This equation will change how you see the world (the logistic map)

The logistic map connects fluid convection, neuron firing, the Mandelbrot set and so much more. Fasthosts Techie Test competition is now closed! Learn more about Fasthosts here: www.fasthosts.co.uk/veritasium Code for interactives is available below...
Animations, coding, interactives in this video by Jonny Hyman 🙌
Try the code yourself: github.com/jonnyhyman/Chaos
References:
James Gleick, Chaos
Steven Strogatz, Nonlinear Dynamics and Chaos
May, R. Simple mathematical models with very complicated dynamics. Nature 261, 459-467 (1976). doi.org/10.1038/261459a0
Robert Shaw, The Dripping Faucet as a Model Chaotic System
archive.org/details/ShawRober...
Crevier DW, Meister M. Synchronous period-doubling in flicker vision of salamander and man.
J Neurophysiol. 1998 Apr;79(4):1869-78.
Bing Jia, Huaguang Gu, Li Li, Xiaoyan Zhao. Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns Cogn Neurodyn (2012) 6:89-106 DOI 10.1007/s11571-011-9184-7
A Garfinkel, ML Spano, WL Ditto, JN Weiss. Controlling cardiac chaos
Science 28 Aug 1992: Vol. 257, Issue 5074, pp. 1230-1235 DOI: 10.1126/science.1519060
R. M. May, D. M. G. Wishart, J. Bray and R. L. Smith Chaos and the Dynamics of Biological Populations
Source: Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 413, No. 1844, Dynamical Chaos (Sep. 8, 1987), pp. 27-44
Chialvo, D., Gilmour Jr, R. & Jalife, J. Low dimensional chaos in cardiac tissue. Nature 343, 653-657 (1990). doi.org/10.1038/343653a0
Xujun Ye, Kenshi Sakai. A new modified resource budget model for nonlinear dynamics in citrus production. Chaos, Solitons and Fractals 87 (2016) 51-60
Libchaber, A. & Laroche, C. & Fauve, Stephan. (1982). Period doubling cascade in mercury, a quantitative measurement. dx.doi.org/10.1051/jphyslet:01.... 43. 10.1051/jphyslet:01982004307021100.
Special thanks to Patreon Supporters:
Alfred Wallace, Arjun Chakroborty, Bryan Baker, DALE HORNE, Donal Botkin, halyoav, James Knight, Jasper Xin, Joar Wandborg, Lee Redden, Lyvann Ferrusca, Michael Krugman, Pindex, Ron Neal, Sam Lutfi, Tige Thorman, Vincent
Special thanks to:
Henry Reich for feedback on earlier versions of this video
Raquel Nuno for enduring many earlier iterations (including parts she filmed that were replaced)
Dianna Cowern for title suggestions and saying earlier versions weren't good
Heather Zinn Brooks for feedback on an earlier version.
Music from:
epidemicsound.com "What We Discovered" "A Sound Foundation 1" "Seaweed" "Colored Spirals 4"
ve42.co/Artlist "Busy World" "Children of Mystery"

[@tanvisharma6903]

Once the video reaches a certain rate of complexity, our brain starts to understand only periodic parts of it. Until it's all chaos and you throw your phone away.

[@realdanrusso]

yeah i made it 8 minutes in --with so much struggle-- and then blew up

[@ashfujiyama]

this is such an underrated joke

[@caturlifelive]

up

[@gingaming_gg]

Pretty much 😂😂😂

[@depressedguy9467]

Go back

[@AustenAlexander]

Me: struggles with elementary algebra Verisatrium: Feigenbaum constant stays relative to the bifurcation diagram Me: go on

[@nickreed7277]

Damn dude. your elementary had algebra? i didnt get to that till highschool

[@Matt_10203]

@@nickreed7277 or he means that it is elementary level algebra, not in relation to schooling, just that he struggles to understand the basics of algebra

[@Edlundd]

Bro I dont even know what algebra means

[@JoeyLovesTrains]

Lol I got algebra in 8th grade in Middle school But my class was the only one, everyone else had pre-Algebra

[@luppa79]

Verisatrium :)

[@user-qx5cn1si1q]

Things like these actually make our world more magical than any science fiction

[@glowerworm]

We're living in a simulation and the 4.669 number just happened to be one of the global variable definitions.

[@ihatemicrosoftsobadly3188]

@@glowerworm Dear matrix guys don't use global variables it's bad practice

[@glowerworm]

@@ihatemicrosoftsobadly3188 lmao, maybe it's destiny that every programmer, no matter how powerful, will write bad code.

[@ihatemicrosoftsobadly3188]

@@glowerworm so true ... ;)

[@justpassingbyy]

@@glowerworm Shoulda been 4.2069 imo

[@brucefoote540]

Your 3 dimensional animation of the rotating Mandelbrot Set graph conveys so very much information and complexity. Amazing!

[@ingridfong-daley5899]

YES! Thank you for such well-constructed visualisations. They really are like compact little gold mines of information. :)

[@3orM00Rrecharacters]

I’m an artist and took screenshots of the set graph and logical equation as inspiration. Data really is beautiful.

[@ingridfong-daley5899]

OMG i took screenshots as well :) Agreed--data IS lovely.@@3orM00Rrecharacters

[@HiManLOL]

people out there proving existance of god with Mandelbrot fractal

[@convolutionality]

its so pretty my inspiration too

[@ColorwaveCraftsCo]

The Mandelbrot Set in 3D is probably one of the coolest things I've ever seen

[@erteple2647]

looks like a deformed pringle

[@benhsuan4817]

That and the "shadow" of a 4-dimensional cube passing through 3 dimensional space are the coolest mathematical animations I've seen. Matt parker shows it at the end of his talk at the Royal Institute if you're interested.

