Sum of n squares | explained visually |

[@Jabrils]

mindbending, but beautiful; the 3B1B army has arrived!

[@KnThSelf2ThSelfBTrue]

Oh snap, my people.

[@Gravitraxer_AangCZ]

@@KnThSelf2ThSelfBTrue OUR people

[@tooljockey2777]

yes it has!

[@themaverick1891]

No.

[@alexhales9520]

3B1B?

[@ThinkTwiceLtu]

If you enjoyed this video please leave a like or comment below, so that more people could see it :) thank you so much

[@skoockum]

That's one way. Another way is to type 1,5,14,30,55 (ie the first 5 sums of n^2) into the search bar of OEIS and scroll down thru the results.

[@monicanagaraj1828]

I like all your videos!!!

[@zhengqunkoo]

Visual proofs are so intuitive. I wonder if there's a general way to prove everything visually, e.g. graphical linear algebra.

[@iliasant.4632]

well, "visual proofs" are not really proofs. They are not rigorous by nature and they are highly dependable on our geometric interpretation. While they're awesome and necessary to our understanding of mathematical concepts, when you' re aiming to prove non-trivial and "deep" results you have to be formal and precise in order to be (rather) easily peer-reviewed by the mathematical community. Visual aid is always welcomed for clarification and dissemination of the science though.

[@zhengqunkoo]

I get that, I just hope we can find a way to make visual proofs rigorous. Perhaps not all proof explainations can be reduced visually, given some proofs are as simple / visual as they can be already, e.g. graph theory visualized as literal nodes and edges.

[@AdrianReef]

"Visual proofs are not really proofs"... Are you sure about that statement? How can a "proof" not be "a proof"? You're basically saying AA (which is false). Because visual proofs are the most profound and researched in the entire mathematical field. Indeed, you might need a formalization in case you should ever reach out for a reviewer,but it doesn't mean that a visual representation isn't a proof only because of the lack of that (if it really is undeniable).

[@sigurfeanaro3356]

Adrian Reef, perfect answer! HA HA HA huahuahuahua

[@trulyUnAssuming]

I guess because a visual proof can end up being wrong because of optical illusions. And because it is hard to actually prove that something generalizes. Every visual proof is basically just a demonstration by one example. And while humans think in examples and we usually understand a concept a lot better with examples, maths is not about examples but generalization. Humans work a bit like statistics. You throw down a lot of data points and then you can get a good idea how the underlying mechanic/function looks. Just think how plotting a graph works. In contrast we can't really understand the formula of a function even though the selected points we choose to plot the function does not represent the entire function. So while the actual truth is the function behind the plotted points, the intuition comes from the selected points we plotted. But maths wants to deal with the truth behind which is the function not just a couple of examples. And if you select the wrong examples you get the wrong idea, of how the function actually looks. Like when you select too small of a scope or too large. So while visual proof helps understand the concept you should still be able to write down the proof. I mean the entire epsilon-delta continuity/differentiability stuff (i.e. calculus) works that way. You can think of it as epsilon and delta balls in 3d/2d space and most of the things like continuity have a very intuitive visual representation that way. But when you write it down you should generalize it to metric spaces. And then the notation we use and the implications we allow act as a handrail, because humans can't really imagine those generalized concepts. EDIT: my analysis professor was really good at teaching us to draw the intuition of the proof and then use that sketch to write the actual proof.

[@JWentu]

I love the fact that you explained it in the time of a Bach prelude

[@normanhenderson7300]

Bach prelude? How about a Miles Davis prelude; that is not credible as well?

[@isaacanwarwatts8844]

@@normanhenderson7300 the music in the background was a bach prelude

[@DorFuchs]

Great animation!

[@ThinkTwiceLtu]

DorFuchs thank you:)

[@malingamurali]

I think it would be nicer if, instead of cutting the cubes in half and getting 1/2, to just double up that whole shape and stick it on top. Then the 2n+1 seems to come more intuitively, that's my opinion anyways haha

[@jeffrussert448]

I agree, then you don't have to factor out a 1/2, which isn't very visually intuitive anyway.

[@normanhenderson7300]

Dividing by 1/2 is alternate to doubling by well ha ha 1.

[@nikhilchauhan8277]

Ncert has same solution

[@Shubham_pandey-nk1un]

@@nikhilchauhan8277 You are right 🤣😜

[@drsuper8180]

Agree Fully

[@hp-qx7tf]

I actually shed tears of joy watching this

[@nishantsharma1241]

Same dude

[@omniscientomnipresent5500]

I am going to say it, whoever found that has a functionning brain

[@ZAKINGOFDESPAIR]

Yeah, first time I saw its derivation I was like "dude... How did u even arrived at such a conclusion ?!"

[@rishabhdhiman9422]

* _sheds a tear_ * its beautiful.

[@ely_mine]

3b1b sent me here, zero regrets

[@yw5617]

your awesome dude i just spent 20 minutes scrubbing the Internet to find the source of the animations you make and god am i happy to find you

[@LunaSuJu]

I've seen lots of other proofs but visualization is really something else. It really helps memorising things and makes formulas less intimidating. Thanks!

