Using topology for discrete problems | The Borsuk-Ulam theorem and stolen necklaces

[@anuel3780]

3:56 don't cut or tear the sphere *FLASHBACK TO HOW TO TURN A SPHERE INSIDE OUT*

[@thephysicistcuber175]

[@__-cx6lg]

Yup! That's the general rule in topology: everything has to be continuous.

[@DemonixTB]

That video creepily followed me for years even if i rewatched it. Again. And again. And again. Just please, STOP! I KNOW HOW TO TURN A MF SPHERE OUT OF SOME WEIRD MATERIAL INSIDE OUT LET ME LIVE MY OWN LIFE IN PEACE GAHHHHHHHHH

[@Male_Parent]

You mus'nt tear or crease it.

[@polmarcetsarda]

@@DemonixTB I see I'm not alone

[@mlok4216]

2 thiefs have stolen a 17 jewels-type necklace. One to the other: "Yo, wanna count the jewels and split them evenly?" The other one: "Nah, let's construct 18-dimensional hypersphere to help us out!" xD

[@AlexAegisOfficial]

Then they bought 523425 watermelons.

[@SlackwareNVM]

Won't it be an 18-dimensional sphere? I liked the joke, tho.

[@brightsideofmaths]

That happens if all of your friends are mathematicians ;)

[@ob3vious]

You forgot it to place it from an 18dimensional hypersphere to 17dimensional hyperspace. We only have a little problem... how will we ever find it, if we're using 17 variables. No supercomputer will be able to solve it for you, because it has too much data to work with. You have an infinite amount of points scrolling past and it only can use a few tricks to quickly scan. I donot think any mechanical device is able to solve it for you, sadly.

[@guren1484]

@@SlackwareNVM It should be 17-dimensional because sphere is the surface of the ball.

[@ZoroarkChampion]

This is the first video where I tried to understand fully every single step along the way. It took me nearly an hour to finish the video, but I’m glad I did! Having had no formal math education since graduating high school four years ago, it was harder than it should have been. It gave me an important insight in understanding math I hope someone else will be helped by: to ask with every step why it needs to be the case. If you can’t answer that question, try to figure it out for yourself. This way you will play with the math yourself, which I’ve found to be the only way to truly grasp and enjoy anything. Thank you so much 3Blue1Brown for making these videos and explaining everything so clearly!

[@3blue1brown]

Thanks for putting in the time!

[@Kinslayers0]

3Blue1Brown i was lost in this video sensei :(

[@ScormGaming]

Your active engagement in math is what will take you the furthest, no matter where you started. I'm glad to see a comment with such courage inside the ocean of puns.

[@jonathanharoun5245]

"nearly an hour" I'm a math major and it would probably take me several DAYS to understand this video.

[@williamromero-auila7129]

So it's time to learn to evaluate and steal necklaces

[@eier5472]

10 emeralds!? I know a villager that would give me like two wheat for that

[@wierdalien1]

I'd like some sheep please

[@parallellinesmeetatinfinity]

@@wierdalien1 they are 9 bucks 😕

[@wierdalien1]

@@parallellinesmeetatinfinity have you not played Settlers of Catan

[@artyfarty99changingtohoiii78]

nah he’d give me a stick

[@AceologyUltra]

How about 16 emeralds for 4 planks? Or 29 emeralds for a f***ing rotten potato?

[@pikcube]

2:57 Can confirm, I was assisting a school district with dividing students into cohorts for reduced capacity classrooms, and I used this problem to build my solution

[@ferb1131]

But this doesn't do anything to provide a solution does it? It proves that a solution exists, but if it does anything to help find one that wasn't made clear in the video. Can this same logic provide a solution as well as merely proving its existence?

[@tongshen901]

@@ferb1131 Yes please!

[@AliceYobby]

⁠@@ferb1131yes, if you just do the 3d (or whatever-d) mapping and then find the points that intersect when taking the sphere 1 dimension lower. Knowing how to graph this out with a computer makes it trivial but there are ways to do all that with equations as well.

