All possible pythagorean triples, visualized

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  • čas přidán 13. 06. 2024
  • To understand all pythagorean triples like (3, 4, 5), (5, 12, 13), etc. look to complex numbers.
    This video was sponsored by Remix: www.remix.com/jobs
    Help fund future projects: / 3blue1brown
    An equally valuable form of support is to simply share some of the videos.
    Special thanks to these supporters: 3b1b.co/triples-thanks
    Home page: www.3blue1brown.com/
    Regarding the brief reference to Fermat's Last Theorem, what should be emphasized is that it refers to positive integers. You can of course have things like 0^3 + 2^3 = 2^3, or (-3)^3 + 3^3 = 0^3.
    Music by Vincent Rubinetti: vincerubinetti.bandcamp.com/a...
    Thanks to these viewers for their contributions to translations
    Hebrew: Omer Tuchfeld
    ------------------
    3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with CZcams, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
    If you are new to this channel and want to see more, a good place to start is this playlist: 3b1b.co/recommended
    Various social media stuffs:
    Website: www.3blue1brown.com
    Twitter: / 3blue1brown
    Patreon: / 3blue1brown
    Facebook: / 3blue1brown
    Reddit: / 3blue1brown

Komentáře • 2,9K

  • @felely
    @felely Před 4 lety +4585

    This is hella interesting when you have an English essay due

  • @vib0ng508
    @vib0ng508 Před 4 lety +8721

    imagine being a 1st grader doing their shapes homework and searches up “triangles” and gets this

  • @generalralph6291
    @generalralph6291 Před 4 lety +2426

    I needed this today. I’m building a house made entirely of Pythagorean Triples.

    • @spearmintage
      @spearmintage Před 4 lety +9

      yuki nagato

    • @felely
      @felely Před 4 lety +93

      You’re... you’re what?

    • @sameepdoshi
      @sameepdoshi Před 4 lety +69

      Yeah Build it in front of my school examination hall

    • @Nylspider
      @Nylspider Před 4 lety +33

      Oh cool
      Wait hold up...

    • @beardwright6917
      @beardwright6917 Před 4 lety +22

      Can you pm me a photo of what it looks like as an architectural drawing?
      I’m pursuing civil engineering.

  • @primephoenix1.077
    @primephoenix1.077 Před 3 lety +2271

    Special Thanks to
    1. Pythagoras
    2.Reńe Descartes
    3.Bernhard Riemann
    4.Grant Sanderson
    For this Marvellous Video😄

  • @3blue1brown
    @3blue1brown  Před 7 lety +5141

    As to the "you're" typo at 1:20, I keep telling that second blue pi creature (Randolph is his name) to learn his grammar, but for whatever reason, he just never listens and focuses only on his math lessons.

    • @ganaraminukshuk0
      @ganaraminukshuk0 Před 7 lety +242

      I scrolled down to the comments just to see if anyone caught that.

    • @NikolajKuntner
      @NikolajKuntner Před 7 lety +52

      Hey 3Blue1Blue, thanks for another great video! For fun I've tried out to make Randolph smile (self.play(randy.change_mode, "happy")), but for some reason it wouldn't let me. Any idea why that would be? Moving works fine. Also, I'm gonna do videos on functional programming and logic foundations (no animations) and was wondering how I could do life LaTeXing, as I want to avoid handwriting. Do you have any idea how to approach this?
      Thanks for your math content!

    • @EMEKC
      @EMEKC Před 7 lety +47

      Shame on the second blue pi creature.

    • @ConnorDuzMinecraft
      @ConnorDuzMinecraft Před 7 lety +20

      What are the other ones' names?

    • @harootpashayan
      @harootpashayan Před 7 lety +2

      hey I am an Unemployed Computational Mathematician and help or guidance into getting that Remix gig? h4root.com I obv code and have worked on Drivers, Gaming Industry, Low Latency Streaming etc Ableton Algorithms

  • @onlynamelefthere
    @onlynamelefthere Před 7 lety +1500

    At some point you think you have seen everything, which is to say about a "simple" topic like pythagorean triples. And then comes this video and blows your mind with the elegance and simplicity of it all. And you will be reminded, there is no such thing as "simple topics" and "everything to know".

    • @3blue1brown
      @3blue1brown  Před 7 lety +191

      I couldn't agree more with that last sentence!

    • @selfcentered3406
      @selfcentered3406 Před 6 lety +7

      Truth.

    • @claudiaassis777
      @claudiaassis777 Před 6 lety +4

      onlynamelefthere hey. If you get an already pythagorean triple and Square them, why don't you get a "fermat's triple for n=4"?

    • @theSoberSobber
      @theSoberSobber Před 6 lety +1

      Agreed😊💐💐💐💐👍

    • @SC-zq6cu
      @SC-zq6cu Před 6 lety +16

      Claudia Assis
      Say a,b,c satisfy :
      a^2 +b^2 = c^2
      Squaring both sides :
      (a^2 + b^2)^2 =c^2
      Or, a^4 + b^4 + 2*(a*b)^2 = c^4
      Whereas Fermat's triplet for n=4 satisfy:
      a^4 + b^4 = c^4

  • @sicoree
    @sicoree Před 4 lety +4005

    치직... 한국인...
    깃발 꼽고 경례..

