Chaotic Balls (and other animations) - Numberphile

[@PhaTs00p]

"Before I saw this, I saw lots of red boxes telling me my code wasn't right". This gives me hope

[@yafu2599]

It looks pretty, but so what, there were millions of these kinds of demos on Commodore 64s in the 80s, they were *so* *common*, the Amiga even launched with a bouncing ball demo Nothing new here at all.

[@mwu365]

@@yafu2599 and?

[@edmoore]

@@yafu2599 I think you need to learn the difference between teaching and research.

[@yafu2599]

​@@edmoore this teaches nothing and provides no new research nothing to actually show for his years on this earth and his government funded position except this video with 26year old technology Yeah FRO!

[@edmoore]

@@yafu2599 The other youtube comments disprove your first point - some viewers have learned something. Your second point implies that this is his life’s work which is clearly nonsense. Finally, his bio says he’s an engineer at Apple, not government funded. So, everything you’ve just said is wrong.

[@jasonbraun127]

I love the sound design on these. So serene yet mystical.

[@OrangeC7]

Reminds me of the sound effects in mini metro

[@OliverUnderTheMoon]

My friend made a circular puzzle platformer game called Oco which had similar animations and dynamic sounds I think, maybe you'd like it.

[@planetsoccer99]

Thanks to the Maestro Alan Stewart

[@eugeniusz6735]

Makes you wander whether chaos theory is related somehow to music theory (total amateur in both fields here, so rly idk)

[@eugeniusz6735]

I would also guess the sound effects wouldn't be there if it weren't for the author being aware of such relationship.

[@ilikaplayhopscotch]

Slightly disappointed that the animations didn’t play for like 5 minutes each but that’d be asking for a bit much.

[@Reidemeistermoves]

you should check out Nils Bergland's channel, it's super trippy but also has some really awesome physics in it

[@peterwhitey4992]

@Lakshya Gadhwal - It wouldn't make any difference for the computer.

[@MrJdsenior]

@@peterwhitey4992 Yup, that simulation is a no op compared to a real time six DOF FEA of a car running into a wall at 50 mph, or whatever. Or a very complex asynchronous circuit with all the part parameters fed in running a thousand or hundred thousand iterations with parameters randomly or even full swing best/worst cases. Or a HIGHLY complex fluids problem, like air flow within an aircraft turbine and outside of it or the flow across the entire aircraft at all angles of attack, velocities, etc. I think it was intended to be a joke, though...but JIC. :-/ :-)

[@peterwhitey4992]

​@@MrJdsenior - That's not even the issue. The processing for this simulation doesn't become more complex over time.

[@PracticalAI_]

@@peterwhitey4992 It's made in Mathematica, it's too slow for real-time (terrible language)

[@AdarshSingh-wv4ff]

I'm thankful to you for bringing Matt Henderson to us!

[@alexanderbilbo]

On the question if the chaotic circle would ever form a circle again, and Matt says that there's a theorem that if you let the simulation run long enough it will. Shouldn't this be fully predictable from the beginning? given the determined variables, the entire run (indefinite if so may be) should be predictable from the start. posted here so someone can see it

[@Varun2799]

@@alexanderbilbo I think it is related to the pigeon hole theorem which is if there are finite number of states in a system then after some n state changes, there will always be a loop. An interesting result from this theorem is the fact that if you start with the solved state of a Rubik's cube and if you repeat a certain pattern of moves, then eventually you will return to the solved state again because: 1) Each move transforms one state into a different one 2) There are a finite number of states Therefore there will eventually be a loop This same principal applies to this pool table as well.

[@Sanjay-ub7eq]

@Naman the one who programmed these chaotic codes

[@alexanderbilbo]

@@Varun2799 Yes if there is a cap on the n of combinations it will eventually repeat, but when programming this simulation, you could theoretically create infinite amounts of combinations, and if you have the intelligence in the hardware, sort out those who repeat and those which never repeat, correct?

[@Triantalex]

??

[@elliotkeil6063]

Shoutout to my fellow numberphile-phile’s

[@Ana_crusis]

And into the infinite regress...

[@nkanyezihlatshwayo3601]

Number(-phile)²..??

[@concernedcommenter8258]

Yo

[@Nawmps]

shoutouts to numberflips

[@PotatoShirts]

Word.

[@rene0]

Hey, a new face if i'm not mistaken? Nice! Get him more often too, we love chaos!