[@AliceYobby]

@@ferb1131 for the video, it proved that *every* case has a solution, but it used a single specific example. the parameters that change aren’t just how many variables (jewels) and divisions, but also the number of each jewel (the fractions), which determine how the mapping is done and where the points intersect.

[@badgermcbadger1968]

​@@AliceYobbythis only proves a mapping exists, not what it is. Finding the mapping is the hard part

[@ehtikhet]

This channel is sooo wonderful, the “poetry and literature” made accessible to those of us who struggle with the “grammar”!

[@ts4gv]

good analogy 👍

[@benjones6273]

I love how you took advantage of the symmetry between the two recipients of the jewels and related it to that between the positive and negative square roots. Absolutely fascinating!

[@ool647]

Grant, you should really do a ‘essance of topology’ series. It would be perfect for it’s a complicated topic, really hard to visualize 🙂🙂Like to make grant see this comment!

[@elliotwilliams7523]

I have been asking for this too YESSSS!!!!!!

[@windowslogo3577]

agreed. This is one of things you just can't find explained normally on the internet. Even though it is elegant and beautiful as was(and will be) demonstrated.

[@BigIndia942]

Yes please

[@BatterflyHigh]

I would love this!

[@safakhan1375]

I'm an architect and I would sploosh so hard

[@ckannan90]

Please continue making shorts. I’ve been following you for years, but the shorts always introduce me to older videos I’ve missed

[@3blue1brown]

Good to hear! I was a bit worried it may be bothersome to bombard people with excerpts of old content.

[@izzykaplan9961]

This channel has to be the best that I have seen. I have watched virtually all the videos on it and it manages to explain many concepts either not taught in high-school or not taught nearly as well. I was first introduced to this channel in the summer and have only just finally watched everything on it. I'll miss binge-watching after school, but I'll still be watching every new video soon as I can. The proofs in this channel have provided a new way of looking at things, and the series on things like Calculus and Linear Aldabra demystified them and made them understandable. The series on Neural Networks contained enough information (after watching like 2-3 times) to program a Neural Network for reading handwritten digits, and it's many other series gave me the fundamentals needed to get a heads-up on Calculus and Linear Algabra. Thanks @3Blue1Brown for creating this amazing channel and keep up the good work!

[@Realsheepsoft]

"You're probably a mathematician at heart" Thanks for the vote of confidence but I have my doubts lol

[@aryanbhatia6992]

i would have liked this but it would have distorted the equilibrium of this world

[@GhostGlitch.]

I mean your on a 3b1b vid sooo

[@ethannguyen2754]

@@GhostGlitch. *you’re

[@GhostGlitch.]

@@ethannguyen2754 you understood my meaning, so why exactly does it matter?

[@GhostGlitch.]

@@ethannguyen2754 also "sooo" isn't a word and I didn't use a period, you going to point them out too?

[@JustinMasayda]

Can you imagine if we had teachers in many other disciplines just as excellent at decomposing inaccessible material as he is? What a much more curious world we would live in. I think the ability to clearly animate each component of complex concepts is what makes this channel so effective. We need more skilled teachers who can animate, as visualization is such a powerful method to facilitate learning.

[@therandomthoughtsofaninsig5492]

I find the Feynman Lectures to be the 3blue1brown equivalent in physics.

[@ClancyXanecrest]

3:00 "trying to minimize sharting" Generally a good idea

[@xyzct]

Lol. (He actually said "sharding," but your version is funnier.)

[@cristianeering]

god i love the internet

[@fernbear3950]

@@cristianeering I don't.

[@DOROnoDORO]

@@fernbear3950 "I don't" -🤓

[@cxpKSip]

​@@DOROnoDOROWell, it gave us the cheese grater image...

[@bikinibottom2100]

3b1b thought me nothing is too difficult to grasp. Every mathematical concept, even the most subtle and abstract ones, are fundamentally intuitive. Not easy, but definitely intuitive. That information is priceless.