  • @maane28
    @maane28 Před 3 lety +302

    "The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.”
    - Henri Poincaré -

    • @seanleith5312
      @seanleith5312 Před 3 lety +2

      He studies the topic that provides fund. Many scientists study global warming, not because it delights. They know that's a bunch of lies, but that's easiest to get money from.

    • @pranaygupta6688
      @pranaygupta6688 Před 3 lety +12

      @@seanleith5312 climate change denier? 99% of scientists, especially climate scientists, believe in climate change. AND, climate science by far does not make the most money... What about medical science (doctors, pharmaceuticals) or engineering (especially for companies like Boeing and Lockheed Martin that get military contracts)?

    • @seanleith5312
      @seanleith5312 Před 3 lety +1

      @@pranaygupta6688 All you know is repeat the propaganda from your school and liberal media. Do you have a brain to think for yourself?

    • @Hobbit_libertaire
      @Hobbit_libertaire Před 3 lety +5

      @@seanleith5312 And why don't you believe in climate change ? Have you any proof to sustain your belief ?

    • @seanleith5312
      @seanleith5312 Před 3 lety +1

      @@Hobbit_libertaire Who said I don't believe climate change? Climate change happened since the earth existed, it's always changing, it will be forever. What I don't believe is: Man-made CO2 is the driver for climate change. There is no evidence to CO2 plays any meaningful way. And it is theoretically close to impossible that CO2 play any meaningful role. You are indoctrinated to believe in this religiously. It is disgusting to use science as a political tool.

  • @nathanielsharabi
    @nathanielsharabi Před 7 lety +1858

    >has final exam in 2 days
    >*sees 3blue1brown uploaded new vid*
    >"the bloody exam can wait"

    • @FacultyofKhan
      @FacultyofKhan Před 7 lety +24

      It seems that the meme-arrow trend I started last week has carried over to this video as well! Good, good, muahahaha

    • @evanoc
      @evanoc Před 7 lety +60

      Faculty of Khan What? Greentext arrows have been around for years, lol

    • @FacultyofKhan
      @FacultyofKhan Před 7 lety +8

      I meant using meme-arrows in the comment section on 3b1b's videos. I made a comment last week on the pi/prime irregularities video using meme-arrows, and was (rightly) made fun of for it. It's amusing to see the trend continue here.

    • @zoellazayce6796
      @zoellazayce6796 Před 7 lety +4

      Further Maths right

    • @da_bes
      @da_bes Před 7 lety +48

      don't kid yourself, you didn't start shit

  • @johnrickert5572
    @johnrickert5572 Před 7 lety +1003

    Absolutely beautiful! I have a Ph.D. in Mathematics and have never seen a discussion of Pythagorean Triples in terms of complex numbers before. Thanks for this great video!

    • @anonargentum9135
      @anonargentum9135 Před 7 lety +32

      John Rickert Doctor, i'm interested in your profession since i'm going to study and become an applied mathematician and I wanted to know how it has been to be a mathematician :), greetings

    • @johnrickert5572
      @johnrickert5572 Před 7 lety +75

      Thank you for your reply. Well, I was in Pure Mathematics instead of Applied. I believe that Applied Mathematics would give you very great flexibility. Academia may or may not be the best environment to be in. Even though I no longer work as a mathematician professionally, I still study mathematics and find it fascinating. I have never regretted the time and effort I have put into it. I hope that you find it rewarding.

    • @nucleartree8159
      @nucleartree8159 Před 5 lety +2

      @@danielwylliel.rodrigues1015 you know we are both a year late. CZcamss recommendation algorithm is retarded

    • @wacamac1006
      @wacamac1006 Před 5 lety

      @@nucleartree8159 even more for me

    • @Meminjo
      @Meminjo Před 5 lety +3

      Would you mind sharing what you wrote your doctorate about? Thanks!

  • @theseal126
    @theseal126 Před 4 lety +597

    You should make an ”Essence of topology” series. Topology is very visual but can be hard to describe with just numbers. I think ur animations would make a great fit for teaching topology
    You could cover topics like: Projective space, Equivalance relations or quotient space, affine geometry, hyperbolic geometry.
    And then u can end of the series by briefly giving an understanding to the poincaré conjecture.

    • @glitchy9613
      @glitchy9613 Před rokem +10

      I'd honestly love for 3b1b to talk about hyperbolic geometry

    • @theseal126
      @theseal126 Před rokem +5

      @@glitchy9613 ikr, hope he notices how many people that have liked this comment so that he makes a series

    • @glitchy9613
      @glitchy9613 Před rokem +4

      @@theseal126 Wait shouldn't it be called "Essence of geometry"? most of those topics relate more closely to geometry than they do topology.

    • @theseal126
      @theseal126 Před rokem +2

      @@glitchy9613 Oh, true!! Essence of geometry sounds better. Though maybe some people might get the wrong idea so maybe essence of non euclidean geometry

    • @mihailmilev9909
      @mihailmilev9909 Před rokem

      @@theseal126 this sounds like a beutiful idea, I need this

  • @soheilsanati1941
    @soheilsanati1941 Před rokem +70

    In Euclid’s Elements there is a description of all the possible Pythagorean Triples. Here’s a modern paraphrase of Euclid.
    Take any two Odd Numbers m and n, with m < n, and relatively prime (that is, no common factors). Let A = m x n; B = (n^2 - m^2)/2, and; C = (n^2 + m^2)/2. Then A:B:C is a Pythagorean Triple.
    For instance, if you take m = 1, and n = 3, then you get the smallest Pythagorean triple 3:4:5.

