How about you just take the equations (or approximations of them that require less computing power) and run them as a particle system in real time? Then you get a unique wallpaper everytime!
I think I just found my senior thesis topic. Thank you so much. Edit: My Senior Thesis was on the Lorenz Attractor. My professors gave me an A! Thank you for the inspiration!!!!
Bruh I come from a math HS and my skills are so lacking yet "existing" enough to grasp a very tiny bit of this, thanks for reminding me I've gotta study more and more and more given how beautiful what awaits me could be 👀✨❤️
I just finished an applied math PhD and if I could go back and talk to myself in high school, I'd say "there's so much more beautiful math that you can't even begin to imagine.."
Hi again! I noticed that a lot of people recognized and enjoyed Debussy's "La Cathédrale Engloutie". I just wanted to let you know that besides studying data science, I'm also a cellist for a local orchestra in St-Maurice, Switzerland. We support young musicians by hiring musical coaches to accompany our projects, and performing two concerts a year for our communities. I've been with them for seven years, and I feel so lucky to be able to perform the works of great composers of the past. If you would like to support the orchestra, please click a "I'm a fan" at our crowdfunding page www.lokalhelden.ch/os-m and share our project on your socials (unfortunately, the webpage is not in English). Thanks again for coming around to appreciate beauty together, EDIT - The crowdfunding goal has been met, thank you so much! Orfeas
I'm going to be the curmudgeon being off-put by the Debussy (especially the volume of it)! I'd always rather prefer Ravel, it's more attractor, err, attractive. Or why not some cello music then? (It doesn't even have to be Bach's unaccompanied cello suites, some Geminiani, or the Trio in Eb by Schubert). Yes, of course, I turned off the sound. I know how to Internet, but it would have been nice! :)
very thoughtful music choice that added a lot of depth to the video. Debussy's pieces often "chaotically attract" around different tonal centers, and unpredictably. With a sense of delicacy and intentional grace and symmetry. I appreciate Debussy for very similar reasons the chaotic attractor visualizations are so interesting to look at
@@mikeciul8599 All of Debussy is stunningly captivating and beautiful! and gymnopedie is actually Satie :) Check out Valse Romantique from Debussy for an interesting sort of Satie-inspired sound mixed with beautiful Debussy harmony
@@alexandramuller9055 Hi, sure! I don't know how familiar you are with maths but - it was just a university assignment regarding dynamical networks where each node represented an oscillating complex number and there was diffusion in the network so eventually nodes would synchronize if the system was stable and attain the same real and complex parts. The value of each node was described by and ODE with a forcing term causing oscillations and a diffusion term including differences between the current node and each of its neighbouring nodes. In the images in this video synchronisation is happening when alll the dots converge to the same point and move together! This happens mathematically when the diffusion difference term in each ODE equals 0, hence all nodes have the same value! Tell me if you don't understand anything and I could simplify haha :)
@@Dabnaait is generally speaking the process of some quantity spreading from a region of high concentration to a region of less concentration. Such occurs everywhere at all times, and includes things as liquids, gasses and even humans. So, seeing the value of a node in my example as a value of concentration, or density, diffusion will raise the value of neighbouring nodes if a node has a high value.
there’s something so philosophical to this kind of math, it’s so mystical in its abstractions whenever you derive some sort of physical idea from it, it feels so much better than just saying it.
If you ever feel like your computer has been getting too complacent, this video was an otherworldly experience that felt much longer than 6 minutes and I would love to see some of these go on for an hour or more, especially with more classical music!
This might sound crazy, but can a given Lorenz attractor’s initial coordinates be reverse calculated from x number of iterations back to the start, by integrating the equations in some way ?
I think so. One can reverse "forwards" and "backwards" by putting a minus sign in every every one of the differential equations. Then it is just a matter of evolving the system the normal way. There is one caveat: these are differential equations. They are notoriously difficult to "solve exactly". You can approximate the evolution of such a system by numerical integration (use a computer, and take small steps in time), but as far as I know, it is impossible for most differential equations (or systems of differential equations) to determine exactly what the state should be after a specific amount of time.