[@mizomint4197]

But... Humans can't see in 3D... So... Have you really seen it?

[@bestawdomx6835]

@@erteple2647 there is infinite?

[@steveskeletonne-7394]

kzbin.info/www/bejne/b4aTiYmaq7aqd6s This is essentially just a bunch of 3D fractals.

[@youmaybebusy]

"Oh cool, the Mandelbrot-Set. I have seen that before." *turns Mandelbrot-Set in 3D* "What the ...?! That's illegal!"

[@DrIridium]

haha same

[@scurvofpcp]

I know, right

[@IRuinEvrything]

relatable

[@smoothe14]

This is getting out of hand, now there’s two of them... wait four of them. Hold up. Eight of them... wtf a chaotic number set of them??!? This really is getting out of hand. Oh wait no just two of them.

[@Uberkist]

mandelbulbs are gunna blow you away bro

[@zanityedpo1398]

When I was young, I always paid attention to my leaky faucets. I would mess with the flow since we were already wasting water, and I noticed this happening. Warms my heart to see it explained 💜

[@strayspark1967]

i've had a similar experience, now explained 30 years later......strange world we live in, everything makes sense, eventually

[@dmajorvgm8735]

The video didn’t really explain the behaviour though, it just made an observation/comparison to the mathematics they were talking about. I’d be interested in a more in depth video that actually explains why we see this behaviour so often

[@nu1x]

@@dmajorvgm8735 Well, noone knows WHY, people just noticed where it matches the algorithm. And why may be, I strongly suspect, some extremely fundamental basis on which the whole reality has existence, hence why this video is quite startling.

[@joekelley5121]

I've had courses in Algebra, Calculus and Trigonometry, but never really had these concepts connected to actual real-world phenomena, so it always felt very... mechanical, very transactional. If they incorporated these concepts into the teaching, I believe it would make it much more relatable during the process.

[@mikeinjapan2004]

Good point. But there are people who can actually connect those things naturally. I think this is also a skill on its own - we don't need to be thought because we see those patterns interacting each other and what have presented seems to be trivial...

[@stiefjac5082]

100% i always thought that we would have leaps and bounds of breakthroughs in understanding the entire structure of the universe if we taught students in school basic math... and then showed them crazy concepts like these in the hopes that one of the billions of people on the earth will see them and go "huh that reminds me of x, what if we apply it here?" rather than pushing them through advanced mathematics to get degrees not only that but if you look up slow motion explosions on youtube and look closely the inital explosion looks like the mandelbrot set, and you can see that as it expands there are little bits of contraction in every single explosion if you take it frame by frame i watched this video when it came out 3 years ago and i've been obsessed with the concept ever since

[@ADUAquascaping]

​@@mikeinjapan2004You probably notice all of the 25+ functions within plasma/flame 😮😂

[@va941]

Guess you never did psychedelics while in school then😂😂😂

[@piercelane7757]

A lot of this video is centered in something you might find in a numerical analysis class. I just took one in college and we went over a lot of this kinda stuff.

[@entropy_7827]

Everyone's gangsta until Mandelbrot set gets rotated on the Z axis.

[@FunTheMentalist]

when he rotated the Mandelbrotset on the z axis, i felt that

[@kumarsaieshrane9692]

😂😂😂😂

[@masterpsk]

I did the "Keanu whoa" follow by a long "Duuuude!"

[@gangsta6730]

I was like "hol up-"

[@eusebiorozco]

Hahahahaha

[@feynstein1004]

Numberphile: Makes video about the Feigenbaum constant Veritasium: Makes video about the Feigenbaum constant but includes the z-axis

[@unathimatu]

it was a bifurcation

[@BinyaminTsadikBenMalka]

@@unathimatu lol, perfect response dude!

[@BasiC7786]

and then there's 3blue1brown telling you what PI has to do with all of this

[@cezarcatalin1406]

Feigenbaum constant: Works for all recurrences that come from functions with a single hump. A mathematician: Well, we have to demonstrate that A computer scientist: Give me the Summit supercomputer for a few days to find a counterexample

[@larrysal8866]

_mind blow_

[@madeeb04]

The fact that “chaos theory” is a real thing absolutely blows my mind. Even pure chaos can be mathematically modeled. Incredible.

[@falcxne]

For anyone wanting to study the math of this, I recommend Steven Strogatz - Nonlinear Dynamics and Chaos. That was the textbook I used in undergrad.

[@simrans3675]

So, basically, 'Chaos' is a misnomer now. There is clear method and math to the madness. Are we really living in a simulation :-)? Sometimes, it fascinates me to see all the advancements made in math, science. probability. stats etc with a simple thesis to put some structure/framework to quantify the world around us. It has helped us explain a few phenomenons, create products, even predict stock prices, human behavior (looking at patterns and trends etc) but the other part of me is conflicted - for life and its energies cant be modeled. It's too fascinating to be modeled out and the vastness of the universe and my spiritual beliefs are a complete anti-thesis to my scientific side. Heck, even science is ever changing - now, we are discovering qubits - particles that carry both -ve and +ve charges, which has deep implication to our view of physics and the world around us! While amazing, how much can humans really 'figure out' vs realizing there is something more to life that what even our more prolific attempts at math/AI/ML will ever get us to.. Or, perhaps, there isnt a higher power :-)?