    • @null_pointer_deref
      @null_pointer_deref Před rokem +6

      It's essentially the same formula that we get when generalizing the squares of complex numbers for these triplets. It's incredible how many proofs you can do with complex numbers, even in things you wouldn't normally expect them to appear!

  • @shiladri007
    @shiladri007 Před 7 lety +492

    This is quite simply the best Maths learning resource on the interent...a service to humanity!

  • @jacheto
    @jacheto Před 7 lety +754

    I LOVE THE FACT THAT YOU ARE POSTING VIDEOS EVERY TIME PLEASE NEVER STOP

    • @jacheto
      @jacheto Před 7 lety +47

      i also love the fact that is in 60fps so thank you

    • @Talaxianer
      @Talaxianer Před 7 lety

      Why? Do you watch in 0.5x speed?

    • @error.418
      @error.418 Před 7 lety +2

      It's subjective, not axiomatic

    • @Treegrower
      @Treegrower Před 7 lety +3

      60 FPS / 1080 P MATH WHAT THE FUUUUUUUUUUU

    • @TheLastScoot
      @TheLastScoot Před 7 lety

      Higher framerate means more data. Also, at a certain point, some people can't tell the difference. Barely any humans would be able to tell the difference between 1000Hz and 2000Hz, so doubling the amount of data used serves no purpose.

  • @bazboy24
    @bazboy24 Před 3 lety +59

    Mathematics displaying its beauty, taught by someone who is in love with its beauty

  • @dodobow
    @dodobow Před 2 lety +490

    이런 영상을 볼때마다 수학의 신비함에 대한 인식이 점점 커져가는 거 같아요. 참 끝이 없고 흥미로운 학문이 수학이 아닐까 싶습니다. 흥미롭고 재밌는 영상 감사드려요!

    • @samgrattan5465
      @samgrattan5465 Před 2 lety +46

      That is awesome and good for you! I’m replying in English because I know CZcams has a translate function, so I hope you can understand this message clearly. Math can truly be a beautiful subject to explore, and videos and visualizations like this make it possible for everyone to experience it. I get excited just thinking about the future of math education, since I know that people like this will be able to make even the most esoteric topics approachable.

    • @lanerutledge6850
      @lanerutledge6850 Před 2 lety +9

      Exactly Dude. I hope google translates this correctly. But really math is crazy because of the way that hundreds of equations can make such organic and natural shapes

    • @dog6705
      @dog6705 Před 2 lety +24

      한국인이다!!

    • @ryanchowdhary965
      @ryanchowdhary965 Před 2 lety +7

      I like math, I listen to math every night to cure insomnia.

    • @Toby-em4vr
      @Toby-em4vr Před 2 lety +6

      @@samgrattan5465 Bad news: Google is really bad at translating English to Korean, and idk why.
      Anyways, I completely agree to your comment!

  • @OskarElek
    @OskarElek Před 7 lety +346

    The beauty of maths is that you can take something seemingly trivial and boring, and make it extremely intersting by digging deep enough.
    The beauty of 3b1b is that he does it for us :)

    • @hreader
      @hreader Před 3 lety +8

      This is exactly what schools should be doing but a lot of them don't.

  • @Joe72521
    @Joe72521 Před 7 lety +203

    Does anyone ever feel saddened by the beauty of these videos? It's not just, "I wish math was taught to me this way", it's that I now think there's got to be this beauty in so much more, and my eyes are just not open to seeing it.

    • @jyothidudupa240
      @jyothidudupa240 Před 5 lety +1

      Exactly! Well said!

    • @magicianwizard4294
      @magicianwizard4294 Před 5 lety +2

      For sure. Normally I'm there trying to cram my head with as much math as it can fit in for some test I don't give a crap about, and I don't like the math at all. But there is hidden beauty waiting to be discovered, and I am waiting for me to discover that I CAN discover the hidden beauty in mathematics.

    • @vencedore1000
      @vencedore1000 Před 4 lety

      I usually feel saddened while watching these videos when I realize just how little I know, and worse yet, how I’ll never be able to know everything there is to know in maths. Not only because we lost a lot of valuable information as time went on, but also because it is such a broad field.

  • @hammerfall321
    @hammerfall321 Před 2 lety +5

    I love how the students get angry when the teacher introduces complex numbers.

  • @nitinmadan4009
    @nitinmadan4009 Před 4 lety +33

    What an amazing visualization. A few years back, I tried coming up with a proof to find an elegant proof for finding Pythagoras triplets. Didn’t succeed.
    But this video just gave me a whole new perspective.
    Cheers!

    • @peter10003
      @peter10003 Před 4 lety

      Yes, I thought the Pythagorean triples from Sumerian times (1,000 years before Pythagoras lived) were found by trial and error. I never guessed that there could be an algorithm for it, let alone a simple(?) algorithm as described by this video.