As far as I'm aware, you can integrate forward or backward, but it will be equally susceptible to tiny variations in starting conditions either way. Notice that these points all separated considerably after enough timesteps despite starting almost on top of each other; trying to integrate backward in the same way would yield a wide variety of spread out starting points even if your ending points were all very close to one another.
you can, but it will be different each time you do it, if the initial xyz are even 0.0000000000000000000000000001 changed. its cuz the iteration is a composition of nonlinear fcns
I love mathematical art like this, but I feel like you could have done more to make the structures of these more understandable. I feel like picking a bunch of points really close together to start makes it hard to follow some of them, though it looks awesome on the Aizawa one, some of them don’t seem to be entirely on screen, like the 3-cells CNN, Newton-Leipnik and Bouali, and there should be some way to show more of the 3D structure of them (for example, points closer to the camera are larger). Ie, maybe you could start by showing the divergence from a cluster of points like you do here, but then fill the attractor out with some more evenly divided points and start moving the camera around like how you initially showed Lorentz. Also, some of the names for these are hilarious, especially Nose-Hoover xD.
I've struggled with maths for a while now, and avoided it to boot, but lately, I'm discovering more and more of an interest in it, despite being less than novice. The visuals and the equations that bring them to life are fascinating, and I was wondering where I should start, if I wanted to learn the mechanisms behind attractors. Thank you for making this video, it has ignited in me a curiosity for math I didn't know existed!
I know this was posted over a year ago, but the branch of maths related to this phenomenon is called "chaos theory". It's part of a larger branch called "differential equations". In brief, differential equations deal with how systems respond to change. For example, if you set a block on a spring into motion, how would it move over time? Or, like in the Lorenz Attractor case, if you put an air particle near a flame, how does it move over time? At the start of the video, the three equations that were shown were three differential equations. dx/dt represents the rate of change in x position over time (i.e. x-velocity) and then there are equivalent equations for the y and z dimensions. How these equations govern the position of the air particles over time drastically change based on the initial conditions of the air particle, leading to the beautiful patterns seen.
i love that these are utterly deterministic, and reversible, just like fluids shear mixing in a thin film between two concentric cylinders (a Couette cell)
If possible, I'd like to see a higher "Resolution" on the lines, there are some minor jagged egdes when the particles go very fast, such as at 3:28 on the left side. Also, it would be nice for some of them to get a more zoomed out view. Other than that very nicely done! Also I love your voice, it feels like it really fits with the visuals and music.
This is very cool! I wonder what it would sound like if you added sounds whose pitch, volume, and frequency corresponded to the position, speed and velocity of the points
These simulations are nice to look at, though I admit I had some trouble determining if a point was moving towards or away from a camera, what with the viewport being at a fixed position. Perhaps if the dots got bigger as they got closer, and shrank when they moved away? Oh well, it's a minor nitpick at most. Good video.
I turned on cryochamber's live stream and this at the same time and muted this ( but with captions ) and it matched waaaaay better than I was expecting lol great.
why do we possess the ability to find beauty in a concept so abstracted from the nature we evolved in and from? why did we evolve the ability to receive pleasure upon observing certain mathematical functions?
Chaos and Love. You have found an amazing discovery my friend. I am a very creative person, and this video strikes at the cores of my abilities. So thank you, just for sharing this. Beautiful find.
Adding dx, dy, dz to x,y,z repeatedly is called vertex integration. It is an easy way to break down and simulate a lot of complex interactions. Often used in games. (I was a game developer) in the early days every game was it's own engine. These days the 'game engine' takes care of all this 'complex' mathmatics. But if you have the brainpower for it, it is amazing to play around with. You'll never look at the world the same. I made (and sold) about 200 of these early games-that-are-game-engines in my life. Very thankful for that experience. Attractors are important, but they are just the tip of the iceberg.. most things are not really expressable in language.. possibly in maths.. currently trying that..
The Tinkerbell map is a fun one! Just had a 4 week crash course in chaotic dynamics. Was fascinating to see how laminar fluid flow transitions into turbulence. Got an A* on my report too ;)
The new Fantasia looks great! Honestly though, this was super cool. These would also make for some phenomenal energy effects in movies, TV, or Video Games.