[@danelyn.1374]

I'm ngl most times I see a 3b1b video my brain feels huge, but not because it actually is, just because I can actually understand the usually complex topic that's given in a really amazingly well defined way. I remember struggling w/ so many things in school just because the simplest problems weren't explained well, and it's actually insane to see how well the combination of visual animations and expertly crafted explanations can make so many complex topics seem palpable. I love this channel lmao

[@minecraftmovieman1]

The genius of the presentation of this video allows me to be so engaged as presenting the fact that seemingly unrelated ideas will lead towards one solution actually gets the mind thinking about how such things could come together and it feels so much like I am finding the solution for myself in my head.

[@3blue1brown]

By the way, Brady Haran recently started a numberphile podcast. I had the honor of being its first guest, and I'm looking forward to listening to some of the mathematicians he has lined up here. Go take a look! www.bradyharanblog.com/blog/the-numberphile-podcast

[@alexgabriel5877]

Probability series waiting room :) is it coming?

[@egilsandnes9637]

Absolutely recomend listening to the podcast. It went really fluently.

[@gabrielfair724]

Thank you for revisiting this. I understood more this time

[@billrussell3955]

Very cool.

[@billrussell3955]

I'm considering n+1. N=5. It's combinatorics and topology. 1,2,3,4,5,6...

[@soshiasamiei]

You are the awesomeness in visualizing math. Now I understand why they give a Radio frequency pulse wave to the Hydrogen atom in MRI modality, such that the flipping of the function from a 90 to 180 gives us an echo signal, which is the equivalent of the signal that the proton gives when 90 degrees excitation on the transversal plane. Nobody has ever explained it as u did from topological point of view. Amazing job.

[@FrankBria]

I love topology! Dive into algebraic topology and things get even more awesome! My favorite version of Borsuk-Ulam: "you can't comb the hair on a billiard ball." It involves the ability to create a non-vanishing vector field on the sphere if no antipodal points are the same. (Basically, that g vector function never vanishes and can be used to create a tangent vector field.)

[@Ema-ih1cl]

If I'm gonna start studying physics it's mainly because I started to understand a lot of things with your videos. It all started with your Calc I series, which blew my mind and let me to believe that this is something that I CAN understand, and it MAKES TOTAL SENSE. From that point YOU CAN NOT SIMPLY JUST IGNORE THE EXISTENCE OF MATH IN THE WORLD, EVERYWHERE. And this is all because had the patience to make every single of these videos. You are by definition the best explainer I've encountered, and you gave me a reason to keep existing over here. Thanks again man

[@ChaitanyaDamu]

Lovely video. I love the way you bring soul to math (I'm and engineer, so I find it difficult to follow the books written by mathematicians for other mathematicians and at some point I just give up). I watched it a couple of times in past week, trying to understand each segment separately and today I pieced everything together, and I completely agree that this is indeed a beautiful piece of math. Nice work, keep it up!

[@Are3.14]

I think that if you're the " math friend " and people around you do not understand how can you like math, this is a perfect problem to show them. One of the things I like the most in math is how two ( or more ) seemingly completely unrelated problems can somehow have a useful connection between them, and I think that property of math could amaze pretty much anybody.

[@amardexter9966]

"Lets color each segment of line instead of jewels". me colorblind: wait what?

[@nataliawrozek]

I love when things translate onto others so gracefully. I'm amazed, thank you Grant

[@mitchkovacs1396]

Just finished the new vid, this is definitely an improvement! Understanding this one felt effortless :)

[@ramonafrombarcelona]

thinking midway through about the fact that both g and n are even, paused to think about an example of an even function (cosine). And suddenly, you mention that the path is a 180° rotation of an open path that's continuous where both halves' endpoints meet, and then my mind was blown... and there's the necklace problem atop of it. math is simply beautiful, and never ceases to amaze me.

[@macronencer]

I remember the original of this blew my mind. Not sure exactly what changes you've made, but all I can say is that it's still utterly beautiful.