  • @macmos1
    @macmos1 Před 7 lety +104

    You have an incredible intuition and perspective on mathematics. Please never stop sharing your knowledge with us!

  • @Treegrower
    @Treegrower Před 7 lety +203

    Watching this high is the craziest shit ever

    • @klipslip1977
      @klipslip1977 Před 6 lety +1

      FACTS

    • @petermarquez949
      @petermarquez949 Před 6 lety +5

      DAMN IM BOUTA DO THIS

    • @returntolifeband
      @returntolifeband Před 6 lety +3

      holy fuck if Bob Ross blows my mind I can only imagine what this will do

    • @6884
      @6884 Před 6 lety +9

      username checks out

    • @prabhindersinghsahni3015
      @prabhindersinghsahni3015 Před 5 lety

      ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ v

  • @blockyhour4224
    @blockyhour4224 Před 2 lety +29

    The fact that I finally understand what he's talking about makes it SO much more interesting

  • @matthewao
    @matthewao Před 4 lety +11

    He literally blew my mind with the animation in the first 15 seconds of the video

  • @swurviie
    @swurviie Před 7 lety +123

    Fantastic visualization of the Pythagorean theorem in the intro

  • @hugosales8102
    @hugosales8102 Před 7 lety +802

    "What's you're favorite proof?"

    • @ypey1
      @ypey1 Před 7 lety +111

      he is better at math then grammar

    • @HolmAdrian
      @HolmAdrian Před 7 lety +257

      than*

    • @GamerFilesnet
      @GamerFilesnet Před 7 lety +24

      than*

    • @ericespinoza1548
      @ericespinoza1548 Před 7 lety +8

      I was wondering if anyone else noticed that lmfao

    • @RedTriangle53
      @RedTriangle53 Před 7 lety +24

      I love the one sentence proof for the laplacian operator in polar coordinates. "trivial and left for the reader as an exercise."

  • @brucefoote540
    @brucefoote540 Před rokem +5

    I have a problem breathing every time I watch a 3b1b video because the concepts exposed there are breath-taking!!! Thank you Grant!

  • @ryanchowdhary965
    @ryanchowdhary965 Před 2 lety +7

    I like math, I listen to math every night to cure insomnia.

  • @joefagan9335
    @joefagan9335 Před 7 lety +19

    Grant, you are simply amazing. I've a life long passion for maths and took an M.Sc in maths just for fun. Thank you so much for these videos. Imagine if Einstein or Feignman or even Euler or Pythagoras could have seen your videos, they would have been blown away. You're taking the beauty and structure that they could see and shown it to the masses. You are the ultimate pedagogue. Thank you.

    • @Ir77iridium
      @Ir77iridium Před 2 lety

      I bet Euler saw this when he became blind

  • @aresharesh8671
    @aresharesh8671 Před 7 lety +13

    This is absolutely beautiful. Thank you so much for posting these videos. It is such a great pleasure to watch and learn the topics here with your incredible visuals to lead the way. I look forward to more amazing content in the future.

  • @brotherseraphim9700
    @brotherseraphim9700 Před 3 lety +2

    Very grateful; just what I was looking for! Had a suspicion that Pythagorean Triples to All Triples were as Rational Numbers to All Real Numbers, but wondered how to get at showing it. Thank you for the missing clue of using the Complex Plane, and for the unusually clear and nicely paced presentation!

  • @hobby_Betelgeuse
    @hobby_Betelgeuse Před 3 lety +42

    和訳確認しながら英語のリスニングも鍛えられるし、数学の知識も深められるしで良い動画

  • @Kolinnor
    @Kolinnor Před 7 lety +47

    Those animations are outstanding.

  • @MegaMoh
    @MegaMoh Před 5 lety +22

    For anyone who wants to graph the intersecting parabola, the general equation for each parabola is x=[+/-](y^2 / 4(n)^2 - n^2) where "[+/-]" is plus or minus and "n" represents the nth parabola away from the origin. In latex, it's written as:
    x=\pm\left(\frac{y^2}{4n^2}-n^2
    ight)
    for those who want it written neatly. The straight line equations are as simple as taking each coordinate that from the intersection (a,b) and making the equation y=b/a * x or y= \frac{b}{a}x in latex
    NOTICE: A parabola written in the form of ax^2+bx+c has a=1/(4f) where f is the focus. I noticed that the focus for those parabolas using the equation is n^2 so that the focus of all of these parabolas is it's number squared. then noticed that the focus changes when the "c" term changes in the equation, then the focus get translated by "c" and what turned out is that the "c" term in the above equation is also n^2! so n^2(the focus) - n^2(translation by "c" term) gives 0. so that all of those parabolas have their focus at the origin and each one is away from the origin by n^2 distance! Let's work together to figure out why this equation works with these givens

    • @StarNumbers
      @StarNumbers Před 7 měsíci

      A side note: The creation of the parabola equation started by trying to determine the trajectory/path of a cannonball. The framework takes the parameters of gravitation and the earth below but the earth must be flat. Yes, the earth is flat (and stationary), while thinking of the ball earth as "Close enough for govt work" is just that.

    • @kimjiwoo9557
      @kimjiwoo9557 Před 24 dny

      THE GOATTTTTT

  • @tommiweck8660
    @tommiweck8660 Před 3 lety +21

    It's fun how CZcams recommends me this just after a math competition where I could have used this information and saved some time.