[@mayabartolabac]

yeah i don't think we've seen him anywhere else on numberphile

[@danielliao265]

"I love chaos"-mathematician joker

[@MrJdsenior]

Everyone loves chaos!...oh wait.

[@davidappelgate320]

11:25 through 11:30 the blue ball fits PERFECTLY with the music. So satisfying.

[@quinn7894]

Underrated comment. Best drum solo.

[@ayushdeep7900]

I know him from Twitter, he's Matt Handerson, he is a great animator, Grant Sanderson introduced him on Twitter

[@drenzine]

The names sound similar

[@Triantalex]

??

[@dominicdelprincipe2583]

I really appreciate the sounds added to these impeccable animations. They enhance the experience by a lot. Thanks for that, and the artistic presentation in general

[@rickseiden1]

Matt is obviously going to be a wonderful addition to the channel!

[@SquirrelASMR]

I need more of his animations! He has a Tumblr blog where he used to upload them all

[@FLS96]

I love the audiovisual animation in the end. Music combined with maths, even if chaotic, is always interesting.

[@nikolayrayanov2895]

That was awesome! We want more of Matt! I've been thinking and imagining these things in my head, but seeing them visualized is a completely different story.

[@vtron9832]

I also adore the sounds that accompany the animations and bounces. Very soothing.

[@seth333]

Thanks!

[@beantown_billy2405]

The videos with Matt are great. Please keep them coming.

[@SquirrelASMR]

More of this guy, his animations are so interesting, I need more! And also his Tumblr blog has lots of cool ones.

[@Thesebji]

The movement of centre of mass of Lorenz's wheel blew my mind.

[@Mrfrog2024]

I love the visualisation, its on par with the weird and wonderful graphs. Its encouraged me to actually code some maths so, thank you!

[@HAL--vf6cg]

therapist: nerdy henry cavill doesn't exist, he can't hurt you nerdy henry cavill:

[@ColinBroderickMaths]

I think Henry Cavill is nerdy Henry Cavill

[@RedRad1990]

At first, I read "nerdy *herdy* cavill" 😂

[@avadakedavra80]

I was thinking the same, he looks likes him.

[@erickgarcia6494]

I was just thinking he looks like Henry cavill

[@andy-kg5fb]

"But before that i got a bunch of red boxes telling me my code is wrong". Hashtag relatable there. You can't say you coded something without getting the red boxes

[@Veptis]

It's one of the greatest Twitter accounts you can follow. There is beautiful animations every few days really beings abstract concepts down in a visual (and sometimes aural) systems that our brains have spent a lot of generations on to understand.

[@Mutual_Information]

It’s really nice to see Matt here. Can’t think of someone more well deserving.

[@wonder_platypus8337]

Absolutely love your videos. Im not great at math myself but I'm working on it daily. My job definitely helps too as I work with a lot of numbers.

[@Maazin5]

I love this! Showing the actual code used in the animation demystifies it and makes it easier to understand

[@matthenderson1672]

Amazing stuff - really well explained. Can’t wait to see more from you on Numberphile

[@Kilroyan]

Incredibly interesting, and beautiful to look at. I'd be glad to watch more of this!

[@ClementinesmWTF]

I really want a part 2 on this with more animations and visualized situations that can be described like this. Maybe some 3D stuff or more "complex" (while still simple) systems. And definitely longer animations. This was so interesting to see.

[@Nick_Trevino]

I've been following Matt on Twitter for a while now, his animations are great

[@georgehumphreys7450]

The billiards examples were beautiful!

[@IwoIwanov]

The world needs this kind of content. Great (as usual). Keep up this outstanding work.

[@VanDerPol]

The konvex pool table is an absolute fantastic and descriptive visuslization of the sensitivity of chaotic systems on initial conditions.

[@guilhermereis2303]

could you share the Mathematica codes ? I am learning about how to do Mathematica simulaitions and it have those codes to look would be great. Awesome video by the way !

[@TheSucread]

code is visible in the video. I've wanted to paste it here, but for some reason KZbin is blocking/deleting my comment.

[@Dalroc]

Cool Matt Henderson! Love his animations!

[@sciencedoneright]

Yay, a new member to this wonderful math channel ❤️

[@brady1123]

I am surprised that the WhenEvent condition never misses a bounce out since he's doing floating point math and using strict equality with an integer. Maybe Mathematica is smart enough to use an epsilon neighborhood when evaluating equality between floats and ints?