[@singerofsongss]

This made me understand why topology is a part of math at all. To say it blew my mind would be an understatement.

[@VivekSingh-zl8ke]

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing-one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing. ... They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That's what mathematics is to me.” ― Paul R. Halmos

[@youngjin8300]

interlinked.

[@totaltotalmonkey]

Shame that it has to be inconsistent. en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems

[@ganondorfchampin]

+totaltotalmonkey That's not what it's saying...

[@JanischMaximilian]

totaltotalmonkey Gödel‘s incompleteness Theorem cleary *does not state* that maths is inconsistent, but rather that (quoting from your article) no consistent system of axioms whose Theorems can be listed by an effective procedure is capable of proving all truths about the arithmetic of the natural numbers. So it is rather *incomplete.*

[@ichdich2332]

I'm always amazed by the incredebly high quality and how he can explain it in such a way that even I undestand the basic Idea, as someone whose math-skills could be described as squareroot -1. Imaginary.

[@borekworek69]

Love that intro! It's so satisfying to watch. 0:27 for instant replay

[@luisgomes1814]

My favorite KZbin channel. Always feel enlightened after every video. This guy is simply amazing and probably sets the benchmark of how math needs to be taught.

[@VitalSine]

I think this is my favorite 3blue1brown video yet! It's such a beautiful proof! Who knew higher dimensional spheres could be practical?

[@123sendodo4]

I remember how I stop watching when you said about the temp and pressure on the globe, thinking how impossible that could be Now I watch the video a year later and finally understood it. Thank you!

[@qaz123amangupta]

I am from India Cant Thank You enough for making these videos, i could never learn in class because they do not show the actual Spiral of mechanics that goes around , the original idea of how the problem was first formed and how things are connected. teachers never understood what i was talking about but finally i can see now in your videos everything clearly

[@fatsquirrel75]

No wonder the teacher's couldn't picture what you were trying to say if you yourself weren't able to see it clearly until now.

[@chiranjitray760]

Wow just wow...I haven't studied topology, but still I get the basics and the way you have correlated is non intuitive and so such awesome

[@egilsandnes9637]

For a brief moment I thought it was a remake of my favorite 3B1B video, the one about using topolgy to prove that you always can inscribe a rectangle in a loop. 3B1B is the best popular math channel there is!

[@RaviSingh-qd5pz]

I just got to know about topology and was very intrigued by this topic but did not find a beginner's video about this. Thank you man for making this video.

[@KasranFox]

What is a sphere? A miserable little pile of coordinates of equal metric. But enough talk!

[@Irondragon1945]

HA

[@Zosso-1618]

Thank you for remaking this. I had a hard time following the original and even though this version is shorter, it feels so much less rushed. Now I understand this problem completely. Thank you Grant!

[@algorythmis4805]

Math is deep 42 This, my friends, is the day when peak awakening was reached.

[@moetard9581]

Wok af

[@atallguynh]

Math = 42... How am I just now hearing about this?!?

[@error.418]

@Toby M Sucks that the UK, the origin of The Hitchhiker's Guide to the Galaxy and the mythos of 42, uses the word MATHS which is 61 instead of MATH which is 42... so clearly the UK should switch to the word MATH instead of MATHS. QED.

[@moadot720]

1. I was going to say that, but I didn't feel like it... 2. OMG AWAKENING IS ONE OF MY FAVORITE WORDS EVER...!!!!

[@fouadnaderi1806]

You are really making math easy to understand. Excellent job. Thank you

[@OverQuantum]

2:30 - you do not need 2nd cut (from the left), 1st sapphire could go down, 2nd and 3rd - up

[@wild_lee_coyote]

Another way to think about it is how taking the square root of a number always has a positive and negative value. Square root of 4 is +2 AND -2. So the antipodal of any point on a n dimensions sphere is always going to be its exact opposite signed coordinate. This is because it is the only value that the square of that value is exactly the same, without the position being the same. +n^2 = -n^2 at the same time as -n is not equal to +n. Very nicely done

