Square Orbits Part 3 (Finale): The Grandeur of Fourier Series

[@3zan6bel9]

this serie grew in a ridiculous monster. I truly hope there is a homer simpson cycle somewhere in the universe

[@wvhdogg]

really cool that the coefficients can be calculated efficiently, even for 150 data points, without having to resort to brute force which would be computationally infeasible

[@AllThingsPhysicsYouTube]

I'm using an FFT (Fast Fourier Transform) routine to calculate the coefficients, so it is pretty efficient. And there are 251 vectors, not 150! 🙂

[@coreC..]

Can you imagine that this person,Jean-Baptiste Joseph Fourier, was able to imagine all of this in his mind only? Without any help of computers.. That is truely amazing and exceptional.

[@sumsizzurp]

The entire pacing and narration of this video is great !

[@AllThingsPhysicsYouTube]

Thanks! Please consider subscribing and sharing with others who might like it!

[@spamfriendly6489]

Best physics teacher on youtube !!!

[@zacharynovak2180]

I love this channel, it needs way more subscribers

[@AllThingsPhysicsYouTube]

Glad you like it. Please help spread the word!

[@alantaylor2694]

People much smarter than me...'Did you know that time itself can speed up?' Me 'Yeah, I can get behind that.' The Fourier series shown in exquisite detail. Me 'Witchcraft!' Seriously it just blows my mind. Great vid. Thank you.

[@AllThingsPhysicsYouTube]

You’re welcome, and believe me you’re not the only one who feels like this is witchcraft! 😊

[@arctiid4494]

1:30 I appreciate this transition

[@AllThingsPhysicsYouTube]

That was a last minute idea, and I was pretty happy with how it turned out!

[@harriehausenman8623]

@@AllThingsPhysicsKZbin Totally woth the effort!

[@MarcRenault]

@@AllThingsPhysicsKZbin I bet you had to try for hours to position your body exactly within the outline! Just kidding :-). Excellent video!

[@AllThingsPhysicsYouTube]

@@MarcRenault 😄

[@geenes4120]

Aaah yes another part! I wonder how the picture would look like when all, or just some random orbits were shown. My first suggestion would be, that the image gets sharper and sharper from the first to the last orbit. Do the individual points collide?

[@AllThingsPhysicsYouTube]

Good question. I don’t know the answer, but my guess is that the individual points will not collide. That just feels like it would introduce some kind of redundancy.

[@harriehausenman8623]

Wonderful! 🤗 I actually just started to watch the video because he's a cool dude - not expecting anyting new, but oh boy was I wrong. It's the first time I could appreciate the beauty of the dance. Reminds me of my first mandelbrot zoom long time ago. And the structure has an almost organic nature. Thanks man! 🤩

[@AllThingsPhysicsYouTube]

Glad you enjoyed it!

[@endrawes0]

Thank you for this video. Very illuminating

[@AllThingsPhysicsYouTube]

Glad you liked it. Feel free to share with others you think might like it.

[@jayantnema9610]

you make awesome videos !! Thanks for the effort 🤗🤗

[@AllThingsPhysicsYouTube]

Glad you like them! Please help spread the word!

[@noj5393]

Absolutely beautiful

[@AllThingsPhysicsYouTube]

Thank you! 😊

[@josiahtaylor8714]

Keep it up buddy, we all know you can continue to make such amazing content

[@AllThingsPhysicsYouTube]

Thanks. I appreciate the vote of confidence!!

[@JohnLRice]

Excellent as always Dave, good to hear you play guitar too! 👌

[@harriehausenman8623]

I nearly missed that due to acute out-awedness 🙂

[@AllThingsPhysicsYouTube]

Oh my goodness, it's John L. Rice! I learned this song for my father-in-law's memorial service and fell in love with it, and when I was putting this video together I thought, hey, that song would work pretty well in this video!

[@JohnLRice]

@@AllThingsPhysicsKZbin 😎👍

[@sk4lman]

How can one possibly calculate all of these arrows and rotations, to come up with a certain pattern? One thing I can't grasp is, does it start simple and then you keep adding arrows and fine tune each one as you go? Or can you paint yourself into a corner that way, and have to erase dozens or hundreds of arrows and change something close to the beginning of the chain? Like, if you'd want two additional adjustment screws to the guitar, how much of the chain would you need to change? How would you even know what to change..? It boggles the mind..!