If feasible, it would be really cool to see the starting equations (like you provided for the Lorenz attractor). They could stay on just a bit and then disappear so as not to muck up the mappings.
Wouldn’t the attractors make the equation and point less chaotic? Is it only called chaotic because it’s unpredictable to when the point will come near the attractor?
![Diggy Graves - Red Vineyard [Official Lyric Video]](http://i.ytimg.com/vi/8Ro085T0mnQ/mqdefault.jpg)
this is amazing. these attractor graphics with a higher framerate and longer runtime and you've got the perfect screensaver ngl
Made a program like this in Processing a while ago, used it as my live wallpaper on Android. Could definitely be done on PC
@@EggKiddo I never knew you could use processing to create live wallpapers 😯
I'm going to actually dive into it this time!
How about you just take the equations (or approximations of them that require less computing power) and run them as a particle system in real time?
Then you get a unique wallpaper everytime!
@@EggKiddo Please share it! I'd absolutely love to see them! I dont have a coding background otherwise I'd do it myself haha
@@rpyrat There is an app called APDE which can export your sketches as a live wallpaper
I think I just found my senior thesis topic. Thank you so much.
Edit: My Senior Thesis was on the Lorenz Attractor. My professors gave me an A! Thank you for the inspiration!!!!
Congratulations! That's so amazing!
CONGRATS STRANGER THATS CRAZY 🥳
'A' is for attractor!
@@orfeasliossatos welp the comment is kind of late
@@egebabus3423better late than never
It’s amazing how each Attractor never fails to appear harmonic and synchronised
Chaotic Composers, Everything is music, art, creativity, same thing but different initial inputs and expressionism different output.
Bruh I come from a math HS and my skills are so lacking yet "existing" enough to grasp a very tiny bit of this, thanks for reminding me I've gotta study more and more and more given how beautiful what awaits me could be 👀✨❤️
these are all little systems of PDEs
I'm really glad you liked it, stay curious my friend!
I just finished an applied math PhD and if I could go back and talk to myself in high school, I'd say "there's so much more beautiful math that you can't even begin to imagine.."
@@complexobjects sure gives a lot to look out for thanks man!! :D
@@complexobjects I've yet to encounter math that is beyond me, could you cite an example so I can push myself?
This is a beautiful video, and I think the choice of Debussy is befitting.
Always finish on de Bach, never on Debussy.
Family guy
The Debussy fits so perfectly
@@benjamintoulouse7052 weirdly specific but kinda relatable at the same time
@@alexanderbayramov2626 relatable pfp to insert into this comment chain de Bach
what is debussy?
Hi again!
I noticed that a lot of people recognized and enjoyed Debussy's "La Cathédrale Engloutie". I just wanted to let you know that besides studying data science, I'm also a cellist for a local orchestra in St-Maurice, Switzerland. We support young musicians by hiring musical coaches to accompany our projects, and performing two concerts a year for our communities. I've been with them for seven years, and I feel so lucky to be able to perform the works of great composers of the past. If you would like to support the orchestra, please click a "I'm a fan" at our crowdfunding page www.lokalhelden.ch/os-m and share our project on your socials (unfortunately, the webpage is not in English). Thanks again for coming around to appreciate beauty together,
EDIT - The crowdfunding goal has been met, thank you so much!
Orfeas
I always enjoy Debussy 😂
ah yes Debussy
had to do a double take reading this
뿅
I'm going to be the curmudgeon being off-put by the Debussy (especially the volume of it)! I'd always rather prefer Ravel, it's more attractor, err, attractive. Or why not some cello music then? (It doesn't even have to be Bach's unaccompanied cello suites, some Geminiani, or the Trio in Eb by Schubert). Yes, of course, I turned off the sound. I know how to Internet, but it would have been nice! :)
Brilliant video. Thank you for this and using your appreciation for music to make an adroit partner of performance of dynamics and sound.
very thoughtful music choice that added a lot of depth to the video. Debussy's pieces often "chaotically attract" around different tonal centers, and unpredictably. With a sense of delicacy and intentional grace and symmetry. I appreciate Debussy for very similar reasons the chaotic attractor visualizations are so interesting to look at
I enjoyed hearing a less familiar Debussy piece! Gymnopedie and the arabesques are lovely but this one was a surprising gem!