[@roshnirana2098]

2019:- so this is aerogel 2020:- let's throw a little chaos at children and each other

[@AngelOrtiz-wt7xt]

Roshni Rana aqq msm

[@zperdek]

Chaos is magic.

[@PavR376]

Hey we have the same last name, first time I've meet one online

[@user-pj8mt2kd4v]

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[@BAgodmode]

When you turned the Mandelbrot on it’s side to show the bifurcation, I’ll be honest, it blew my mind.

[@jingalls9142]

That one made me stand up an pace frantically for awhile....mind blown..

[@shadeop6773]

@@jingalls9142 same..... It blew my mind.

[@asparagii]

yeah that was awesome

[@carriersignal]

Same here. Had to stop it and sit back in my chair. Perhaps this is well known, but this is the first time I have ever heard of it.

[@r0flcats]

Its now my wallpaper!!! That image its insane

[@maegodragon]

Only got to 5 mins and 18secs and already getting Heart Flutter! This video animations, graphing bifurcations, and the questions you pose prior to clear explanation is absolutely AWESOME! Thanks for the Thrills!

[@marco-salfi]

You a rabbit by any chance

[@hugojj101]

You're heart chaotically fibrillated, you maybe dying haha. Nah that's mad though. If you're heart did actually double beat, that was literally an example of the video. So strange, and even stranger that complex numbers that exist on a plane we can't see are linked to all our hearts and the rest. One day we'll understand maybe. hah

[@BanditMatt]

Currently writing a report on chaotic behaviours and bifurcations, remembered this video existed and im so grateful now

[@Chord_]

"I'm so excited about chaos" -man one month into 2020

[@maumau9466]

Lmao😂😂😂

[@norpriest521]

@@maumau9466 What that means

[@starryepidemic2532]

@@norpriest521 it means laughing my ..... Off and no im not gonna fill in the blanks

[@koborkutya7338]

Well, it just tells us someone is messing with the parameters big time.

[@MrSleepyweezul]

When things are chaotic, you focus on chaos. And when you focus long and hard enough, even chaos can make sense.

[@garfieldturismo]

Everyone here were gangsta till he turned the Mandelbrot-set 3D

[@cfneal1459]

Explain? Please.

[@doctorquantum3364]

@@cfneal1459 r/woosh

[@jinx5349]

@@doctorquantum3364 you can't woosh someone for wanting an explanation.. 🙄

[@rohangaikwad255]

Hahaha

[@maxlank539]

I'm dead 😂😂😂😂

[@jeff_tj]

That book is amazing - and you have done it justice. That 3d graph was awesome!

[@jawadfarrukh6736]

Do you know which software did he use?

[@Peace2051]

Thank you for making some fundamental mysteries of math/nature at least approachable if not fully understandable to the general public. Great job!

[@TheMsksk]

"How did I get to be 37 years old without hearing about the Feigenbaum constant", thank you for lowering that bound to at least a 24 year old. Your videos always leave me inspired!

[@SergioEduP]

It lowered it for an 18 year old too =P maybe there are even younger people watching, and if there aren't we could share the video to them

[@incription]

@@SergioEduP 17, beat that

[@arvintis2293]

InCrIpTiOn 16.

[@incription]

@@arvintis2293 I think krish beat both of us

[@tomifiju]

@Krish Kalra yet here I am during meiozis I

[@edstervedster]

6:40 When the animation showed the Mandelbrot set being related to the bifurcation diagram my mind was blown...

[@elee9056]

edstervedster I WANT THAT MODEL AS A CHANDELIER. SO. BAD.

[@carloshogar2]

Same here. I am fascinated by fractals and never knew this relationship

[@airpolygon2714]

@@elee9056 you should make that idea a reality! I'd buy two

[@rianby64]

The same happened to me! not sure if really they're related... must rewind and check it again.

[@Cloud7050]

Crowdsourceeee

[@Sagitarria]

You really did a good job with this one. This is something that’s been fascinating to me for a long time, and there was even a point a few years ago when I was trying to explain to my mathematical friends, the connections between the logistic map and the Mandelbrot set and they didn’t believe me! This stuff seems so fundamental, but it doesn’t seem like it’s tot as a unifying principle when it could be. It’s just nice to see somebody acknowledge all of these different pieces of the puzzle so thank you and thank you for making some thing I can .2.

[@magic.marmot]

I've been studying emergent behavior since the 80's, when I first read Conway's Game of Life. Back then, it was treated as a nuisance, irrelevant, and annoying. I love watching the break into oscillation as your R-value changes .That's the same behavior as a damped spring, or feedback in an audio system. The transition to chaos between states makes a lot of sense. I love the visualization of the Mandelbrot set, that gave me the perfect starting point to understand it in another dimension. Wonderful.

[@jemert96]

*_rotates the Mandelbrot plot_* Everyone: "wait what's he doing" Me: "he's beginning to believe"

[@futurefacts6909]

hahahahaha

[@futurefacts6909]

is it normal this video gave me goosebumps ?

[@londonkoster3888]

@@futurefacts6909 yes

[@jvm53]

yeah, you are special...

[@londonkoster3888]

@@jvm53 she is

[@DanielSmania]

I am a mathematician, and I do study phenomena associated with the Feigenbaum constant. You did justice to the topic! Excellent video!

[@mackhomie6]

i want this guy on my team after a nuclear holocaust. me and a bunch of mathematicians.