[@AllThingsPhysicsYouTube]

It does boggle the mind! I provided a "derivation" in the square orbits part 2 video. The critical insight is that to actually calculate each of the Fourier coefficients you need to use the entire shape you want to draw. Thus, the actual shape is "embedded", in some sense, into each of the Fourier coefficients. Therefore, if you wanted to change the picture in some way, it would require you to change *all* of the Fourier coefficients (though some of them may end up being quite similar to the originals). I hope that makes some sense.

[@neur303]

You start with a function that draws the shape. In this case you use polar coordinates. Then the Fourier transformation is done on this function. The interesting part of this transformation is that you look at the function at a certain frequency one at a time, each gives you 2 coefficients, that show you a direction and a length of an arrow (called phase and amplitude), you kind of see them in the video at t=0 (the start of the drawing) It's important to know that each arrow has its own speed, which is constant. I imagine the process in a way tjat you rotate the whole "world" with a certain frequency and only the one arrow you are looking for is standing still. All others keep rotating and when you sum it up those cancel out, because they point in all directions with the same probability. Hope it gives food for thought.

[@neur303]

Ah sorry, polar coordinates mean that you make a function that returns a distance for an angle. This also limits the shapes you can draw to a degree, as you need to be able to draw a continuous line, circling around the origin.

[@AllThingsPhysicsYouTube]

@@neur303 Wow...sounds like you know a thing or two about Fourier series! Well said!

[@fordprefect859]

I need this as a screensaver.

[@AllThingsPhysicsYouTube]

I would mind having it as a screensaver myself!

[@JoeSmith-cy9wj]

The ramifications of this are astounding. Perhaps our lives, and everything else, is deterministic.

[@WoolyCow]

as usual...i understood none of the math but i still love this stuff! getting into complex numbers now at school so hopefully might fare a little better when your next video comes out haha :D great vid and gorgeous graphics!

[@AllThingsPhysicsYouTube]

Glad you liked it!

[@kenthedawg6383]

Love your videos man!

[@AllThingsPhysicsYouTube]

Thanks!

[@gregbell2117]

The omitted hat-tip to 3blue1brown who likely inspired this video and also created the animation software used: kzbin.info/www/bejne/qGfWeIqKeLKtaM0

[@AllThingsPhysicsYouTube]

Please watch Square Orbits Part 2, as this is where the (lengthy) hat-tip takes place!

[@dracus17]

Such a smooth ending

[@AllThingsPhysicsYouTube]

Heh, I thought it would be fun to use the star in the guitar system in the solar eclipse logo!

[@realcygnus]

Yup, amazing indeed ! One of math's top mindblowers imo & used extensively in engineering. 👍

[@AllThingsPhysicsYouTube]

You are absolutely right that it gets used a lot! I’m sometimes surprised at how useful it turns out to be!

[@realcygnus]

@@AllThingsPhysicsKZbin IKR, if only Joe could see his brilliant ideas in action as conceptual machines realized in our modern world.

[@AllThingsPhysicsYouTube]

@@realcygnus heh…Joe! I do think Joe would be pretty amazed to be able to visualize things like this. Can you imagine what he might have accomplished with today’s tools? (On the other hand, perhaps today’s distractions would derail him.)

[@realcygnus]

@@AllThingsPhysicsKZbin Yup, who knows ? Though it would probably have only enhanced their work. its also amazing just how far back some BIG thinkers go. Euclid for instance, just wow. With only pencil, paper, compass, straight edge & a 🧠to work with. & I forget who said it but also just how "unreasonably effective" math even is at describing our world in the 1st place.

[@AllThingsPhysicsYouTube]

@@realcygnus I believe that was a Eugene Wigner paper (on the unreasonable effectiveness of mathematics).

[@Josdamale]

2:10 Great videos. One thing, however, you don't properly explain here is the order of the vectors as you join them together head to tail, which is actually extremely important. That was the key that got glossed over.