[@Bouldest1]

If it works like MATLAB, then it does some smart variable length stepping and looks for sign changes to know if it has passed an event!

[@DrBrangar]

Mathematica is capable of symbolic math, not just numeric. The code he has written solves the differential equations symbolically (because they are easy equations), and so he implicitly has a symbolic function to solve for when the next impact will be. There will be numeric precision issues down the line, as the 16th place errors propagate. This is actually visible in the lack of energy conservation that should be there with these differential equations.

[@brady1123]

@@DrBrangar That's interesting. So Mathematica solves for the analytical solution of the DEq up until the WhenEvent condition occurs, at which point it applies the 'reflection' function and then re-solves the DEq again?

[@mathgeek420]

@@DrBrangar It looks like he's using NDSolve instead of DSolve. Isn't that going to be a numerical solver then?

[@Nukatha]

Yes, but the Wolfram devs have spent decades making sure boundary conditions behave as one would hope. Perhaps excessive speeds and huge timesteps would get you into trouble, but it just works nearly all the time

[@txe9113]

Really satisfying animations, I would love to watch more of these.

[@vervok]

Look up Sebastian Lague if you're interested in these kind of animations. Especially the Ant & Slime animations video!

[@MichaelDeHaven]

Nils Berglund channel also has lots of cool videos of billards you might like.

[@rohitchaoji]

One of the best Numberphile outros

[@tkimaginestudio]

Awesome work by Matt!

[@VICTORdoomm]

every time the blue ball hits the edge of the circle, the circle rotates clockwise, and rotates counter clockwise when the yellow ball hits the edge. With the tracers and counters on, which ball hits the edge more and which direction is more.... bonus footage :)

[@adielubbe4200]

another quality video. Thanks Numberphile. :)

[@555stevens]

Yay, Matt Henderson!

[@stylis666]

I'm loving the drums! It's music to my ears

[@jw41538]

this is maybe the most calming numberphile video.

[@thepacpac2398]

Really love this topic. Impossible to fully understand, but that's the beauty of it

[@Titanic-wo6bq]

I love this. More of this guy please.

[@benjaminramsey4695]

I feel like you just scratched the surface of this!

[@PhilBagels]

Excellent! I especially loved the waterwheel example!

[@WereWade]

Great stuff. I enjoy thinking about all sorts of mathematical curiosities and making my own math problems.

[@geekjokes8458]

and if you want to see (a lot) of these animations, there's a channel from a guy named Nils Berglund not only sinai billiard simulations, but also a gas in weird chamber shapes, waves in fractal ponds and a lot of other chaotic systems he's a physicist (i think?) and started posting them only a couple of months ago, and "the algorithm" favoured him and now he gets a couple thousand views a video

[@Arikayx13]

Omg, glad I’m not the only one hypnotized by those videos.

[@gutzimmumdo4910]

more of those, those are the coolest.

[@DrBrangar]

Other fun fact about that bouncing ball problem, it happens to demonstrate very cleanly that some numerical integration methods are not energy conserving, and this system is constantly leaking some out

[@jackwilliams1468]

Absolutely love that you shared the mathematica code

[@LincolnChamberlin]

I like this guy, can we have more from him?

[@dehb1ue]

Nice job with the audio addition to the animations.

[@robelbelay4065]

Excellent explaination, first time I understood code :D Thanks!

[@Wheau]

The rapid bounces at 11:26 are pretty cool!

[@eugene887]

What a sick beat at the end!

[@mzadro7]

petition to make this a series, kinda like the graph series with Neil Sloane 👇

[@mathoc5273]

Yes! Love seeing new hosts.

[@keatonwright5764]

Phase space and attractors and Fourier spaces and lots of cool ways to analyze chaotic motion

[@LordHengun]

Yo, the blue ball did a sick drum fill at the end

[@allyourcode]

I love that the code is shown and explained (even a Matlab hater can sort of follow along). Elegant!

[@stoatystoat174]

Loved this, Would love some half hour youtube videos of these animations running. Not sure why they are quite so attractive so they must be art or summat

[@WhatAreDrums729]

Outro sounds like a lot of my drum lessons. Pure chaos.

[@jeroenrl1438]

Loved the drums in the end. We really need a band using that as drum track for something!

[@topkek9563]

That outtro. Banging.

[@macronencer]

The most important question I want answered is: how do I get a job like Matt's?