[@MoorganHart]

Yeah, another well done video. I am left wondering the relevance of the Mandelbrot Set though. I'm not a mathematician, and I've never heard of it before. I'll do a web search, but would have liked to hear about where the equation came from within the video.

[@Polarwhisper6]

I think you meant: "I am 'A' mathematician"

[@christianmathison5892]

@@Polarwhisper6 he didnt claim to be an english major

[@Sergiuss555]

@@christianmathison5892 to be fair there could be girls among those mathematicians.

[@rianschoeman559]

James Gleick's book is really good. Once in a post office que I observed someone putting a fan on a scale. The reading bifurcated. I asked her to set the fan to a higher speed. The reading bifurcated again between four readings. I was blown away.

[@Prophet_Isaiah]

So cool

[@penguinswithpencils2333]

I ran across this video about a month into my discrete dynamical systems course. It has been super helpful to have something that gave me some intuition surrounding these patterns before I encountered them formally in class. Sharing it with my professor now :)

[@ShiftK]

**slightly opens kitchen faucet* Me: I'm somewhat of a scientist myself

[@bilalthefighter829]

lmao

[@treewizard4508]

Have you written a scientific paper?

[@Hwywbwjw]

Tree Wizard wat

[@ginoyesano5649]

Your comment fits your profile picture so well 😂

[@Hapasan808]

*slightly opens kitchen faucet * Me in a Joker voice: "I'm an agent of chaos."

[@blobishlybelfer2717]

thanks for showing me that as a child i had absolutely no chance of predicting when a water drop will drop.

[@K3zz21]

"when a water will drop" a SINGLE water

[@who-ow5ix]

😂😂

[@plaverbach]

@@K3zz21 When the water would drop?

[@zacharyhall7466]

Chaos theory?

[@haydenbellis3047]

.Turn to United Brethren Church. Using the Lords name in vain is a sin and so is cursing upon your enemies. And so is saying anything hateful, premarital sex and lust are all sins..⁰

[@amanuka291]

I can't even describe how amazed I am. Thank you for this video! It was a huge help with my studies and such an entertaintment!

[@jaianupamvarun8722]

I truly am humbled by the way you present such a complicated topic in such a simple way that even i who isn't a fan of mathematics got interested immediately 😃❤️💕

[@warrenchinn4114]

Magnificent. I am a 53 year old professional entomologist and this is utterly pertinent to invertebrate population biology. You state you are 37 years old in this presentation and I feel very humbled. Thank you for restoring my confidence in a world that seems bent on science denial, superficiality and facebook banality. Please keep up these exceptional presentations of important and complex concepts in nature, top marks !

[@warrenchinn4114]

@@shyamkarthikeya4769 That was my point ... :)

[@supteg2.495]

@@warrenchinn4114 again a smart man..

[@genericusername4206]

Warren Chinn true

[@bfern053]

@@shyamkarthikeya4769 delete this

[@DickEnchilada]

I learned about this constant (and the equation) when I was taking a System dynamics course 2 years ago. Absolutely loved that course. Stay learning dude!

[@JohnSmith-gs4zv]

13:57 "What _really_ *IS* a faucet?" - Vsauce music starts

[@impanthering]

ok John Smith

[@calix451]

I miss him

[@geekjokes8458]

But his video would actually *start* liake that And finish by talking about *UNIVERSALITY*

[@DrSpooglemon]

"This is water dripping from a faucet. OR is it?!"

[@ricardopieper11]

I read the first 3 words of your comment in michael's voice before acknowledging the word "vsauce" in it

[@nsfa19]

This has got me really interested and fascinated as well. Congrats on such high quality top-notch videos! Thank you very much!

[@shamsamir1698]

One of the most beautiful videos on youtube! keep getting back to it every time.

[@DreamLoud05]

THat was definitely one of the shortest 18 minute video I have ever watched on youtube. Very fascinating

[@FnaticKiller]

Now that is a compliment.

[@IRL.johnny]

this is frighteningly true... I watch it at 1x while my wife brushed her teeth... how's that possible?

[@diabl2master]

wtf, I did not realise that was 18 minutes, I was totally enthralled

[@pdutube]

If you observe a subject watching this video at a specific fraction of the speed of light, you could see it at 18 minutes no matter what the subject's playback speed was.

[@liqwiz]

I felt Exactly the same. Only by reading your comment I realised "wait... That was almost 20 minutes?"

[@sloanehowell6166]

“When this baby rotates on the z-axis, you’re gonna see some serious sh*t.” - Dr. Emmett Brown

[@davememelandcanada6722]

We don't talk about the Doc, since he f**ked up the timeline for 2020.

[@chop-daresistance7514]

Wow doc.. that's heavy

[@chop-daresistance7514]

@@davememelandcanada6722 I don't think that was doc.. I think that was biff when he stole the DeLorean

[@anasaamir5595]

wht is z asix

[@Leosayshi]

@@anasaamir5595 If I had to guess it’s probably another axis like the y and x axis except it’s used for 3D shapes.

[@samuellafleur890]

Having just watched your video on the ubiquity of 37, I couldn't help but notice that was the first example you provided when taking about period doubling sequences of any length!

[@nfc14g]

Feels like when you are using exponential equations your inputs are very important and deserve a lot of explaining why you chose it, regardless of real value or best estimate value

[@briancherry8088]

8:43 - Boom. Mind blown. We've been missing the 3rd dimension of the Mandelbrot set this whole time.

[@mikeciul8599]

Actually, I think there might be a 4th dimension, because for some complex values of c, the equation could converge to complex numbers.

