

I would love to see this channel grow to the size of 3b1b...please feel free to share with others who might enjoy it! The formula I present in the video does allow one to calculate the masses and orbital radii for all the moons, but in terms of actually being possible I'm afraid the close encounters will render this system unphysical.

[@xamarinmaster1403]

This was a truly awesome video!

[@itsmebougie]

This is surprisingly helpful for musical synth development

[@AllThingsPhysicsYouTube]

I used to play in a progressive rock band back in the late 70’s/early 80’s. I know a little about musical synths, and you’re right!

[@Kahjal11]

Wonderful walk through the process, thank you for your work! Cool you got so inspired by 3B1B, the look also seemed familiar :D

[@AllThingsPhysicsYouTube]

Thanks for watching!

[@MolecularMachine]

I've never had much of a brain for math, but in spite of that, I can never get enough of natural science content. Even if I can't quite grok the details, it's at least nice to know that there are all these logical steps leading from 2+2 all the way to orbital mechanics.

[@Darthvanger]

I've been always dazzled how a combination of sines and cosines with different frequencies can approximate any function. Moons upon moons makes it so much more intuitive and easy to grasp! Just the rotating arrows, which compensate each other, instead of the weird sine/cosine wavy shapes :)

[@AllThingsPhysicsYouTube]

I too have always been fascinated by this topic! Glad to hear you liked the video!

[@werner134897]

EXCELLENT explanation! Bravo.

[@AllThingsPhysicsYouTube]

Glad you enjoyed it!

[@fordprefect859]

This needs to be a screensaver. it's so mesmerizing.

[@AllThingsPhysicsYouTube]

It is very mesmerizing, isn’t it?

[@melendroach6331]

Grant is a wonderful person of course, and his videos are great and beautifully explained, but I have to say: your video really made me understand the Fourier series, much better than 3b1b's. Thanks.

[@AllThingsPhysicsYouTube]

Wow…that’s high praise. Thanks so much!

[@ryansamarakoon8268]

It's beautiful to see that you were inspired by 3b1b. Maybe one day you'll help inspire the next generation of maths/physics educators on KZbin 👀

[@alexsappington1587]

Another great video with fantastic animations and explanations for most everything. The one part I question though is your assumption that we can treat each sub-moon as a two body system with it's parent. I think in the original problem with 3 bodies of vastly different masses, this assumption makes sense, as proved by the viewers' 3 body simulations you mentioned. But here I don't think it's safe to use this assumption and not check it's validity. Is it possible to determine the masses needed for each of these orbital periods and radii? I would think it is. From there wouldnt it make sense to compare those masses to see if m1>>m2>>m3... is still true? My gut feeling is that this orbital system with so many (say 12) moons would not actually be stable when accounting for the effects between all objects at once. Then the next question could be "how many moons can we have and keep the orbit stable?" I think things would have to be very spaced out and with large mass differences at each stage, but you could have many layers of orbits, similar to how our sun orbits the center of the galaxy and our galaxy orbits something else.

[@thedeep697]

I thing there is no theoretical limmit to the number of objekts, but a praktical one. If the objekts become increasingly heavier and the orbits increasingly bigger, will at some point relatity come and make everything even more complex. The speed of the gravitaitional waves becomes relevant, as well as the maximum speed of the objekts itself (speed of light), and the gravitional force of other heavy nearby objekts and once that happens it becomes extremly unstable and thus unlikely to be able to exist, even if you try to build it yourself.

[@AllThingsPhysicsYouTube]

I didn't actually check to see if the masses obeyed the assumptions and whether the higher-order moons would lie within the Hill sphere's of the parent moons. But even if everything checks out there is still the problem that two of the objects will almost certainly experience a close approach, and this will likely ruin things.

[@yahccs1]

This makes me wonder - is the whole observable universe orbiting around something else? (Maybe appearing to expand just because it was all orbiting at different speeds?!)