@@mikeciul8599 All of Debussy is stunningly captivating and beautiful! and gymnopedie is actually Satie :) Check out Valse Romantique from Debussy for an interesting sort of Satie-inspired sound mixed with beautiful Debussy harmony
DE WHAT
@@marselo1316 lmaooooooooooooooooooooo
@@marselo1316 you can't call yourself a man if you don't like debussy...
the music fits beautifully with the movements... simply beautiful!
Been sitting in matlab all day staring at such black dots flying around trying to understand a system. This was quite a bit more entertaining!!
what are you working on if you don't mind me asking?
@@alexandramuller9055 Hi, sure! I don't know how familiar you are with maths but - it was just a university assignment regarding dynamical networks where each node represented an oscillating complex number and there was diffusion in the network so eventually nodes would synchronize if the system was stable and attain the same real and complex parts. The value of each node was described by and ODE with a forcing term causing oscillations and a diffusion term including differences between the current node and each of its neighbouring nodes. In the images in this video synchronisation is happening when alll the dots converge to the same point and move together! This happens mathematically when the diffusion difference term in each ODE equals 0, hence all nodes have the same value! Tell me if you don't understand anything and I could simplify haha :)
@@cha4kn whats diffusion?
@@Dabnaait is generally speaking the process of some quantity spreading from a region of high concentration to a region of less concentration. Such occurs everywhere at all times, and includes things as liquids, gasses and even humans. So, seeing the value of a node in my example as a value of concentration, or density, diffusion will raise the value of neighbouring nodes if a node has a high value.
@@cha4kn wow, thanks, that's really interesting stuff
I love how this reminds me of that one courage the cowardly dog episode with cosmos squids
and here I thought I had seen every episode
there’s something so philosophical to this kind of math, it’s so mystical in its abstractions whenever you derive some sort of physical idea from it, it feels so much better than just saying it.
Anyone who uses Debussy to set the mood for complex beautiful animations gets my upvote automatically.
Debussy??? 🤨🤨🤨😳😳
Curious to hear what it would sound like if each point was assigned a midi number/event on a synthesizer
How would you assign it for 3d coordonates?
@@killianobrien2007or infinite points
The music and the colours turn these math equation into videos of awe and the very definition of beauty itself.
You can take any moment of that animation and turn it into a beautiful wallpaper
6:20
Maths and Music; order out of chaos. What a divinely inspired video. 👏👏👏👏👏👏
If you ever feel like your computer has been getting too complacent, this video was an otherworldly experience that felt much longer than 6 minutes and I would love to see some of these go on for an hour or more, especially with more classical music!
I'd love to see these in VR, the added depth perception would make it look even nicer I bet.
This is certainly an inspiration ..that tomorrow I'll be writing my exam on dynamical systems.!
The Aizawa and Halvorsen Attractors are just so breathtaking.
Halvorsen Attractor remind me Treyarch emblem or Penrose triangle
Very fascinating. Looks like energy being simulated.
Such beautiful visualizations of mathematical graphs... Mesmerizing work, well done!
Just got this recommended and im glad it did. I have a feeling this will blow up
This is - on a surface level - some kind of art. You've done an amazing job!
Why on a surface level? :)
The symmetry in Halvorsen's attractor is majestically beautiful ! Well done !
Ah, math fireworks. Every day can be the 4th of July when you love math.
love this comment!
Absolutely Beautiful, thank you for taking the time to create this. 🙏👏👏
Bro makes a single video and it is incredible
So beautiful! This was so satisfying to watch! Great video 👏🏿😍
This is absolutely beautiful, thank you so much for posting!
This might sound crazy, but can a given Lorenz attractor’s initial coordinates be reverse calculated from x number of iterations back to the start, by integrating the equations in some way ?
I think so. One can reverse "forwards" and "backwards" by putting a minus sign in every every one of the differential equations. Then it is just a matter of evolving the system the normal way.
There is one caveat: these are differential equations. They are notoriously difficult to "solve exactly". You can approximate the evolution of such a system by numerical integration (use a computer, and take small steps in time), but as far as I know, it is impossible for most differential equations (or systems of differential equations) to determine exactly what the state should be after a specific amount of time.