[@DemonSwrd]

now imagine the 4th dimension.

[@briancherry8088]

@@DemonSwrd - no. The world will explode.

[@hammerth1421]

​@@mikeciul8599 It is the real part of the complex value returned from the complex logistic map. If the result was also complex the Mandelbrot set would be 4D.

[@brando3342]

@@DemonSwrd We can't imagine that. That's like asking someone to imagine a one dimensional point. Yes, some cosmologists want to say that was what the singularity was at the beginning of the universe, but think about it, it doesn't make sense.

[@andriypredmyrskyy7791]

As a control systems engineer, I've seen this behaviour before in several systems when control goes awry. Now that I know there might be something I can do about it, this opens up all sorts of possibilities to new control methods. I'm freaking out a little bit.

[@Rotem_S]

Actually the first physical chaos measured was an engineer using a simple circuit with a transistor and a harmonic input, where he sometimes saw one period, sometimes two, and sometimes all of them

[@austinwessels]

I was thinking about controls too for most of the video!

[@revenevan11]

Wow, you could control something back from the edge of chaos or even from already started chaotic behavior (like an airplane already entering an aerodynamic stall) using the methods that those scientists used on the rabbit hearts! I hope you're looking into that paper right now lol.

[@brocktechnology]

I'm no engineer, but I'm right there with you. I was hit by the realization that this little bit of math is the reason PID is so difficult to do well.

[@revenevan11]

@@brocktechnology I'm super excited for the possibilities of better control algorithms by using something analogous to the research in restabilizing heartbeats, the implications are awesome for aerospace engineering.

[@harshildobariya6773]

One of the most mind-bending concept I have came across. I love your channel, it is one of the best.

[@Khpr666]

Bro! I love you! Thank you! Your work is increasing the growth rate of knowledge.

[@sidsixpoint7]

Jokes on you, this didn't change the way I see the world because I'm dumb.

[@HerGatiox]

at least you see that. You know that you don't know anything. congrats

[@gabeharrison7683]

@@HerGatiox a bit condescending

[@Zain69z]

Lmao

[@aaroncruz9181]

You see math without numbers

[@BenDover-vh9rs]

@Majin Buu then how tf does he know that?

[@MrTigershark11]

6:37 One of the best plot twist I've seen in my entire life

[@snowfloofcathug]

Hereizer Oh that pun is wonderful

[@madalinspoiala1249]

You need to be higher. Great pun

[@theshocktart]

Such a legendary comment, bravo...bravo!

[@justarandomdood]

Dad joke levels are through the roof

[@NiranjanDixit23]

Get out lmao

[@potat8089]

I remember watching this video when I was about 14 years old and being fascinated by this, but my understanding kinda dropped off when the Mandelbrot Set came into the video (I knew nothing about it). Rewatching this video at age 16, I can really appreciate it so much more, now that I understand it. Thank you for introducing me to the Logistic Map!

[@matgeezer2094]

Again, did not expect to be blown away but am utterly. Superb video, bringing quite a 'dry' subject alive.

[@belgianbushrc7934]

I am intelligent enough to find this interesting but not quite smart enough to fully understand... Great video!

[@eletinalex]

covid-19= this is control over the rabbit population

[@naturelovingfroggy6348]

Well said, me too x

[@shadowpresident4203]

That's probably the ideal situation. If a tutor were to customize a lesson for you specifically, they'd probably aim for a difficulty level you'd find interesting, and a lecture that you could mostly follow. You'd also be very slightly outside your comfort zone, to give you room to "grow into" the lesson. It would also be a bit challenging and would leave you with a few questions to mull over, thereby slightly increasing the chance of watching another video, or of reading a book or article on a similar topic. Your comment would probably be music to the ears of a lot of educators.

[@eletinalex]

@@shadowpresident4203 thx

[@comet1954]

I feel your pain! I truly do....

[@TehmasKhan]

You know the video is good when it’s 18:39 minutes long and you still didn’t want it to end

[@za012345678998765432]

I didn't even notice until you pointed it out, thought it was like 9 minutes

[@MartinBuzon]

@@za012345678998765432 I was going to say 9 too. Damn witchcraft.

[@za012345678998765432]

@@MartinBuzon Next thing you know Derek comes here and tells us that this equation can explain that as well and it was all planned from the start XDD

[@setsugennoao3137]

@@za012345678998765432 same

[@KillaahMusik]

It felt like 4,669 minutes

[@namakoro_osu]

since the start of the pandemic i have come back to this video time and time again and it was the first video i watched of yours and every time i rewatched it i under stand just a little more thank you for making many complicated topics over the years bite size and understandable

[@IIronyy]

Dude I have absolutely no idea of anything I just heard but I still listened all the way through it takes a very talented type of person to intrigue his audience that much.

[@bradleyewoodworth]

Out of all of the KZbin channels that I don't understand, this one is my favorite.

[@zaphodsbluecar9518]

ROTFL - Nice work Brad! :-)

[@AzTech71]

Lol

[@Streamwalker1000]

LOL ! I'm with you on that.

[@simphiwenxumalo4387]

@@Streamwalker1000 LOL me too

[@johnmartin9451]

I'm sure you understand more than you thought, otherwise you wouldn't be drawn in. It's about patterns and randomness and how the two shall not meet.

[@g0g0duck20]

Me: Watches video because I think I'm smart My brain: Hold on there buckaroo

[@paratame105]

I feel attacked

[@AverageAlien]

Don't worry we're all retarded here. Welcome to the club.