  • @kaspersolberg1938
    @kaspersolberg1938 Před 4 lety +7

    Even as a mathematician, this channel is mind-blowing and so well animated and explained. Thanks a lot.
    If only I had 3B1B when I studied complex analysis back in the 90´s.

  • @DiscoMouse
    @DiscoMouse Před 7 lety +64

    love the peeved pi at 6:00

  • @quantummath
    @quantummath Před 7 lety +7

    Dude, I love your channel, keep up the great work.

  • @tasiemiecuzbrojony
    @tasiemiecuzbrojony Před 2 lety +2

    Rewelacyjne opracowanie problemu, doskonałe wizualizacje, jestem pod wrażeniem... Zawsze ciekawło mnie ile jest tych trójek pitagorejskich i jak je szukać. Dziękuje, pozdrowienia z Polski

  • @Amr-Ibrahim-AI
    @Amr-Ibrahim-AI Před 4 lety +1

    Wow! This is amazing and mind blowing! Thanks for your mind-stinulating videos 🙂

  • @SSJProgramming
    @SSJProgramming Před 7 lety +50

    Seriously ... unbelievably amazing content.
    Keep it up!

  • @Cesariono
    @Cesariono Před 7 lety +22

    Oh my God.
    One of my biggest motivations for studying programming was precisely this: a visualisation of all of the pythagorean triples. I can't believe you've done this. Thank you.

    • @GalacticSlayer
      @GalacticSlayer Před 7 lety +2

      Mithra and now you studied programming for nothing
      jk
      programming = low effort, high reward

  • @djyoon123
    @djyoon123 Před 4 lety +1

    Thanks a lot, great description, inspired video. Wow! The square of every integer pixels except those at diagonal go to Pythagorean triple. It shows us a fabric on how complex plane and complex number is defined.

  • @keremardicli4013
    @keremardicli4013 Před 4 lety

    This channel never ceases to amaze me.. unbelievably good...

  • @jimsmind3894
    @jimsmind3894 Před 7 lety +4

    So elegant and beautifully​ illustrated.
    I remember noticing parts of this when looking at triples, it seems so obvious now!

  • @sketchartyst
    @sketchartyst Před 6 lety +5

    This is honestly so incredibly beautiful. Seeing this made me emotional

  • @jinseokkim2586
    @jinseokkim2586 Před 4 lety

    probably the best video from your channel.
    great

  • @vardhanshah2810
    @vardhanshah2810 Před 4 lety

    Only this channel has till now made me able to visualize a plane with complex numbers. I feel so different in the inside. Amazing vid

  • @duffyoxopatt3950
    @duffyoxopatt3950 Před 7 lety +32

    Man i love your videos!
    I was pretty bad at maths in school, but you explain so well i can understand everything.
    And your voice would cure cancer.

  • @mamalittlefoot1491
    @mamalittlefoot1491 Před 6 lety +4

    This is so beautiful! Thank you for sharing your knowledge and time to produce this aesthetic video :-)

  • @yash1152
    @yash1152 Před 3 lety +2

    0:12 nice rearrangement of those 9 and 16 cubes

  • @orlybuchbinder3585
    @orlybuchbinder3585 Před 3 lety

    Thank you for the most beautiful video.

  • @drddff9788
    @drddff9788 Před 7 lety +9

    This is one of the most beautiful things I've seen in a while

  • @merp1998
    @merp1998 Před 7 lety +5

    Watching this video was a magical experience. Thank you 😄

  • @Pablo360able
    @Pablo360able Před 3 lety +26

    I came up with an entirely different way to generate Pythagorean triples in middle school, though much less visual, using the property that x^2=∑(1≤i≤x)2i-1, i.e. that squares are the sums of odd numbers: Any expression of a number's square in terms of a sum of squares that does *not* start at 1 corresponds to a nontrivial Pythagorean triple, where the hypotenuse's square is the sum when the sequence of odd numbers is extended down to 1. You can generate such a series by choosing the number of odd numbers to add, which can be any factor of x² with the same parity (both even or both odd) (there's a valid interpretation for when n>x, though it's a bit weird), then choosing the *middle* of the sequence to be x²/n. Someone check my math.

    • @sidharthghoshal
      @sidharthghoshal Před 7 měsíci +2

      Ah so basically if for some j != 1 we have that 2j+1 + 2j+3 + 2j+5 ... = m^2 then obviously 1+3+5... 2j-1 = n^2 and 1 + 3 +5 + ... 2j-1 + 2j+1 + 2j+3 ... a square number as well. That's a nice observation!

  • @winterglue274
    @winterglue274 Před 4 lety +2

    The sound and animation are soothing
    really chill math

  • @noa.leshem
    @noa.leshem Před 7 lety +33

    you're on fire WHAT IS THIS INSANE POSTING SCHEDULE

  • @jibran8410
    @jibran8410 Před 7 lety +5

    The amount of work it takes to make these vids....You deserve more subs man and you don't even put ads in ur vids.wow

  • @math3usyb
    @math3usyb Před 3 lety

    your videos are always so amazing. I can see clearly why plato had correlated geometry in his cosmology

  • @shashanksingh3594
    @shashanksingh3594 Před 3 lety +1

    your explanation and video is so awesome that after watching the first 6 minutes, I immediately wrote a python script which generate these pythagorean triples

  • @raza8442
    @raza8442 Před 6 lety +4

    Your visual representation is the best, as I have seen ever.