[@vtron9832]

Topology is one of my favorite maths, the idea of surfaces changing against the laws of physics and making new mathematical properties with it, it's awesome! Also, I think that another way of combining these two piece of math is to close the necklace into a circle, and finding a way to flatten it, so that both segments have the same number of jewels

[@meghanto]

No, that probably wouldn't work but you're encouraged to dry run it

[@mysteryman7877]

Meghanto you can fold the flattened necklace, so there is something here

[@poisonoushallucinations3168]

Would flattening the necklace still work for three or more jewels?

[@vtron9832]

Poisonous Hallucinations perhaps not, but there is still a connection

[@vtron9832]

Poisonous Hallucinations perhaps not, but there is still a connection for two jewel types

[@filiphes2991]

Wow! this fact about temperature and pressure absolutely blew my mind

[@aliensconfirmed3498]

Same theorem applied to 1D case says that if you travel between 2 points A and B for a day and travel the same path from B back to A on the next day then there is a point on the path which you reached at the same time of the day on both days. Not as interesting as temperature thing but nice way to look at the concept.

[@calebdunham1789]

I'm blown away by how beautiful that proof is! You've given me something to take to Thanksgiving to dazzle my family with! All credit will be given of course, but more people need to be aware of how incredible math is!

[@longdonsilver8149]

Thanks Borsaks, Ulams and 3blue1brown!! S-phere sphere SPHERE sounds great when I’m a little giddy!

[@QueenFondue]

Okay, now do three thiefs!

[@alexbrodbelt297]

I also have to thank this channel for inspiring me to pursue mathematics as a career. I am sure this is the best choice I could have ever made. Currently I am exploring Galois Theory and might even use this opportunity to make a video of this style to help me and others see the beauty of Galois Theory, after all teaching content like this properly feels like teaching how to paint like Van Gogh or to compose like Bach. Thank you Grant, you are a great inspiration to me. Hopefully one day I can help you make mathematics accessible to everyone and more importantly recognise the story-like elements maths has!

[@akshaysachan6029]

I smiled when he said "You and your friends want to split the booty evenly". Great video btw

[@RyanTosh]

Sharing is caring ( ͡° ͜ʖ ͡°)

[@stylis666]

I'm not a mathematician at all. I do love these videos. They make sense, even to me. And even though I don't use the maths presented and the lack of practice makes it impossible for me to reproduce any of it or explain it to someone else, it does seem to get easier to understand the mathematical problems I didn't understand before. Not too strange though. It seems that after 20 years I remembered most of the harmony studies I never used, so it makes sense that some of the maths should have made its way into my memory as well. But that's only part of my point. I may not practice with the maths I learn, but I do practice a lot with abstract ideas like those in maths or like this one, that shows a simple to understand similarity between the antipodal points and how you can divide the pearl necklace. It's simple things like that that I love about abstract puzzles. I also learned that necklace isn't spelled neckless and that there is a good reason for that XD

[@sophieward7225]

Every day you post is like a surprise Christmas

[@henryg.8762]

Yeah. Bewarb of those fake math channels. They're no good.

[@mischiefmanaged1045]

Grant, this is seriously one of my favorite videos ever. The feeling I get when I see the connection... Wow.

[@Lorenzo23910]

Please make an “essence of algebraic geometry”!!! You are the hope of mathematics education!

[@xyzct]

It would be helpful for those who can't tell their a$$ from two holes in the ground.

Absolutely brilliant. Yes, I do remember your previous video on the problem, and this new version is just as fascinating. The proof feels genuinely correct.

[@gabrieleciccarello4876]

You'll never stop to surprise me. This is wonderful. Your amazing work is like fuel for the flame of my curiosity. Your videos make me love math even more. It's amazing what math modelling can do. More beautiful than a piece of art.

[@tomasroque3338]

You always remind me of why I love math, which is why I love your channel. Well, I'll have to deal with it pretty regularly if I'm going to study theoretical physics in college.