[@Darthvanger]

Works like a clock! Connect the springs in a highly thought-out manner, and spin! :) All the springs straighten out and join their forces to make a twist just at the right moment, based on the intricate tuning of the radiuses and spin directions. It's crazy how any force is basically an invisible spring. Of course kx is much different from 1/r^2, but the mechanism is so similar: just push and pull 🤔

[@xaytana]

I wanna see this in 3D orbits, as a moon won't always share its orbit's plane with its planet's orbit's plane. Outside of the obvious 3D spirals that form around the parent body's path, I'm curious what shapes could come out of it. This could also allow for point of views that aren't just top-down, modifying the viewpoint could also be interesting. Elliptical orbits would also be interesting to see.

[@biogoo]

Easy to follow and understand. Well done!

[@AllThingsPhysicsYouTube]

Thanks!

[@junkcrafter122]

19:33 Interestingly, no matter how many times you iterate, it will never look like a perfect square - due to the optical illusion of the circle under the square tricking your eyes into thinking that the square's sides are curved!

[@jwsjacobs]

Nice video, also love the background music

[@AllThingsPhysicsYouTube]

I'm glad you like it...a few people have commented that they don't appreciate the background music.

[@krogerbrandtrailmix3281]

here before this channel blows up!! so much great content

[@AllThingsPhysicsYouTube]

I hope you're right! Please feel free to share with others who might find it interesting!

[@benrex7775]

As an electrical engineer I approve of this video. We had all of that in school, but not put in this way. The only thing I would have a bit of a problem with is the finding out the formula for the rectangle. That would require some googling for me.

[@AllThingsPhysicsYouTube]

Glad you liked it!

[@yahccs1]

Thanks that was fascinating. I vaguely remember learning about Fourier series and transforms at university. You explained it really simply and clearly! A good reminder! Nowadays the only time I come across anything remotely like it is when editing audio files and it has an option to remove noise using spectral analysis - or you can look at a Fourier transform of the music track showing how the volume of each frequency changes over time. I had forgotten the mathematical details involved! Maybe in reality a moon of a moon is hard to come by - and a moon of a moon of a moon.... I think the mass ratios would need to go down exponentially! Perhaps an asteroid could be redirected to become a moon of our moon, and a smaller one orbiting that, and a small artificial satellite orbiting that, and a microscopic camera orbiting that - then they would get so small they might just have to be attached by threads instead of gravity?! It reminds me of the ever decreasing sizes of creatures (parasites on parasites.... on fleas on dogs or whatever)- or like the camp song we learnt about "the frog on the twig on the branch on the log in the hole at the bottom of the pond."

[@AllThingsPhysicsYouTube]

I was thinking about trying to work in the quote about dogs have fleas and fleas have fleas, but couldn't make it work all that well with the video. And yes, Fourier stuff is usually done in "one dimension" with a time varying signal (such as music). I'll probably do a video on this topic, but it will be well in the future as I already have a number of other topics already planned out.

[@HebaruSan]

Hill spheres are 3D, but so far you've worked entirely in 2 dimensions. I wonder if there are any interesting 3D shapes that could be traced out with this method. In other words, what if the inclinations aren't constrained to 0 or π?

[@AllThingsPhysicsYouTube]

Oh I'm sure there are all kinds of interesting 3D possibilities, but that's a MUCH more difficult problem.

[@Holobrine]

Part of the problem is in two dimensions, we can use complex numbers, but no such luck with three dimensions. Perhaps quaternions would offer an approach? Or perhaps the appropriate generalization can be found with geometric algebra?

[@feedbackzaloop]

@@Holobrine yep, just add another angle, integrals turn into doubles, coefficients remain independent. With Green's theorem one can lower the order of integration back to one

[@Holobrine]

@@feedbackzaloop I’m struggling to understand how Green’s theorem fits, and what extra angle is at play. Is it Euler angles?