  • @LorJSR
    @LorJSR Před 7 lety +5

    3Blue1Brown - This videos are incredible, and I love them. There must be so much work that goes into making one of these, I can't even imagine. I'd love to see a behind the scenes video about how you go about planning, writing, voicing and finishing these things.
    It's a thing of beauty and a joy forever, it must be like making a porcelain vase - incredibly complex and time-consuming, and producing something outstanding. =O

  • @Leyonad
    @Leyonad Před 3 lety

    These animations are clean. Great job!

  • @Lenny2Lux
    @Lenny2Lux Před 3 lety

    I'm addicted to these videos. He just keeps blowing my mind!

  • @milojacquet7507
    @milojacquet7507 Před 7 lety +19

    I remember discovering this method a few months ago and being amazed about how is generates these triples. When you showed that it generates multiples of every triple, that was incredible! I had no idea that it generated every triple.
    Also we met at that café at Stanford completely coincidentally, remember? That was amazing.

    • @3blue1brown
      @3blue1brown  Před 7 lety +10

      +Milo Jacquet Oh I remember. Hope all is well!

    • @milojacquet7507
      @milojacquet7507 Před 7 lety +2

      Yep! Recently I've been learning about a continuous function that is nowhere monotonic. It's quite strange!

    • @SpaghettiToaster
      @SpaghettiToaster Před 7 lety +1

      Milo Jacquet the weierstrass function? 3b1b could make a cool video on that I bet. It has a pretty badass look to it.

    • @rudboy9599
      @rudboy9599 Před 7 lety +2

      SpaghettiToaster that's the one that's like an infinite sum of cosines right? It's all jaggedy when you zoom in. It's also continuous everywhere but differentiable nowhere, right?

  • @One_In_Training
    @One_In_Training Před 5 lety +7

    You sir, are a truly gifted genius. These videos are so beautiful, they make me tear up.

  • @crnbr
    @crnbr Před 4 lety +375

    아 자막 반만 만든거 실화냐.. 똥덜닦은기분 후.. 어차피 이해못할거라서 참는다..

  • @taesan0512
    @taesan0512 Před 4 lety

    this video is such like an art
    i think you must have to feel beauty of that way to visualize them

  • @themeeman
    @themeeman Před 7 lety +150

    Please do a full video on fermats last thereom and how it was solved. I have read up on it, but I think that a video from you would make it simpler to understand.

    • @ptyamin6976
      @ptyamin6976 Před 7 lety +12

      all i know is that it has something to do with modular forms which is connected to algebraic geometry. in any case, thats a lot of deep background material and thats why i think it would be impossible to understand in even an hour long video

    • @dudeman3981
      @dudeman3981 Před 7 lety +19

      Clingfilm Productions There's a reason why it took the worlds greatest mathematicians over 350 years to solve it.

    • @Angel33Demon666
      @Angel33Demon666 Před 7 lety +29

      Dude Man Nah, its solved by Fermat himself. It's just that the proof is too large for the margin to contain. :')

    • @burthpinmc5489
      @burthpinmc5489 Před 7 lety +3

      Angel33Demon666 Oh not again!
      You sneaky fermat

    • @Nothing_serious
      @Nothing_serious Před 7 lety +14

      Apparently his proof is too long to contain in a video.

  • @erichschmidt1328
    @erichschmidt1328 Před 5 lety +3

    I am always surprised by a 3blue1brown clip. And I am always a little bit frustrated that I never saw these interesting things for myself, although I had complex numbers, calculus, linear algebra and so in during my study. Congratulations for your fine Clips and your beautiful animations.

  • @my_me_my
    @my_me_my Před 3 lety +90

    중간에 자막이 없는 건 한국인 난이도에 맞춰서 영어까지 직접 해석해야되는 교육계의 참된 뜻인가

  • @user-us3ph3gt3m
    @user-us3ph3gt3m Před 3 lety +1

    2021년에 듣고있는데, 정말 유익한 영상이네요 감사합니다

  • @kantaki
    @kantaki Před 7 lety +54

    A video about quaternions would be amazing.

  • @Arithryka
    @Arithryka Před 7 lety +7

    3:01 never questioning the validity of the complex plain again, this is just too brilliant.

    • @vari1535
      @vari1535 Před 4 lety +1

      It is the complex _plane_ that is valid, not the complex plain, you moron

    • @denelson83
      @denelson83 Před 4 lety

      @@vari1535 Besides, "complex plain" is an _oxy_moron.

  • @cerwe8861
    @cerwe8861 Před 4 lety +49

    You can also do the Pythagorean tripple Generator algebraicly:
    a²+b²=c²
    a²=c²-b²
    a×a=(c-b)×(c+b)
    a/(c-b)=(c+b)/a=u/v
    ¹ (c-b)/a=v/u
    ² (c+b)/a=u/v
    ¹+²=³... Just Kidding
    ¹+²:
    2c/a=(u²+v²)/uv
    c/a=(u²+v²)/2uv
    ²-¹:
    2b/a=(u²-v²)/uv
    b/a=(u²-v²)/2uv
    Now we can say that numerator= numerator and denominator=denominator
    So we get
    a=2uv
    b=u²-v²
    c=u²+v²
    The same result.