[@joelproko]

That water wheel exists in physical form at the FHNW in Windisch, Switzerland. I walked by it every time I went to the train station when I still lived in Windisch. Sadly, the buckets aren't *quite* big enough and if conditions are just right, the water can fall between the buckets directly into the basin below. That means people can hold the wheel in that position until all the buckets are empty, which halts the motion of the wheel until a strong enough gust of wind puts it in motion again (either by pushing on the buckets asymmetrically or by blowing the water stream sideways enough that it lands in a bucket) or until it rains.

[@vijaypanchalr3]

Sanderson and handerson loves animations.

[@EinLucas]

Such an interesting visualization.

[@timanb2491]

One of the most inrtesting and unexplored area in math!

[@kovlo9853]

Chaos theory 🔥

[@egodaha]

wow. This is amazing. Great work!

[@alexandersmith4796]

cool dude, bring him on for more.

[@austynhughes134]

It has to be a good sign, that the morning of the day I am getting married (Today, 7/24) a new Numberphile video is posted!

[@FromTheMountain]

Have fun today :)

[@RSPikachuAlpha]

Congratulations!

[@m136dalie]

Would love to see more of this

[@S2I2]

I want to watch a video where this plays out for an hour

[@YuanLiuTheDoc]

This is a wonderful demonstration of an aspect of chaotic behavior that is not often discussed, i.e., what about linear systems? If I am not mistaken, all mathematical study of chaotic systems begins with nonlinearity. By definition, a linear system, even a diffusive one, is deterministic. In other words, none of the animations is truly chaotic. However, Henderson asked a really important question (I'm extrapolating his question here): When you are observing very-long-period motion that has lost the initial simplicity, what can assure you that the motion is periodic? If we accept that any observation introduces uncertainty, the answer is unknowable. As a consequence, we have to accept that even a perfectly linear system is chaotic in reality. (To think, fundamental quantum mechanic equations are linear.)

[@david203]

Observation or measurement is actually not a part of quantum mechanics, in the standard (Copenhagen) interpretation. That is actually one of its many flaws. A linear system, including a quantum mechanical one, cannot show chaos. That is one of the beauties of QM: that its predictions are precise.

[@phasm42]

Takes me back... when I was in middle school in the early 90s, I had some books on chaos theory and would simulate various chaotic attractors in GW-BASIC 😅

[@michaelpaullane]

My dude is playing with screen savers all day. I remember when the pipe screensaver came out- "it was so random and fascinating"

[@RedStinger_0]

7:35 Cool album art!

[@brunofa4559]

I could watch these examples and animations for hours haha

[@SaveSoilSaveSoil]

Beautiful animations! Very cool!

[@damiendavenport8899]

The fact that he has CONTROLLED points for the first two collisions of the circle makes this completely NOT random.

[@onebronx]

Chaotic != random

[@monkerud2108]

Just normal billards with straight edges but with rotation effecting and being effected by the angle, pretty sure that introduces chaos as well.

[@HypnosisBear]

Nice 👍, I love Chaos!!!

[@steveb1243]

Excellent exposition.

[@onlyeyeno]

@Numberphile Thanks for another great video. And please please, if at all possible, feature more of Mr.Hendersons work on the channel Best regards.

[@Lightning_Lance]

Whenever there's a video about chaos theory I tend to think "oh I know about this, it's cool, but how interesting can it be to watch more on it?" and then the video turns out to actually be super interesting.

[@1122slickliverpool]

This would probably be a fun front end project to make.

[@ferociousfeind8538]

What an amazing drummer

[@Saladassu]

a screensaver or a wallpaper of the left bouncing ball one would look dope, you could have it so the ends fade at the same rate as the lines are drawn

[@L0j1k]

YO I LOVE THIS DUDE!!!

[@TheVectorious]

I could watch the circle billiard problem for hours.

[@ammaleslie509]

Wow I'd pay a lot to have that for a screen saver, with the sounds

[@internetuser8922]

In 1994, my parents said the Jurassic Park movie was too scary for me to see in theatres. So they got me the original Michael Crichton novel instead (which happened to be way more violent than the movie). However, it did introduce me to Chaos Theory, and I've been interested in it ever since. The book even shows iterations of a dragon curve fractal as separations between groups of chapters, which was one of my first exposures to fractals as well.

[@JMUDoc]

Chaos is not about being unpredictable; it's about being beyond the practical limits of predictability.

[@babytextor]

When Clark Kent goes to college instead of working for the Daily Planet

[@tommythecat4961]

A whole video filled with perfect album covers... Pink Floyd would be proud of you!