[@feedbackzaloop]

@@Holobrine it fits as we have double integral over the area, will turn in into single over curcumference... As per angles, kinda but not exactly: Euler angles apply to the body, we want to describe only the vector (so only two of three apply), so basically spherical coordinates with representation of the vector with exponent to the summ of coresponding to the angles powers

[@PJulianC]

This is similar to 3 blue 1 brown’s video on drawing with circles and makes me think is it posible to have orbits that can draw out paintings? I am sure it would be a near Imposible orbit but it would be cool if we found a star sistema that is flipping us of

[@samfallon2568]

this video is amazing very interesting to know what calc 2 is used for later down the line

[@AllThingsPhysicsYouTube]

Glad you liked it. Stay tuned for more!

[@forgettenartmath7670]

to be honest, I think you've exceed 3b1b's Fourier explanation, but I watch this after that so I had some pre-knowledge. Thank you.

[@AllThingsPhysicsYouTube]

Thank you so much. That’s high praise indeed, and as much as I want to believe you, I suspect the pre-knowledge played a part.

[@Michaelonyoutub]

From my knowledge on general relativity and it's version of gravity, the moon tracing the square wouldn't actually experience any "whiplash" at the corners due to appearing to change directions very quickly. In general relativity there is no force of gravity so there is no force to suddenly jerk you in another direction, you just pass through warped spacetime in a straight line. That doesn't mean you would be perfectly fine on that moon though, likely there will be some insane tidal forces due to crazy amount of warping in spacetime which might rip you and the moon apart.

[@nos9784]

In the first vid, The corners are also the apogees of an eliptic orbit, right? meaning the slowest point. Not sure how the fact(?) it is a rotating ellipse figures in... According to my (limited newtonian) understanding, there cannot be any whiplash, because why would it? Gravity is not some kind of tether, it is a vector field with no sudden changes, And velocity of the moon and everything on it is governed by the moons freefall around its parent body- simple momentum. No sudden changes, just gradual ones, and mostly in direction? Unless you hit something, that is. For whiplash to occur, you'd have to be huge- creating significant tidal forces/ an accellaration gradient along your spine despite the large distance to the parent body?

[@simonmultiverse6349]

In the introduction, you mention the stability of the orbits. This suggests that you should optimise each collection of orbits for: (1) The desired shape of orbit; (2) The stability of the orbit; (3) The speed of the moon along each part of its orbit. I guess that (1) is obvious, (2) is required for the long term and (3) might be helpful depending what you want to use the orbit for.

[@GoldphishAnimation]

Most hype sequel drop of the year

[@connorowenmetz3107]

Hi David, very nice video. I'm curious how did you parameterize the non-constant speed series angular frequencies? Did you prescribe an omega(t) or let it take on free values? Curious how that led to faster convergence. I wonder if this same effect could be accomplished with eccentric orbits.

[@AllThingsPhysicsYouTube]

Actually, I took the easy way out and constructed a square out of (x,y) data points that were not uniformly spaced along the square (more points at the corners and fewer on the straight segments). Then I found the Fourier coefficients numerically, so I don't have a specific formula for them. But it might be possible to do it analytically.

[@connorowenmetz3107]

@@AllThingsPhysicsKZbin Oh, I see what you mean, that's a very cool way of doing it! thanks for the reply. keep up the good work

[@Hello-mi6gj]

I few months ago, I fell upon one of your videos regarding Newton´s first law that left a great impression on me, and I have been a great fan ever since. I congratulate you on this wonderful video with an easy-to-follow and very well articulated explanation on this curious aplication of the Fourier series. However there is one thing that I have not been able to understand: How do you obtain the function depicting a square? I understand the initial parametrizations but the jump from those to the formula is a complete mystery to me. I have tried to parametrize a function drawing a triangle (using t= 1/4, t= 7/12, and t=11/12 and designating their corresponding coordinates using a line of length l that stretches form the orgin to these values) but have been unable to proceed any further. I would be very grateful if you could elucidate this step.