[@paintingjo6842]

After watching this, I legit ran to my parents screaming "IT'S ALL CONNECTED"

[@hierkonnteihrewerbungstehe5636]

What was their reaction?

[@jannegrey]

I marvel at how late 19th, early 20th Century Polish school of mathematics (Like Sierpiński, Banach etc.) was so good that it was still cutting edge in 1950, and still very relevant today. I usually don't revel in my own countrymen achievements that much, but the best guide for calculus, and the best guide to complex numbers are short books by one of less known Mathematicians that I found in repository (they are free BTW, although in Polish), and while there are some, historical differences to modern math, I went from not understanding Calculus at all to complex differentiation in matters of 2 hours. Thank You for revamping your video, and adding a little bit of new- I was going to say that Mathologer also did a great video about it, but of course you guys know each other, so You've put link into description. Although You missed one opportunity that people asked you for in former version of this one - to make it also available to color-blind people :( but I know it would be a lot to make new shapes etc.

[@manshal467]

I am a simple man I see 3blue1brown's video... I click

[@KnakuanaRka]

Manshal Chawre If I had a nickel for every time I heard that comment, I could retire. If I read all those comments, I would gain absolutely nothing. You don’t need to post just to hear yourself talk; do you have anything to actually add to the discussion?

[@techieswew]

@@KnakuanaRka There is no discussion and there is no need for condescending intellectuals like you to waste your time in futility trying to clean up the festering fecal stain that is the KZbin comments' section. If people like the post, it will be more likely to be shown up at top.

[@smivan.]

I've seen the previous version of this video before, but man it's still fantastic to watch.

[@hiqwertyhi]

can we just take a minute to appreciate the beautiful music at the end though? this vincent rubinetti guy knows what's up edit: just listened to some of the 3b1b album, it's really nice. kinda got a bit of classical meets steve reich meets old school runescape music vibe going on

[@mykevelli]

Very clever. I just love seeing how "complicated" math can actually be so relatable. People think of mathematicians as being strictly analytical but you have to be so creative to think of ways to reframe your problems. It's always a fun journey when you take us down that line of thought. It was great to see you at ThinkerCon, by the way. Safe travels back home!

[@RecursiveTriforce]

This video was first called: "Who (else) cares about topology? Stolen Necklace Problem"

[@TheLuckySpades]

No wonder I got confused when looking for it again

[@sivetcr]

finally, i proved a single thing to myself before watching a proof (the opposit point thing)... i'm so proud of myself for finally not absoultely sucking at everything 🎉

[@uzKantHarrison]

It's weird that until university I had no interest at all in this kind of topics and I enjoy them so much now. If my high school teachers were like you, I would be probably studying mathematics instead of computer science now. But CS seems an appropriate choice anyway

[@ReallyLee]

Starting about 12 years ago, I bought "Formal Knot Theory" by Louis Kaufman. I will watch your topology videos and see if Kaufman's very simple knots and linear equations is explored by you. The author points out that some obvious theorems and conjectures of the knot labeling and enumeration activity were not rigorously proven. Heady stuff to come out of simple pencil lines, labels and finger counting linear equations on paper. Thank you for your wonderful videos, I have a mostly liberal arts background with a library of college bookstore and thrift store math books. I take delight in your presentation of beautiful ideas and your flexible mental good humor. Right now I am doing problems from Hal Varian's Intermediate Microeconomics. I am revisiting his section on consumer transportation choices. I am not trying to pass a college course, I want to jack-hammer away at the macroeconomic framework and assumptions that continue to drive our society to burn fossil fuels.

[@aero-mk9ld]

Bro is a yapper bro nobody cares 😂😂😂

[@MeriaDuck]

0:26 Math is deep -> I would love a T-shirt with that!