[@AllThingsPhysicsYouTube]

I guess I did sort of gloss over that. Because these are vectors, order doesn’t matter. You can add the in any order and you’ll (almost magically) get the same shape!

[@n20games52]

Amazing

[@AllThingsPhysicsYouTube]

Thanks!

[@scobrado]

Wow! Once again, amazing video and presentation. What is the render time for the whole outline, and what kind of hardware/software is involved? Is the software commercial or proprietary?

[@AllThingsPhysicsYouTube]

The software is freeware...I'm using manim, a Python package developed by Grant Sanderson of 3Blue1Brown. The render time for these depend on a number of factors, but 4-8 hours is typical for a highly detailed and high-resolution simulation.

[@harriehausenman8623]

It would be interesting (and potentially equally "shicadelic") to mess around with the phase and show what the guitar looks like, if the arrows don't exactly phase up. A little "phase-space-exploration" video 🤩😵‍💫

[@aneyesky]

Mind blown

[@adrianbiro2167]

Nice!

[@ltd5480]

Cool video! ❤ Can I have the source code please? Are you using Manim CE?

[@AllThingsPhysicsYouTube]

Regarding the video: thank you. Please feel free to share with others who might be interested. Regarding the code: I am using manimce, but the code is not in "sharable" condition and I don't really have the time to clean it up. Sorry.

[@SebSN-y3f]

Mega cool!

[@AllThingsPhysicsYouTube]

Thanks!

[@toddzimmerman4407]

Are the arrows ordered by frequency?

[@AllThingsPhysicsYouTube]

Hi Todd! Yes, arrows are ordered by frequency (0, -1, +1, -2, +2, etc.), mainly because it was convenient. But it's also generally the case that the arrows tend to be longer for the lower (magnitude) frequencies, so it turns out to be a good way to do it visually as well. Of course, you'd get the same final image regardless of what order you use.

[@toddzimmerman4407]

I wonder if there would be a different ordering that might lend itself to some insight

[@AllThingsPhysicsYouTube]

@@toddzimmerman4407 Interesting thought. I have very little intuition about this, so I can’t really hazard a guess, but if I had to guess I’d say no.

[@allanwrobel6607]

so whats the 3rd dimension or 'n' dimenation (n > 2) equivilent?

[@jlfqam]

you can answer this, Some exoplanets with the size of Jupiter display orbits on the order of less than 10 days. My guess is that they are sunspots (starspots) instead. The models to describe orbits consider the star as one dimensionless dot, which is not realistic when a planet is close the star. The sun appears to be non homogeneous and that should be observed as a variable gravity pull depending on the distance to the sun and even along the different radial dimensions. For me, Jupiter orbiting the sun closer than Mercury would not survive, would not be a stable orbit. Remember Oumuamua asteroid, its unusual path could be described if the object experienced a uneven distribution of mass within the sun, also other unusual paths of small objects that have been found in the Solar System. Is there a realistic model of the Solar system where the sun is a 3D sphere where all its defects can be tested?

[@chadissimusrex8038]

Is there a way to estimate the number of arrows needed to create a specific shape?

[@AllThingsPhysicsYouTube]

Not accurately. Qualitatively, you can sort of estimate the number of arrows needed by the smallest scale features of the shape. When there are large changes in the shape over small changes in time, it will require a larger number of arrows. The smallest detail (in time) that can be "drawn" with a particular arrow is approximately given by 1/n, where n is the frequency. Thus, if the shape changes on a timescale tau, then it will requires approximately 2/tau arrows (the factor of 2 coming from the fact that there are both positive and negative frequencies). I'm not sure if I explained that very well. Think of it this way. Higher frequencies will oscillate more rapidly in time, so if the shape has lots of changes in a small amount of time it will require higher frequencies.

[@chadissimusrex8038]

@@AllThingsPhysicsKZbin So, a bit like a higher frequency radio wave can transmit more data in a smaller amount of time. Except for detailing, about the "general shape" you cannot tell much at first glance, i understand. Well to conclude, your video's have great visualisation and the right kind of energetic presentation. They look like a work of love.