[@AllThingsPhysicsYouTube]

Thank you for the kind words. One way to do the parameterization for a square is to begin at the point (1,0) at time t=0. Next, we want to move to the point (1,sqrt(2)) while t runs from 0 to 1 (or some other vale). Assuming the parameterizations are linear, we have x(t)=mt+b and y(t)=nt+c. Thus, we just need to set x(0)=1 and x(1)=1 to find m and b, and y(0)=0 and y(1)=sqrt(2) to find n and c. This will give the parameterizations x(t) and y(t) for the first eighth of the square. Then you just repeat this process for the other three sides and once more for the last eighth of the square. Does that make sense?

[@Rudxain]

I studied a little bit of waveform synthesis because of music production, and I knew the frequencies had to be odd integers because that's how a square wave is synthesized. I actually got surprised by the fact that some frequencies were negative

[@AllThingsPhysicsYouTube]

Yes, the negative frequencies are not what one would expect based on waveform synthesis.

[@dreupen]

Your videos are great. Thank you.

[@AllThingsPhysicsYouTube]

Glad you like them!

[@michaelharrison1093]

I am currently working on a similar video as part of the Summer of Math project being coordinated by Grant Sanderson who runs the 3B1B channel. My video specifically relates to Symmetrical Components which is a theory used in Power Systems engineering. There is some very interesting math behind all of this that is not widely known developed by Charles Fortescue.

[@AllThingsPhysicsYouTube]

Sounds very interesting. I'll keep an eye out for your video (I also entered the SoME2 contest).

[@michaelharrison1093]

@@AllThingsPhysicsKZbin the theory that Charles Fortescue developed gives you a neat insight to the harmonics that appear when you have rotational symmetry. Take for example your square orbit example if you assume that the fundamental is rotating in a positive direction and the orbit has a rotational symmetry of 4 then according to Fortescue the only harmonics that will be present will consist of the positive sequence components 4n+1 and the negative sequence components 4n-1. The positive sequence components rotate in the same direction as the fundamental and the negative sequence components rotate in the opposite direction as the fundamental

[@geenes4120]

Nice to see you again!

[@AllThingsPhysicsYouTube]

Welcome back!

[@Gunbudder]

I've heard of a super ellipse, but this is a super DUPER ellipse

[@leesweets4110]

11:40 Why does the frequency of each moon about its parent have to be constrained to an integer? We know that the entire multi-body system taken on whole has to form a closed loop; does this necessarily imply that each subsystem must independently form its own closed loop?

[@AllThingsPhysicsYouTube]

Good question. To be honest I haven't really thought through the details of this, but my intuition says that they *would* all need to independently form closed loops. But my intuition has been wrong before, so maybe not.

[@leesweets4110]

@@AllThingsPhysicsKZbin My intuition says that if they were related through rational numbers, then they would form a closed loop eventually, even if it takes multiple cycles about the star. As long as there are no irrational relationships...

[@AllThingsPhysicsYouTube]

@@leesweets4110 Ah. If you assume any timescale for the closed orbit, then I'm pretty sure you are correct. I was assuming a closed loop is formed on a one-year timescale when making my comment.

[@bazingatheskialuser3519]

now that we have a square orbit, now we need a circle orbit

[@fry_fr]

"Although it might seem ... complicated remember that all we're doing is [complicated stuff nobody understands]"

[@AllThingsPhysicsYouTube]

Heh...well, it's a little complicated to carry out the procedure, but all we're doing qualitatively is just adding together a bunch of spinning arrows!

[@pbroks13]

4:23 ain’t NO way you made this by accident 😂 great video though!

[@fairnut6418]

🅱️enis

[@HitAndMissLab]

Fourier transformations in under 30 minutes? Can't be bad deal.

[@OlafDoschke]

What if the highest order moon would comply to Keppler's second law, i.e. the vector of its orbit would sweep a constant area over time on its square path? What would be the effect on the Fourier coefficients? Would it be an extreme case? PS: I'm aware Kepppler's laws only describe the simple and idealized 2 body problem, and the laws don't actually apply to an nth order moon orbit around the sun. The only physically possible square path would need velocity=0 at the corners, as that's the only possible condition to change the direction exactly perpendicularly.

[@AllThingsPhysicsYouTube]

Kepler's laws should still apply to all of the moons!