[@potyka1730]

My friend has a T-shirt saying MATH: Mental Abuse To Humans

[@pauselab5569]

for the specific 2 types of jewls problem, you can find a more self contained proof but it doesn't generalize well. note that making any 2 cuts create 3 complementary subsets of the original set of say order n. the 3 subsets need to be seperated to 2 people, and so it has to be 2 to one of them and 1 to the other and so one subset must have order n/2. now think about a string of 0 and 1s that has n digits. start by looking at the first n/2 terms and note how many 0 and 1s there are. if that isn't a solution, move the whole subset to the right by 1 digit. repeating this will make sure that there is a way to seperate the necklace to 2 people.

[@Rohith_E]

To get a better understanding of just Borsuk Ulam Theorem watch Vsauce video on Fixed Points.

[@dankazmarek1259]

that logic was little weird.. Grant's line of logic was clear and capturable

[@helloimnisha]

These videos make me rethink about my changing my majors from Phy to Math❤️

[@nejlaakyuz4025]

Nish K Double major?

[@0xEmmy]

When you make videos in the future, could you please check them for colorblind accessibility? Everything involving the necklace (discrete and continuous) becomes just about invisible in monochrome.

[@user-vw4xp5nt9f]

if it helps, there's not much to miss on the string section. it's hard to see even with the colors

[@Pwassoncru]

@@user-vw4xp5nt9f you may want to get checked for color vision.

[@nikkiofthevalley]

Uh, being colorblind doesn't mean you literally cannot see colors. There is a type of colorblind that does mean that, but it's by far the rarest.

[@GopherpilledTunneler]

I thought of an interesting way to think about the 2-jewel problem. Make the jewels into colored segments, then connect the ends of the string together to make a circle. Then pick a color and find the centroid of all the segments of that color (the circle excluding the segments of the other color). This gives you a point where all of the lines that go through it split the color 50%. Then make a line that goes through the center of the circle and the centroid of one of the colors. This splits the circle 50% and also the proportions of the gems are equal on both sides, so you can mark where the line cuts the circle and undo the string connection. This gives 2 cuts at most for the 2-jewel problem. Maybe this can be generalized.

[@borisdorofeev5602]

Hey Grant, I started following your twitter recently. I saw that you are well acquainted with Ben Eater who is also one of my favorite youtubers. It's really people like you who give me the motivation and curiosity to keep learning. The way you guys present these topics makes them so interesting that I have to try and emulate it. I've watched and rewatched all your videos and will continue to do so. Thanks again.

[@3blue1brown]

Thanks! Ben is great. Anyone who doesn't know his content needs to pop over there right now.

[@christiankohnle6542]

This is the most beautiful piece of math I’ve seen in a long time, good lord

[@odorlessflavorless]

Please upload the EE paper link again. The present MIT link is broken. Amazing explanations as always :)

[@Lucashallal]

That proof for the theorem reminded me of numberphile’s proof of the fundamental theorem of algebra

[@xjdusuau9851]

ALON AMIT INSPIRED 3B1B!!!! My life is hence complete I shall now die in peace

[@TheCarlagas]

Sounds like the biggest crossover in history

[@tesset8828]

Wrong Alon, you're thinking of Noga Alon the one also responsible for combinatorial nullstellensatz. I know this comment is old, but had to include this.

[@xjdusuau9851]

@@tesset8828 umm I'm sorry but who??

[@NoriMori1992]

Tes Set Or, he's talking about Alon Amit, and not Noga Alon.

[@xjdusuau9851]

@@NoriMori1992 correct

[@giovanigarcia5029]

I think it is the most beautiful math I've ever seen!

[@jasertio]

I think he could have also used a two dimensional analogue of mapping a circumference to a line for a simpler visualization of the theorem. It is a lot easier to intuitively understand the mapping of two circumference points to a single point in a line, than to understand the mapping of points of a sphere to a plane.

[@totaltotalmonkey]

You could only be sharing one type of jewel then.

[@jasertio]

@@totaltotalmonkey what do you mean?