  • @Wurfenkopf
    @Wurfenkopf Před 3 lety +1

    THIS.
    Is FANTASTIC!!!!
    I LOVE it!!!
    I can't believe I graduated in maths and still didn't know about this!

  • @nicholasleclerc1583
    @nicholasleclerc1583 Před 5 lety +12

    4:15
    Yeah, that’s because of Euler’s identity: 2+i is basically sqrt(5)*e^(~1.10715i), so you double the angle and square the sqrt

    • @nicholasleclerc1583
      @nicholasleclerc1583 Před 4 lety

      @�̴̀͌̕
      The Euler Identity happens from realising that, if you interpret the concepts of an angle and of an exponent in a weird way :
      r*e^(i*x) = r*cos(x) + r*i*sin(x)
      Where x is an angle *MEASURED IN RADIANS, NOT DEGREES; VERY IMPORTANT*
      Essentially, you just plug in the value of the angle (IN DEGREES) for the x power of e, then you discard the "rad" unit
      And r is the *square root* of _the addition of the squares of the real number and the multiplier of i_
      So we now know that you can rewrite additions of real numbers with a multiple of the imaginary number _i_ with a single term, that is use without having to add 2 or more things together
      So, since 2 + 1*i is such an addition, we can convert this into a single number, "r*e^(i*x)", where, again, r is a square root involved with the real number (2) and the multiplier of i (1); but when we square this "r*e^(i*x)", then we square "r", therefore we square a square root, tus we get the number that's inside, which is, again, _the addition of the squares of the real number and the multiplier of i_ , which is "2^2 + 1^2", or "5"

    • @nicholasleclerc1583
      @nicholasleclerc1583 Před 4 lety

      And x is the angle between the line connecting to the origin of the Real-Imaginary graph and the complex number and the x-axis; if the complex number's above the negative values of the x axis, then the angle's between 90 degrees and 180 degrees; and if the complex number's under the x axis, then the angle's negative

  • @jacheto
    @jacheto Před 7 lety +53

    I was going to ask what is the program you use to animate this, but you actually program it, on python...I also work with python and it is a really easy language but I imagine it is incredibely difficult to make these videos, I mean, you must know a lot! I wonder if there is some video editing software like after effects but with math properties, not only graphical ones, I bet it would be impossible to achieve such precision and amazing visualizations like your grid transformation stuff using any other software

    • @redjr242
      @redjr242 Před 7 lety +30

      That's right. Nothing gives you the same amount of flexibility as actually writing graphics routines to express exactly what you want to visualize.

    • @robertwilsoniii2048
      @robertwilsoniii2048 Před 6 lety +9

      The dude knows what he’s doing. He graduated in CS/Math from Stanford University. I have a friend who took a class with him, he’s a genius type.

  • @edgardojaviercanu4740
    @edgardojaviercanu4740 Před 3 lety

    These videos are beautiful.

  • @rakhananda1737
    @rakhananda1737 Před rokem

    thank you so much man it helps me a lot

  • @Echozkii
    @Echozkii Před 5 lety +185

    This is the part when those kids in class say, "When are we going to use this in our life?"

    • @Idisagreethisisnotanon
      @Idisagreethisisnotanon Před 4 lety +7

      EchoZK - Games - Music - Illustration I mean when are we gonna use this in life?

    • @immortaltitan3839
      @immortaltitan3839 Před 4 lety +1

      But can you answer that question sir?

    • @aathish04
      @aathish04 Před 4 lety +57

      @@immortaltitan3839 The same place where you're going to use your extensive knowledge of . This type of maths is beautiful, but not useful. In that way, it's more like art than science!

    • @southernkatrina8161
      @southernkatrina8161 Před 4 lety +10

      When you want to know how tall something is without going up a ladder. Which happens enough to make it useful. Roof. Ceiling. Tree you want to cut down that you hope will not crush your rosebush. Ladder height long enough to reach roof. Etc.

    • @kuchenteig4240
      @kuchenteig4240 Před 4 lety +6

      @@aathish04 yes, science can be art too!!

  • @alokyes
    @alokyes Před 7 lety +12

    the best animations in the whole universe

  • @neelamverma8167
    @neelamverma8167 Před 3 lety +42

    i wish my maths teacher was so cool as the Korean maths teacher that gave this video as hw to their students

  • @TheMrSamusic
    @TheMrSamusic Před 3 lety

    This is so mesmerizing...

  • @thisisomer
    @thisisomer Před 7 lety +3

    6:15 Euclid's formula for generating pythagorean triples, I remember learning this but I was never taught WHY this is true. This is so simple so intuitive so brilliant, it makes me sad to think I only know it now, years after seeing the algebra behind this method. Thanks you for enlightening me.

  • @TheSkrillexreptile
    @TheSkrillexreptile Před 5 lety +12

    You have the best videos for understanding math, period.

  • @mr88cet
    @mr88cet Před 3 lety

    That’s awesome: I confess I’d never seen that graphical proof before! Thanks!