[@OlafDoschke]

@@AllThingsPhysicsKZbin Well, but only in their orbit relative to the moon they orbit, not in the resulting orbit around the Sun. Do you think this law is consistent on any level? I don't think so. Would be nice, as it gives a way to define the temporal development of the square orbit, but then the corner case (actual corner case) of v=0 at the corner would also imply a sweep rate going to 0 at that point in time, wouldn't it? So it couldn't stay constant, as the law demands. Or from the other point of view: Intuitively I'd say a constant sweep rate on a square means the velocity at the corners would not become 0, even though it's just a momentary 0 velocity. the sweep area is vector length (sqrt(2)/2 R at the corner) times velocity at any time, so a v=0 would break the law, even if it's just momentarily, wouldn't it? I would simply go through calculations and then we'll see.

[@-NGC-6302-]

So like a celestial Fourier Series Nice

[@AllThingsPhysicsYouTube]

Exactly!

[@Yuri_Gagarin44]

A pure delight!

[@AllThingsPhysicsYouTube]

Thanks!

[@violinviolator5841]

This is pretty cool! How about a cube orbit?

[@phoephoe795]

More curious now- how many moons do you need to keep the final moon in a fixed position relative to the central star? I assume you can model this in the software by tracing a very small square that sits adjacent to the star?

[@cameronhaney4892]

Imagine how wild the eclipses on the smallest moon would be

[@JimGriffOne]

Fourier was way ahead of his time.

[@AllThingsPhysicsYouTube]

WAY ahead of his time!

[@rtheben]

Kudos for crediting 3bkue1brown

[@AllThingsPhysicsYouTube]

We’ll, how could I not?!?

[@yahccs1]

I liked the updated animated orbit with the star also moving. It seemed more realistic but with mass ratios exaggerated to be able to see this! You didn't mention "Spirograph" this time, but the almost-square that can be produced that way is so similar to a Squircle (See Matt Parker's video on the area of a Squircle!) that I thought it might have a similar formula, but not quite! Maybe a squircle orbit (the one that's half-way between a square and a circle) would not be too difficult to construct if you start off with the almost-square orbit that looks roughly like a squircle! You also didn't mention Hill spheres this time, and the constraint for the moon of a moon to be within the moon's Hill sphere and that of the planet. If you keep adding more and more moons, the first moon has to be inside a smaller and smaller fraction of the planet's Hill sphere to allow all subsequent moons to also fit inside it. The rotating vector animation was mesmerising and beautiful as you said, but it gave the impression that the moons were all being flung off into space (or orbits around the star instead!) like objects coming off the end of a rope. Have you made an animation of chaotic orbits around binary stars or anything? It gets so complicated with multi-body systems with significant masses.

[@AllThingsPhysicsYouTube]

I thought about discussing the various Hill spheres for the different moons, and their mass ratios, but the video was already longer than I wanted it to be, and I thought most people just wouldn't care all that much. And yes, multi-body systems with chaotic orbits get pretty complicated, and that wasn't the main purpose of this video. But perhaps something about chaos in the future.

[@yahccs1]

@@AllThingsPhysicsKZbin Thanks. You have made some fascinating and complicated topics seem simpler to understand! The animations really help. I just remember the simplest chaotic system demonstrated by an angled pendulum which could swivel around at its joint. Even that would be difficult to model with equations, but maybe someone somewhere's done an animation of it!

[@AllThingsPhysicsYouTube]

@@yahccs1 Actually, there are a number of chaotic pendulums that are not all that hard to simulate. I have a chaos video in mind that involves a driven pendulum, but it might be a while before I get to it.

[@beaverbuoy3011]

Isn't this supposed to be part 2?

[@thyros_]

wait how do you only have 6k subs 😮

[@AllThingsPhysicsYouTube]

I would love to see this go higher. Please feel free to subscribe (if you haven't already done so) and share the video with others you think would like it!