[@totaltotalmonkey]

In the case of mapping a 3d sphere to a 2d plane there are two cuts, that allows two types of jewel to be shared equally, see 15:15. In the case of mapping a 2d circle to a line there is only one cut - only one type of jewel can be shared equally. To share three types of jewel you need to map a 4d sphere into a 3d space. You need an extra dimension for each additional jewel type, as n jewel types require a minimum of n cuts, see 2:23.

[@ConnorMooneyhan1]

Oh my goodness, I am in awe. Understood it much better this time around. Excellent job, Grant, this is among your best work.

[@aneeshukidve]

I was absolutely smiling like an idiot when you showed the proof

[@MrQwint22]

Beautiful math feels like a headache mixed with child-like wonder.

[@zombiesalad2722]

5:21, Vsauce flashbacks

You blowed my mind. I was thinking I am engineer, develooper and math lover. Please don't stop videos.

[@legoguy217]

The link to the EE Paper appears to be broken. Edit: He fixed it!

[@TheTechovision]

As a physics major I'm always blown away by math. We barely scratch the surface in most physics undergrad studies. I love these videos because of the approach. You'd make a fine professor!

[@tatjanagobold2810]

Everytime a new 3Blue1Brown video comes out I almost get a heart attack because I am so excited to be educated!😍😂 If only school was like this haha 😂

[@AquaWet]

t. gobold Do you like 1+

[@baganatube]

If public schools were like this, the society must have become totally different. Just imagine smart and well educated people everywhere you look.

[@henryg.8762]

It is for me!

[@Saltyarticles]

As an structural engineer, your topics are always interesting. I would love to hear your views on finite element methods and matrix analysis and geometry. Keep the videos coming.

[@xjdusuau9851]

So, this is the same video but different?!

[@conoroneill8067]

The proof of the Borsak-Ulam theorem is entirely different. Most of the rest is similar, though. ...Is it weird that I remember what he did last time from memory?

[@alonamaloh]

@@conoroneill8067 I remember as well! This new proof is more elegant, but there is the detail of making sure the wrapping number around the origin is not 0. That is very intuitive, but it's not immediately obvious how you would prove it. In the specific case of a symmetric path in 2D I can use the angle from the origin to finish the proof, but I'm not sure how to generalize this to higher dimensions.

[@columbus8myhw]

In fact, the winding number can be _any_ odd number (but, crucially, not zero).

[@DoReMeDesign]

it's too beautiful for me to handle. This may have convinced me to take maths at uni, as opposed to physics or engineering. This proof also gives some glimpse into the proof for Fermat's last theorem, I guess, as though the topology and relations are completely different, the Taniyama-Shimura conjecture relates a topological (?), higher dimension concept, modular forms, with some pure number theory concept (elliptic curves).

[@ivarangquist9184]

12:30 That line is very thin and the colors are very similar. Unbelievably great video, anyways!

[@TheLuckySpades]

I just listened to your podcast with Brady and hope you read this, even if you don't reply I've always loved math, I've always been fascinated by it and I live proofs and your videos helped me further that fascination and the desire for more. Even without this channel I would have ended up studying math like I do now, but your animations are one way for me to share my enthusiasm with people outside of that. Thank you for making this fantastic channel and making this content, you are great. (P.S. I completely agree with your statement that gruntwork can be enjoyable, for me that's usually proving smaller facts, or calculations, but it is fun in it's own way)

[@quahntasy]

I am a simple man. I see 3Blue1Brown video. I click even though I don't understand lol.

[@readjordan2257]

Grant: ah you thought the necklace and my topology were unrelated, but tying them together after you cut them apart was super easy, barely an inconvenience.

[@chasemarangu]

subtitles at 2:37

[@keyyyla]

Please make a series about differential geometry! There is nothing more needed than such a subject supported by your animations!!

[@johnchessant3012]

9:54 Vsauce

[@muhammadnourhereh4581]

@Alex DO that soundtrack is playing in my head now

[@albertvila5364]

After watching it several times during the last months/years, i can say that's absolutely insane.