[@Dontlagmebro]

Well how do you think (we) get smart?

[@trevsanna]

LOL....You can actually apply the equation to your comment to see how funny your comment actually is! :)

[@rznleagueoflol4545]

Trevor Baylis no u cant

[@MX.808]

This is a work of genius!! Thank you for making this video!

[@vishalrander9805]

Very much obliged to watch your videos. Thank you for making the complex to simple .

[@CHLOCHLOLP]

if they taught this kind of thing to kids in school im sure they would be 10x more interested in the subject. Its not something I would test them on, but just to show them how interesting and versatile math can be, I think it would get people a lot more motivated to learn.

[@CHLOCHLOLP]

@chcpr1 i think its more that no one in power cares to invest in school systems to make them better, but I dont think they are shitty on purpose, just due to neglect. Why would rich government officials care about public schools when are their kids are privately educated anyways, you know?

[@Eden894HisBiome]

Yeah sadly they don’t teach you anything your not tested for. Unless they a good teacher.

[@kaffeetasse9455]

One of my teachers actually showed us the mandelbrot set and told us about chaos teory. :D

[@chrisbarber7291]

Um, they do teach this.

[@sirxobsidian408]

Unfortunately, this simply isn't true. I'm a high school math teacher, and I've shown this kind of thing to my students on numerous occasions, and the response is pretty much always the same; disinterest. If it's not on a test, they see it as an excuse to take a nap, zone out, or work on other classes' assignments.

[@RPS2443]

"The mandelbrot set is numbers that don't blow up" Mind blows up. Seriously the best explanation ever. Then he shows it in 3d. Learned more in 20 minutes than in the last 20 years.

[@userMikeforsure1997]

I felt that.

[@hannesthurnherr7478]

so your saying your mind doesnt belong to the mandelbrot set?

[@kennyogbogu7777]

Now that's funny.

[@RPS2443]

@@hannesthurnherr7478 Well, rules are rules, so apparently you are right!

[@jfidel3943]

I love James Gleick's Chaos book. It's such an amazing book and I always learn something new from it.

[@mihalykort84]

I went to the "Chaos Conference" at Sydney in (I think?) 87, met some pretty incredible physicists and mathematicians, and even got a couple of Autographs :) but mostly felt just awed that this was an interesting thing, and that I knew somehow it also was kind of important. B Mandelbrots lecture was lovely, but some of the different maths topics (from geometry obviously, but Prime number theory, Analytic and Discrete maths and biological mathematics. For me it was a memorable experience. So I get you, and I wish you more power to your arm, because yeah there's something compellingly beautiful and yet still mysterious. Good on you for your work, long may you continue and thanks!

[@Hgulix62]

This is like when you start a new project or new year resolution. Everything is fine and simple at first, then progressively and unexpectedly everything turn to chaos

[@eternalexperience2489]

Probably the rate of degradation is 4.669 :o

[@somethingtojenga]

"How did I get to be 37 years old without hearing about the Feigenbaum constant?" That is an extremely optimistic outlook on what 37-year-olds should be concerned about.

[@djb903]

I turned 37 yesterday and I'm glad I've heard of it thanks to Derek

[@arvind8302]

SomethingToJenga 100%

[@binashah3106]

well im 7

[@spb1179]

Bina Shah good keep learning. You might be remembered for something awesome if stuff like this interests you!

[@binashah3106]

jk im 13

[@KM-co5mx]

We need more videos like this! ⭐️

[@iamlubos]

Great video! It's also a really good subject for a take-home final exam in a scientific computing class I am teaching this semester.

[@t-riz7630]

Me: cannot do basic calculus *sees title of video* KZbin just gets me.

[@BudEnzo]

I taught myself basic calculus from videos on KZbin. Both Khan Acad and 3blue1brown really helped. I was able to skip Math 251 (Calc 1) at my college and move straight to Calc 2 after a 4-year break between studying BASIC algebra in high school and returning to college.

[@tishafeed8085]

@@BudEnzo wait what, you studied basic algebra in high school?

[@BudEnzo]

@@tishafeed8085 I guess lol. My school called it pre-calculus so basically it was algebra+trig.

[@tishafeed8085]

@@BudEnzo what years/classes? in my highschool (9th-11th year) we did trig+calculus

[@tishafeed8085]

@gamecube.enthusiast - - i did, but the fundamentals were studied in years 5-9, then it was trigonometry and calculus. Actually, i just realised it depends on what you call 'highschool'

[@CuteC3]

So basically like when the bass drops, the tempo keeps doubling until it just goes into complete madness.

[@Cher007]

The 101 of Speedcore and Splittercore 😂

[@sethreardon8805]

Something tells me that you might have said the smartest thing in these comments. Music is vibration, vibration is periodic. This equation probably can be found in music somewhere...

[@TheMaxwellee]

I hope a nerdy DJ builds his set this way

[@StormEngineer]

I almost spit my coffee. Well done.

[@RCFrizz]

@@StormEngineer Imma go brew some coffee so I can almost spit it!

[@VisualJoey]

You definitely did this topic justice in my personal experience! Thank you (…) And this is just one topic where you do justice. I don’t practice maths nor more of the topics you have covered. At least, in a professional or study kind of way that is. However, I have found your explanations to be inspiring and fascinating. You really hit a spot of interest of which I have always had an interest for but never got my head to understand due to the complexity. It’s a gift to be able to watch your videos and get a grasp of understanding. Even though sometimes the grasp barely touches the surface.