[@ericcmcgraw]

20:32 I am deeply interested in this idea of how best to re-parameterize a given curve such that a minimal number of terms follows the desired trajectory as closely as possible. Can you explain how you did it for a square, and perhaps a more general way of going about it for an arbitrary path? Every time I see such Fourier animations it seems like a ridiculous amount of unnecessary complexity goes towards the over-constrained goal of following the trajectory at a constant velocity, whereas a much simpler yet potentially more accurate fit could be found if we could somehow relax the constraint on velocity altogether. I would be very interested to see this explored more deeply, as it could allow a given curve to really be compressed down to its most fundamental bits of information.

[@AllThingsPhysicsYouTube]

This was new to me, and I was initially very surprised by the fact that a trajectory that moves more slowly at the corners converges more rapidly. And then I thought that perhaps this makes sense. I mean, consider drawing a square with a pen without lifting the tip. Would it be easier to draw it moving the tip at a constant speed or by slowing down near the corners? I suspect it is somehow "easier" for the series to "draw" the square if it is allowed to slow down at the corners. For this video, I was able to get an analytic result for the constant-speed trajectory but I didn't even try to do that for the variable speed trajectory, assuming it would just be too difficult. Instead, I created a file with (x,y) data points that were not even spaced and then used fft routines to get the Fourier coefficients. To really tackle this issue, I think one would need to try to determine a parameterization that allows for a variable speed that can still be calculated analytically. Perhaps then one could perform some kind of optimization procedure (calculus of variations?) to get the best parameterization for a given number of terms. This may very well be doable, though I'm guessing it will take a significant effort. In fact, I wouldn't be surprised if people haven't already looked into this question. In any event, while I find this question very interesting, I will leave it to those with more expertise in optimization techniques to investigate further.

[@michaelharrison1093]

Not a direct response to this question but somewhat related - see my comment about Charles Fortescue and his theory of Symmetrical Components. In Power Systems engineering you can get harmonic current distortion that results in circular orbits that have cyclic variation in the rotational speed

[@eamonia]

Too cool, man. That is some heavy jive right there, boy.

[@AllThingsPhysicsYouTube]

Glad you liked it.

[@eamonia]

@@AllThingsPhysicsKZbin Liked it!? You blew my mind, brotha. I'm glad I subscribed.

[@Metaldetectiontubeworldwide]

Got me subed budy , brw saw exacte the same mesmorising video of 3 blue one brown .. been folliwing them for like 6 years

[@AllThingsPhysicsYouTube]

Thanks for subscribing! Please feel free to pass this along to others who might be interested.

[@nbooth]

This is great stuff. Can I make a request for a related topic? How about the three body problem and how it relates to chaos?

[@AllThingsPhysicsYouTube]

That's a great idea, but in all honesty I will likely discuss chaos using a different approach. Unfortunately there will likely be a good number of videos before I get to something like chaos.

[@ThomasHaberkorn]

There's also a theoretical way for a line orbit: consider 2 stars of equal mass orbiting each other. Now place a planet in the center of the system, but slightly push it perpendicular to the place of the system. The planet will do up and down linear motion crossing through the plane of the system. There's also a name for this orbit, but I forgot, so if you know please tell

[@pumpkincell]

Got thia in my recommended and its very good that i did

[@AllThingsPhysicsYouTube]

Cool. Glad you like it!

[@Manabender]

OK, sure, you can use a Fourier series to draw any shape you want. But does this hold up under n-body gravity? Applying the Fourier series to a collection of moons assumes that only every pair of *adjacent* moons interacts gravitationally, and disregards the effect of non-adjacent pairs (where "adjacent" means two consecutive terms in the series).

[@AllThingsPhysicsYouTube]

Of course, you are correct, but I believe I made this point clearly in the video. Once you have a good number of higher-order moons there will be "close encounters" that ruin the "two-body" approximations being used.

[@WollaWinkie]

Very interesting, good video!

[@AllThingsPhysicsYouTube]

Thank you very much!

[@olegmoki]

Amazing. I wondered, what program do you use for animations, because i saw it on 3Blue1Brown, and i quickly enough found an answer!