[@AllThingsPhysicsYouTube]

@@chadissimusrex8038 You've got the right idea with the radio wave analogy. Typically, for a picture or pattern we would talk about spatial frequencies instead of temporal frequencies, but in this case the spatial and temporal frequencies are directly related. And yes, these videos are definitely a labor of love. Hopefully, if I can get enough subscribers I will be able to do it full time!

[@timebender25]

One question we had, that this trilogy doesn't answer, is the question, "Are square orbits possible when they're not the furthest orbiting body?" As in, including the gravitational influences of planet orbiting beyond the square one (whatever their orbits may be), are square orbits possible? In the "Sphere the Cow" of the Fourier Series, the answer is "of course, just assume that all further bodies are of negligible mass". But assuming that the mass of further bodies *isn't* negligible, and *is* important tot he resulting orbit's shape. is it still possible?

[@AllThingsPhysicsYouTube]

Fascinating question. All I can say is that according to Fourier it takes an infinite number to make a true square. So based on that the answer would appear to be no. But Fourier series is just the only systematic method we know for determining such orbits, so it certainly seems possible to me. I mean, if you consider the nearly square orbit from the first video, it seems reasonable to suspect that some further orbiting objects with just the right properties might very well result in making the nearly square orbit even better. But of course, this is just my intuition. The million dollar question is how would one go about finding the objects that would lead to this square orbit. And that I don’t have an answer for.

[@Rene_Christensen]

How did you convert the pictures to coefficients? I have tried different software but they have not been that great.

[@AllThingsPhysicsYouTube]

Do you mean how did I convert the coefficients to pictures? I use a Python package called manim, developed by Grant Sanderson of 3b1b. I mention this briefly in the Square Orbits Part 2 video.

[@Rene_Christensen]

@@AllThingsPhysicsKZbinNo i mean where you got the initial pictures from expressed as numbers such that the coefficients can be found in the first place. Did you draw the guitar and then convert it into numbers?

[@AlexLapugean]

4th in the Fourier series

[@noj5393]

5th in the Fourier series!

[@KiviliG]

What graphical software was used to simulate this?

[@AllThingsPhysicsYouTube]

I use manimCE, the community edition of Manim, the mathematical animation python package created by Grant Sanderson of 3Blue1Brown.

[@IndrayudhDas]

Awesome concept but it is a N-body problem clearly and you are taking oscillating perturbation but how do you know these perturbations are stable?

[@AllThingsPhysicsYouTube]

Yes, I address this briefly in the Part 2 video, where I point out that such orbits are not going to be physically realizable. Interestingly, I think I discovered a way of getting a truly square orbit without resorting to hundreds of moons, but it will be a little while before I make a video about it.

[@IndrayudhDas]

@@AllThingsPhysicsKZbin Alright, I was also trying to figure out something with my instructor. I can't wait to find out your solution.

[@Darthvanger]

What if our universe is just a set of bodies drawing a cat shape at a larger scale 😹

[@harriehausenman8623]

And I really think you should put a seizure warning of some kind in. Just to be sure nowadays.

[@AllThingsPhysicsYouTube]

Heh...yes, it's a bit chaotic in some spots!

[@harriehausenman8623]

@@AllThingsPhysicsKZbin I like the chaos. Strange attractors are just a pertubation away 🙂 just some people have problem with flashy-stuffs of varying degrees

[@AllThingsPhysicsYouTube]

@@harriehausenman8623 It will likely be a while, but I will eventually get around to making a video on chaos. It's such a fascinating topic!

[@JonFrumTheFirst]

30th.

[@morchel332]

if ur ever in germany hmu and lets do some lysergamides ;P

[@AllThingsPhysicsYouTube]

Not likely to visit Germany again for a while...Thailand is probably my next big destination...

[@killianobrien2007]

2nd

[@cubbybartles997]

3rd

[@AllThingsPhysicsYouTube]

Cubby Bartles? That name sure sounds familiar. I suspect I know who this is!

[@harriehausenman8623]

@@AllThingsPhysicsKZbin third + first i

[@edmund-osborne]

frist