[@lintonatlas8933]

Outstanding explanation of chaos theory. I especially liked the unifying of the madelbrot set diagram and the doubling diagram, I don't recall seeing that before.

[@infinitiv525]

I am so glad that some lad just woke up one day and decided: "You know what, I should do some research of dripping faucet."

[@TheFredmac]

That would be a lad that doesn't get government funding.

[@blackdaan]

weed is good for some things...

[@rtleitao78]

You mean, instead of fixing it?

[@BenjaminCronce]

you mean "woken up by a dripping faucet"?

[@slaughterround643]

Veritasium: "Students can have a little chaos, as a treat."

[@joshgiesbrecht7060]

literally my teaching strategy

[@AaronRWilliams97]

*little a chaos

[@bengardner8639]

If you're gonna reference the meme, at least do it right. It's "little a", not "a little".

[@marcushendriksen8415]

Make them climb a ladder for it, though

[@AaronRWilliams97]

Ben Gardner *le sigh* He DID say pedants were necessary..

[@preranaprasaddeshmukhbe20b34]

This is one of the most amazing video I had seen on KZbin till now

[@Elletanvilla]

Thank you so much for this video!!! I can’t understand it fully but my God it still moved me to tears

[@vincenzkoop43]

6:43 most underrated placement of the term ‚plot twist‘

[@wedantil]

Pun of the day!!!!

[@kennethsizer6217]

I love you, man!

[@Surya-jz6te]

german?

[@cadosian078]

Top 10 anime betrayals

[@snowstarsparkle]

i don't get it, please explain !

[@locotx215]

6:41 - "The Plot Twist" figuratively and literally

[@aliffikri1888]

Noice

[@himagnamukherjee9382]

pure gold

[@carsonmorris127]

This whole video is a plot twist

[@papa_pt]

bravo

[@DoktorIgnaz]

HOLY

[@DavidMauas]

This is really mind-blowing, I absolutely love it!

[@mikedavies5551]

Brilliant especially the final part about different functions. That is new to me. Thanks.

[@PietchRhum]

I feel so dumb! I have a PhD in Ecological Modelling, I have used this equation - and more complex version of it - to model population dynamic multiple time and I didn't knew of this Feigenbaum constant!!! I had teaching on Chaos and none of my professor mentioned it! Well, thanks Veritasium for teaching us new things!

[@KarasuInaiga]

P. V. I read that as “Executive Meddling”.

[@metanumia]

@@KarasuInaiga 🤣

[@spencerquirk6917]

@@KarasuInaiga 🤣🤣🤣

[@BrandonRx2]

Huh

[@ansh6370]

Bruh I'm just an eight grader. Tomorrow's my maths test and I'm supposed to learn mensuration and algebra but I'm here trying to figure out what this video even is about after watching it for the 7th time.

[@farhan132_]

The more I learn, the more I realize I don't really know anything!

[@mdestwo]

Ah, so you really have been paying attention! That seems to be one of the more important lessons to learn in life. So, here’s to you, me, and everyone else realizing we know less and less each year! 😀

[@farhan132_]

@@mdestwo Well said!

[@glaucovillasboas8212]

That's for sure. I always think I'm smart, but a year later I always remember how stupid and dumb I was, and it keeps repeating

[@avikdas4055]

Exactly man. But now I'm 100% convinced that I know absolutely nothing.

[@Kraflyn]

welcome to the club! :D

[@pudayshankar1036]

Your videos never failed to make me think about what is the purpose of this universe.. why is it there and how it works so perfectly without anyone controlling it..

[@rainthevaporeon7852]

I've not shown any interest in mathematics (or physics for that matter) for a long while, and this somehow sparked up my interest in it again, thanks. Really fascinating stuffs

[@veritasium]

Feedback from UCLA A/Prof Heather Zinn Brooks: "At first, it seems totally crazy that increasing r would lead to cycles... if it's describing population growth, why would a bigger r cause a drop in some years? In fact, the magic is in the (1-x_n) factor of the equation, which models a carrying capacity. The idea is that populations are typically resource-limited, so growth can only be supported up to a certain point. Once you think about that, it makes sense that huge growth rates could result in "boom and bust" cycles in the population, because the population would grow faster than their resources would support."

[@shetty98]

1:30 @m in 🇮🇳

[@NatePrawdzik]

I imagine increasing rabbits also increases wolves, so that probably factors in as well.

[@timezone5259]

Yes

[@gordonrichardson2972]

This topic came up when I was studying applied mathematics around 1980: en.wikipedia.org/wiki/Lotka-Volterra_equations

[@11quinnjet75]

Can this be applied to the expanding universe? because it has a growth rate and if I remember right scientists think that that rate has changed?

[@avrenna]

When the Mandelbrot set rotates: whooaaaahhhhh

[@xiaoxiaoxiao686]

5erif meme echoes in my head...

[@Adam_Bosscoe]

Right!?!?!!

[@valentinbernard8126]

Yeh that was spectacular

[@trickytreyperfected1482]

"You're not supposed to do that!"

[@aurelia8028]

I swear to god, my mind was blown away by that! I had only ever seen the Mandelbrot Set as that 2 dimensional boundary thing, so seeing that it was actually a 3D-thing was insane

[@rajeswaranparameswaran6611]

Amazing quality of presentation! Besides being a scientist with msrcellius 'teaching' skills, you are a classic cameraman too! Grateful.

[@darrenirwin5411]

Fascinating! Thank you for this wonderful video.

[@variable7833]

This feels like a window into a small part of how the universe works. Its kind of scary and exciting.