[@sonicmaths8285]

It also helps in the expansion of mathematical intuition, which builds comprehension for the subject and that is worth barrels of gold.

[@firebad0]

this is good

[@sigurfeanaro3356]

I would say wonderful, but good is almost enough!

[@vanshgarg9560]

One of the best channels on KZbin

[@paulmcmullan9931]

God knows this; that's why genetics works so perfectly.

[@kaziaburousan166]

And here we are...the viewers of 3Blue 1Brown...what a amazing and beautiful channel it is..thanks to 3Blue 1Brown for suggesting this channel

[@Youbetternotrecognizeme]

This is the best thing Ive ever seen

[@bmisawa3750]

I am a simple man. I see Think Twice video, I press like.

[@cleiven3533]

You don't *think twice* before pressing like?

[@kikones34]

A simple and beautiful proof. Very nice animation, good job :D!

[@ThinkTwiceLtu]

thank you!

[@janschetters7720]

I love it. I will show it to everybody I know and not know.

[@DeeEm2K]

Commenting just due to the fact that I learnt a visual proof with beautiful music as well

[@AbuTheEvil]

Thank you very much, after watching your video it's just impossible to forget the formula, and on top of it, it's not due to mere memorization but thanks to a real understanding of how it came to be. And with Bach on top of it, summing all the pleasures of the mind :)

[@stjernis]

If there's anything to be faulted here, it's that it may not be obvious that the three slanted pyramid shapes end up perfectly filling the block. I'd've like to have seen the animation at 0:48 first showing two pieces come together, and show that the remaining empty space fits the third.

[@sunnynegi722]

Wow itna simple way me proof kar diya. me khud se tricks laga ke kafi koshish ki par kuch galti ho jaati thi .Par apne to 1 minute me samjha diya

[@jpf1950]

So nicely done. Thank you.

[@ThinkTwiceLtu]

John Flanigan :)

[@ClassicalRaritiesChannel]

Fantastic job on visualising this, thank you so much!

[@sankarbhattacharya1770]

I have just seen this presentation and I can't esist writing to say how much I was carried away by it. Really it's magnificent-

[@normanhenderson7300]

Dont become too lost now little lad.

[@hasiumcreeper5384]

I feel like putting ALL your videos in my 'Watch Later' playlist just to show to people when I'm trying to explain math.

[@hasiumcreeper5384]

Also, 3Blue1Brown sent me. Hello!

[@sharadsmathhelp2439]

Thank you so much. I really liked your visual proof. I knew the visual proof of sum of first natural numbers but the sum of squares was always hard to visualise. Could you also make a video for sum of cubes of first natural numbers?

[@creedflow]

Just saw your video on Reddit, awesome video you have my sub, I'll watch older and new uploads

[@ThinkTwiceLtu]

Luis Raul Garcia Mendoza I'm glad you liked it:)

[@kennethsizer6217]

Excellent visualization! Excellent choice of music to accompany it! 💕

[@aaronhamilton8997]

Holy crap! I finally understood why!! 3B1B was not mistaken in recommending you!!

[@thevenin26]

This is much better than the proof in my textbook

[@Sam_on_YouTube]

That's so much better than the traditional proof of that.

[@hollow6079]

That's a beautiful graphical way to proof the sum of n squares.

[@anamarijavego6688]

I've watched all of your videos... perfection

[@drsuper8180]

Very elegant! was trying to do this proof that I believe I had done and even drawn out years back with some (orginal wooden!) Dienes Blocks that I have inherited. I just could not remember it even though I was remembering that we needed the step-wise start . I agree with ABrownDude that it would have been even more elegant if you did not switch to algebra and continued making the second copy and placed it on top.

[@ノラ-h3q]

This is so beautiful

[@goofyfoot3063]

Hey there! Loved the beauty and simplicity of your video!! What software did you use to make this video?

[@ThinkTwiceLtu]

Goofy Foot hey:) thanks a lot. I used Cinema 4D to make the 3D animation, I also used premiere pro to edit everything.

[@normanhenderson7300]

@@ThinkTwiceLtu , And the tools you used had to be put together by bits(decimal, octals, hexadecimal- in machine coded interpretation- using s ymbolic based languages to decipher.

[@StarNova9]

This is sooo awesome!!

[@iamgk91]

WoW, beautiful ☺️

[@ThinkTwiceLtu]

iamgk91 ☺️

[@jhfh3112]

Going to show my students this today - lovely animation

[@ThinkTwiceLtu]

jhfh3112 that’s awesome, thank you:)

[@GDLameGames]

I had soo much trouble interpreting this formula and you just cleared everything up! Thank you! :)

[@normanhenderson7300]

I still have a lot of fucking trouble doing so dude. My puny brain cannot grasp such things.

[@bluecrab2]

Amazing visual, thank you!

[@adnanahmadknowledgevideo]

What an amazing visual explanation 🙏

[@mehmetmahirguner7470]

Harika bir şey! KZbin'da bundan daha çok olmalı

[@АлексейАндрюшечкин-е4ю]

It's amazing! So obviously and elegant.