Probably would not work as small errors in calc will not represent correct reversed values.
As far as I'm aware, you can integrate forward or backward, but it will be equally susceptible to tiny variations in starting conditions either way. Notice that these points all separated considerably after enough timesteps despite starting almost on top of each other; trying to integrate backward in the same way would yield a wide variety of spread out starting points even if your ending points were all very close to one another.
you can, but it will be different each time you do it, if the initial xyz are even 0.0000000000000000000000000001 changed. its cuz the iteration is a composition of nonlinear fcns
@@JeffMTX Exactly. Theoretically possible - but in reality not.
I love mathematical art like this, but I feel like you could have done more to make the structures of these more understandable. I feel like picking a bunch of points really close together to start makes it hard to follow some of them, though it looks awesome on the Aizawa one, some of them don’t seem to be entirely on screen, like the 3-cells CNN, Newton-Leipnik and Bouali, and there should be some way to show more of the 3D structure of them (for example, points closer to the camera are larger).
Ie, maybe you could start by showing the divergence from a cluster of points like you do here, but then fill the attractor out with some more evenly divided points and start moving the camera around like how you initially showed Lorentz.
Also, some of the names for these are hilarious, especially Nose-Hoover xD.
I've struggled with maths for a while now, and avoided it to boot, but lately, I'm discovering more and more of an interest in it, despite being less than novice. The visuals and the equations that bring them to life are fascinating, and I was wondering where I should start, if I wanted to learn the mechanisms behind attractors.
Thank you for making this video, it has ignited in me a curiosity for math I didn't know existed!
The RUclips channel "3Blue1Brown" makes great math videos, some are for novices like you and I, and some are more advanced.
I know this was posted over a year ago, but the branch of maths related to this phenomenon is called "chaos theory". It's part of a larger branch called "differential equations". In brief, differential equations deal with how systems respond to change. For example, if you set a block on a spring into motion, how would it move over time? Or, like in the Lorenz Attractor case, if you put an air particle near a flame, how does it move over time? At the start of the video, the three equations that were shown were three differential equations. dx/dt represents the rate of change in x position over time (i.e. x-velocity) and then there are equivalent equations for the y and z dimensions. How these equations govern the position of the air particles over time drastically change based on the initial conditions of the air particle, leading to the beautiful patterns seen.
The animation, the music, the voice, the smoothness. Please make more videos
Great video!
I'm currently experimenting with non curved space attractors. They look similar to CPU architecture.
make a video
@@DiamondSane yes please
Never in my life have I felt the need to have a screensaver. Until now.
i love that these are utterly deterministic, and reversible, just like fluids shear mixing in a thin film between two concentric cylinders (a Couette cell)
This is stunning. The excellent music choice makes it remind me of a Len Lye film. I could watch hours of this set to every debussey piece.
my gosh these are each gorgeous showcases, i think my favourite is the nose hoover attractor
If possible, I'd like to see a higher "Resolution" on the lines, there are some minor jagged egdes when the particles go very fast, such as at 3:28 on the left side. Also, it would be nice for some of them to get a more zoomed out view. Other than that very nicely done! Also I love your voice, it feels like it really fits with the visuals and music.
This is very cool! I wonder what it would sound like if you added sounds whose pitch, volume, and frequency corresponded to the position, speed and velocity of the points
These simulations are nice to look at, though I admit I had some trouble determining if a point was moving towards or away from a camera, what with the viewport being at a fixed position.
Perhaps if the dots got bigger as they got closer, and shrank when they moved away?
Oh well, it's a minor nitpick at most. Good video.
Yeah, would definitely help make the 3D structure more understandable.
Or make it into a tiny program with video game style controls. It would probably fit in less than 50kb
Thank you for making this :)
I turned on cryochamber's live stream and this at the same time and muted this ( but with captions ) and it matched waaaaay better than I was expecting lol great.
The Debussy works so well with the equations. Like a grand fireworks display. Thank you! 😊
wow that was amasing!!
thanks for sharing your work :) My kids really loved it :)))
Good choice of music - _Engulfed Cathedral_ - Debussy
Brilliant and evocative, thank you. We are exploring how these might represent the dynamics of ecological systems.
why do we possess the ability to find beauty in a concept so abstracted from the nature we evolved in and from? why did we evolve the ability to receive pleasure upon observing certain mathematical functions?