[@robologo]

Yeah it's very strange... all these formulas that pop up everywhere. I think there's an explanation though.

[@pablooo8280]

@@robologo have u tried psychedelic drugs? you actually start seeing these everywhere

[@mysonandme8424]

@@pablooo8280 nobody has because drugs aren't real.

[@obiwanjaco]

@@pablooo8280 no

[@obiwanjaco]

@@mysonandme8424 yes

[@lc7ineo]

I found this incredibly fascinating... but understood incredibly little.

[@jakewatson1160]

I was interested in it but i will always hate math cuz ths made me fall asleep lol

[@panpsalt6757]

I understood more than I want, but still less

[@thechappist]

Yup! Make me want to go learn some math!

[@thechappist]

I had to write some things down to look up definitions! But interesting

[@unbiasedhuman1659]

Fibonacci equation.

[@pablog80]

I was already aware of some of this stuff bit had never seen it related so strongly as it is explained here. Amazing

[@zx8979]

Excellent video! A good video to let me know more about chaos after hearing it in class, while my teacher didn't say too much about it.

[@phillipejane8423]

KZbin algorithm - 1st stage - sends to people who are actually interested in math. Less views 2nd stage - sends to related viewers. Less views but still constant. 3 stage - Sends to random people on their feed. Fluctuations in view cycle (no negative views so it shoots up) 4 stage (chaos) - recommends to people like me of all people. Millions of views

[@tomas_horn]

top comment!!

[@norahclarissa6352]

This observation is actually clever, considering that the algo is an AI

[@manictiger]

Recursive view count increases!

[@RahulKumar-sy4xg]

KZbin's algorithm decode!

[@jordanponton255]

Stage 5: people in the Mandelbrot set

[@Max_Jacoby]

- What is the answer to the Ultimate Question of Life, the Universe, and Everything? - 4.669

[@jcornman24]

Nice

[@mmmmmark9751]

It's 42......of course we really don't know what the question is.....(ps. it is 6x9.....which is why it's all fuk dup)....thanks Douglas

[@contessa420]

Nice reference

[@BinyaminTsadikBenMalka]

It's 3 Pi /2

[@RagavPuli]

@@BinyaminTsadikBenMalka 1.5xπ

[@rudyardkipling4569]

Excellent video. The ubiquitous emergence of the Feigenbaum constant is just disturbingly, weirdly beautiful.

[@keyvanmalaie]

That was really a fantastic! Thank you!! I got scared seeing the bifurcation for the first time!

[@liondeluxe3834]

This video made me obsessed. I didn’t sleep last night. This morning I went to my core editor and made a Mandelbrot’s Set visualizer and I watched this video for a second time. I now want to make a logistic map visualizer. Thank you Veritasium.

[@aion2177]

github link please?

[@jamesfield5415]

I got obsessed too it’s mesmerising

[@NovaWarrior77]

please let us know. I for one for sure want to see it.

[@goutamboppana961]

the beauty come when u keep zooming it

[@CrackedCandy]

Oh my, my heart is beating faster now, so, where can I see this art?

[@Jessicalc941]

Me: understands less than 5% of what he said Also me: life changed

[@waterproof4403]

haha same

[@physcedelicanvs]

u nailed it better.i only undestand 0.00001 percent and felt like im the descendant of euklides

[@jerrythemagichamster8586]

The good part is that 5% of the universe is seeable matter sooo.. you understand now about .0025% of the universe.

[@TheBrickagon]

I swear, seeing memes and reading comments make me feel that we all live the same life :))))

[@waterproof4403]

@@TheBrickagon haha some what comforting

[@NuhDhulMitrain]

I attend a lecture about chaos, first it was dull and now it has become very interesting and fun, also numerically solving equations and getting those pictures

[@valiantwarrior4517]

If you look at larger fractal patterns of the Mandelbrot set, you can get a cool “magic eye” 3D effect. Particularly one called “Seahorse Valley.”

[@ShilpanGPatel]

I saw the video title and thought “oh suuuuure this will change my life”, but wow I did not expect to be so taken aback by this. Something so fundamental which I’ve never heard about before!

[@ayaipeeoiiu8151]

Shilpan Patel so you should subscribe to numberphile

[@hidjedewitje]

This concept is actually taught in engineering. Particularly in Control Engineering. It's taught because it gives a lot of intuition in nonlinear differential equations. These differential equations occur quite often in nature and in engineering problems when you have to model the dynamics of a system!

[@DataJYdocs]

ie: the Lorenz Attractor.

[@albertofoti4152]

Yep, I'm here because I'm studying for an exam ahah Non-linear Control and Aerospace applications

[@GdeJ]

Here's another: Chua's circuit

[@albertofoti4152]

@@GdeJ I am from Politecnico of Turin in Italy and we had the pleasure of having a lecture directly from Professor Chua. I think the Lorenz Attractor and the Chua Circuit are the most thought in control engineering degrees. Maybe talking about non linear systems behaviors we also have systems having limit cycles in the phase portrait, for example the Van Der Pol Oscillator

[@GdeJ]

@@albertofoti4152 I'm from Federico II University! I just had an exam on non linear dynamics and control. Other than Van Der Pol oscillator, our professor mentioned also other relaxation-excitation oscillators, such as Fitzhugh-Nagumo and Hindmarsh-Rose models of a neuron.

[@S.Ali_Almosawi]

After this video I was able to see John cena

[@weo9473]

💀

[@moumous87]

After 3 years, watching it again. Love this video!