  • @anotherone3641
    @anotherone3641 Před 4 lety +59

    8:42 6+8i is not possible, but 8+6i well acceptable. The main rule is the real part must be greater then complex becouse u^2-v^2 > 0 must be.

    • @forrest3797
      @forrest3797 Před 2 lety +2

      Interesting, but why does u^2 - v^2 has to be greater than 0 ?

    • @darshdodeja
      @darshdodeja Před 2 lety

      @@forrest3797 Yeah why?

    • @allymacmullin5952
      @allymacmullin5952 Před 2 lety +3

      @@darshdodeja I'm not entirely sure, but I think its because it represents a length/distance, which can't be negative

    • @smiley_1000
      @smiley_1000 Před rokem

      But neither 9 + 12i nor 12 + 9i are hit

    • @gabrielleao2816
      @gabrielleao2816 Před rokem

      ​@@smiley_1000 But 4i + 3 is

  • @merveilmeok2416
    @merveilmeok2416 Před 4 lety +4

    You are a genius (every time I see your videos I have to write that 😁).

  • @JaLikon65
    @JaLikon65 Před 7 lety +253

    *Every 3blue1brown video:*
    1. Take the coordinate plain. Here, our problem can be reframed and explained fairly simply. Our task is to find [x]
    2. Just kidding, throw away the standard coordinate plain. Actually, take the complex plain. Here, our problem looks more complicated, and in some ways it is, but consider how one might solve for [z]
    3. Some mathematical steps later...
    4. As we can see, [z] perfectly solves for [x]
    Moral of the story: Might as well always use the complex plane :P
    P.S. This comment was not meant to be sardonic; it was only a fun observation I had. If you happen to see it 3b1b, please don't take it offensively. I, like everyone else here, absolutely love your videos. Thank you for making them.

    • @fossilfighters101
      @fossilfighters101 Před 7 lety +2

      +

    • @Kualinar
      @Kualinar Před 5 lety +5

      Some times, taking the route that looks harder or more complicated is the best, simplest, easiest way.

    • @matthewto7406
      @matthewto7406 Před 5 lety +15

      Jordan Ellenberg in his book How not to be wrong, the power of Mathematical Thinking:
      Outsiders sometimes have an impression that mathematics consists of applying more and more powerful tools to dig deeper and deeper into the unknown, like tunnelers blasting through the rock with ever more powerful explosives. And that's one way to do it. But Grothendieck, who remade much of pure mathematics in his own image in the 1960's and 70's, had a different view: "The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration...the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it...yet it finally surrounds the resistant substance."
      The unknown is a stone in the sea, which obstructs our progress. We can try to pack dynamite in the crevices of rock, detonate it, and repeat until the rock breaks apart, as Buffon did with his complicated computations in calculus. Or you can take a more contemplative approach, allowing your level of understanding gradually and gently to rise, until after a time what appeared as an obstacle is overtopped by the calm water, and is gone. Mathematics as currently practiced is a delicate interplay between monastic contemplation and blowing stuff up with dynamite.

    • @tychophotiou6962
      @tychophotiou6962 Před 4 lety +2

      You made the complex plane become plain!

    • @vari1535
      @vari1535 Před 4 lety +1

      pLaNe

  • @stickmcskunky4345
    @stickmcskunky4345 Před rokem

    Me: wonders about a concept on my own.
    You: always have a video explaining it eloquently and comprehensively. TY!

  • @soso-rl5hi
    @soso-rl5hi Před 2 lety

    one time i watched a video class because i was desperate and didn't know basic math and now those videos show up on my recommended and i love watching them to see what i may or may not understand and just bc i love hearing smart ppl talk

  • @jeanmarabou9774
    @jeanmarabou9774 Před 5 lety +4

    When I watch these kinds of videos I wonder and imagine how much Pythagore or any antiquity mathematician would have been hyped watching this

  • @user-yn7ue1lk6u
    @user-yn7ue1lk6u Před 3 lety +2

    Очень красиво, спасибо. Я ожидал в конце неких глобальных выводов о распределении точек на окружности, но не дождался, очень жаль. Наверное эта тема ещё ждёт своего исследователя.

  • @effka2660
    @effka2660 Před 4 lety +1

    ... just beautifully emazing ... Thanks!

  • @erikhalvorseth3950
    @erikhalvorseth3950 Před 2 lety

    Just the clip intro pic is immensely beautiful

  • @mr88cet
    @mr88cet Před 3 lety +8

    I “ain’t thunk” through yet the ramifications of this, but I noticed that, although this pattern of interlocked parabolas has a 6-8-10 right triangle but no 3-4-5, it *does* have a 4-3-5 right triangle. That’s a result you get from scaling, as you pointed out.
    So, in other words, if you reverse your axes you can achieve at least some effects of scaling of complex numbers.

  • @seeahcompany890
    @seeahcompany890 Před 4 lety +90

    시작 : 피타고라스는 내가 또 알지
    1분뒤 : 자..자막을 켜볼까?
    2분뒤 : 자..자기전에 보는영상인가?

  • @paulflute
    @paulflute Před 3 lety

    i love these videos.. pretty picture.. soothing voice.. some safe numbers
    and I feel I'm a better person afterwards in a way I can't put my finger on..

  • @thatsmetube
    @thatsmetube Před 4 lety

    Fascinating. Well done.