Chaos and Love.
You have found an amazing discovery my friend.
I am a very creative person, and this video strikes at the cores of my abilities. So thank you, just for sharing this. Beautiful find.
This is beautiful and I'd love to do something like this in VR
Perfectly selected music, thank you for tbis
Thanks man, I appreciate it very very much. I saved this video for later to take notes and research all of those!
Fantastic video! Debussy was a very good choice for music.
Videos like these make great backgrounds
A Brilliant choice of an auditory attractor..🎼
i'm blaming you for making me get addicted to watching these, these are just satisfying to watch.
so beautiful whit the music. excelent selection. thanks
These are incredibly beautiful thank you
Absolutely beautiful! You’re really talented!
Adding dx, dy, dz to x,y,z repeatedly is called vertex integration. It is an easy way to break down and simulate a lot of complex interactions. Often used in games. (I was a game developer) in the early days every game was it's own engine. These days the 'game engine' takes care of all this 'complex' mathmatics. But if you have the brainpower for it, it is amazing to play around with. You'll never look at the world the same. I made (and sold) about 200 of these early games-that-are-game-engines in my life. Very thankful for that experience. Attractors are important, but they are just the tip of the iceberg.. most things are not really expressable in language.. possibly in maths.. currently trying that..
Absolutely stunning--time to read up on chaos theory.
Theres chaos in your dp
If it’s not been done already, these would make an awesome drone display.
I woulve liked the same explanation at the beginning for each light show afterwards, but still a very nice video.
There's just something weirdly calming about this looking organic.
This is what I see when I listen to exceptional music (Vivaldi, Debussy, etc). It’s so breathtaking that it sometimes make my legs wobble.
The second one gave me fireworks wibe from childhood. I hope there is a longer version.
Perfect choice of music!
This was beautiful. Thank you.
The Tinkerbell map is a fun one! Just had a 4 week crash course in chaotic dynamics. Was fascinating to see how laminar fluid flow transitions into turbulence. Got an A* on my report too ;)
We need more of these, please. And we need them a couple hours long. Thanks lmao
Thank you in the deepest sense. : )
They look like they're dancing, it's so beautiful
This is such a quality video! Thanks so much
The new Fantasia looks great! Honestly though, this was super cool. These would also make for some phenomenal energy effects in movies, TV, or Video Games.
I'm not a big fan of math but I appreciate the beautiful things that its created, such as these "attractors" as you call them. Great video mate.
This was beautiful. Thanks for sharing 👌👌
I want to watch this in vr. Seeing this in a three dimensional plane with some soft music would be beautiful and likely hypnotic.
so thats how screensaver were made
I think this i what death would feel like. A forever show of chaotic attractors. Such a delight.
I thoroughly enjoyed watching this. I miss taking mathematics in University. I had to stop due to unforseen issues. I want to go back again.
I like how there were some individual points that would kinda break off from the group but eventually rejoin
Math is the beautiful instrument by which the song of existence is played.
it looks like stars dancing on the canvas of a blank cosmos.
I thank you, earnestly, for sharing, my favorite is the one at 1:48
This kind of geometry math is very useful in game programming, I'm going to save this video and try to use it
Need a masterclass on how to recreate this. It's wonderful.
That's so beautiful thanks for sharing
I don't understand math behind these but chaos is truly amazing and beautiful
If feasible, it would be really cool to see the starting equations (like you provided for the Lorenz attractor). They could stay on just a bit and then disappear so as not to muck up the mappings.
Incredible music choice, reminds me of some trippy VHS we'd watch in class 😆
that was unbelievable, beautiful video
Halvorsen attractor, very sick!
this is some S tier cool math shit i dont understand
this is the most beautiful thing i have ever seen. i cried
Wouldn’t the attractors make the equation and point less chaotic?
Is it only called chaotic because it’s unpredictable to when the point will come near the attractor?
Makes me want a screensaver with a cycle of all of these attractors...
This is great, will done!
beautiful work, thank you
These would have made for sick screensavers back in the day.