[@parthi2929]

It would b beautiful if you could also link each visualized step with theoretical step (which often is greek and latin for many)

[@toasteduranium]

What is often Greek or Latin for many? Poly?

[@ryankang562]

This really helped man i try to visualize everything I do and I was especially struggling with this proof since my textbook only showed the formula for summation, not the proof

[@ThinkTwiceLtu]

I'm glad it helped you:) thanks for the sub

[@thetyler315]

It’s crazy seeing these videos. Apparently I could have made it a quarter of the way through my discrete course using “proof by blocks” 😁!

[@NinjaJacky]

Mind-blowing koi jawab nahi

[@sundhukumar]

Wow…..kudos to sir…. What a beautiful way of seeing math

[@nkanyezitshabalala5256]

I figured out a much simple way to determine the summation formula for this sequence. I came here to see if anyone else had determined it the way I did but so far it doesn't seem so. If you would like to see it I would be happy to show you. Additionally I extended my approach to find the sum of the n terms of the cube sequence and I got the right formula. I think it is possible to do so for higher power sequences.

[@sasoribi1341]

分かりやすい！

[@mohammedal-haddad2652]

Visual proofs are my favourite. Thank you very much.

[@monoman4083]

this i like very much..

[@LoUiSvsMiKu]

this is the most genius proof

[@298chandra]

Beautiful presentation .

[@AngeloYeo]

Truly wonderful...

[@ahmedafifkhan]

Please don't stop!

[@ChristopherLamke]

Beautiful. Thank you for making this.

[@KodamatiUdayaPrabhuKiran]

Oh.. I just love this

[@pebre79]

Wonderful and so elegant

[@abcdefgboom1]

best video on the internet.

[@natejack2292]

The visual sum proofs are very eloquent

[@specialfalcon7427]

Very very very thank you man! please make more vides on these!

[@danii7120]

Amazing video, I love your videos TT!

[@mohammedyaseen2504]

Well done think twice

[@sunsetflory]

lol i've learned that formula during math class but it had no sense to me until now. wow, thanks :)

[@angsonnguyen2430]

This is beautiful, thank you!

[@yamansanghavi]

Beautiful animations. I love your channel.

[@ThinkTwiceLtu]

Yaman Sanghavi thanks:) appretiate your comments.

[@normanhenderson7300]

I was somewhat impressed with the animations; since I am a mathematical novice I was more interested to decifer to pictorial representation.

[@JuanIgnacioAlmenaraOrtiz]

Me encantan las demostraciones visuales!

[@rajatgujrati4057]

Your videos are amazing...

[@jlpsinde]

Great video thanks!

[@ashleylee217]

I like

[@ThinkTwiceLtu]

:)

[@hall0o082]

why are you so good?

[@RiderOfMooses]

this channel is awesome

[@schlaier]

Such a good channel

[@kristinadjurovic]

Most beautiful

[@Kino-Imsureq]

Typo in 0:14 "equvalent" nah who cares about that typo this video is awesome!

[@ThinkTwiceLtu]

Kino -Imsure1200q I'll be more careful with my spelling next time:) glad you liked the video

[@PhilBoswell]

I missed that one but I did spot "height" spelled wrong at 0:29 :-(

[@aayushkarn6198]

Beautiful!

[@vipulpetkar]

This is insane... Now I want to be mathematician

[@junhobbbjeonbbvb1029]

thank you very much, this help me a lot

[@Nostra.Damus14]

This is so beautiful

[@sonusmathsclasses]

I just wanted to say thank you!

[@ThinkTwiceLtu]

Thanks for watching!

[@phanipriyatadapally6543]

U guys are rocked the video by good explanation

[@lyrimetacurl0]

This could also be used to prove how the sum of n squares is only a square if n is 0, 1 or 24. Since it's a cubic equation.

[@AgentMidnight]

Instant subscription

[@ThinkTwiceLtu]

Cubik thanks a lot:)

[@KotMarissa]

Thanks for the videos ! The video is meant to make math fun and engaging, like a magic show with happy music.

[@mrminer071166]

Now make an equally luscious one on Heron's Formula for area of triangle.

[@funkysagancat3295]

That's mesmerizing

[@ananthakrishnank3208]

Brilliancy!

[@ak-8gamer408]

WOW!!! Thanks a lot. Now I'm clear

[@TylerMatthewHarris]

This is great

[@sanjeevraila5161]

great art to help us😍

[@mohammedyaseen2504]

Amazing great work

[@karthikrambhatla7465]

Anywhere we find squares, the squares actually work. I would like to see a cubes generalized formula visuals

[@jihongzhi]

Beautiful animation. But I think at 0:35 you mean "height" instead of "heigth"

[@JaiKumar-ko2tw]

Good explanation

[@anticdisposition5908]

Nice video, what song/music piece is in the background?

[@voorth]

J. S. Bach's Prelude No. 1 in C major, BWV 846.