Using topology for discrete problems | The Borsuk-Ulam theorem and stolen necklaces
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- čas přidán 6. 06. 2024
- Solving a discrete math puzzle using topology
I was originally inspired to cover this thanks to a Quora post by Alon Amit
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The original 1986 by Alon and West with this proof
m.tau.ac.il/~nogaa/PDFS/Publi...
VSauce on fixed points
• Fixed Points
EE Paper using ideas related to this puzzle
dl.acm.org/citation.cfm?id=80...
I first came across this paper thanks to Alon Amit's answer on this Quora post
www.quora.com/As-of-2016-what...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
Music by Vincent Rubinetti:
vincerubinetti.bandcamp.com/a...
Time stamps:
0:00 - Introduction
0:36 - The stolen necklace problem
3:08 - The Borsuk Ulam theorem
9:15 - The continuous necklace problem
13:19 - The connection
17:30 - Higher dimensions
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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
Then they bought 523425 watermelons.
Won't it be an 18-dimensional sphere? I liked the joke, tho.
That happens if all of your friends are mathematicians ;)
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.
@@SlackwareNVM It should be 17-dimensional because sphere is the surface of the ball.
3:56 don't cut or tear the sphere
*FLASHBACK TO HOW TO TURN A SPHERE INSIDE OUT*
Yup! That's the general rule in topology: everything has to be continuous.
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
You mus'nt tear or crease it.
@@DemonixTB I see I'm not alone
10 emeralds!? I know a villager that would give me like two wheat for that
I'd like some sheep please
@@wierdalien1 they are 9 bucks 😕
@@parallellinesmeetatinfinity have you not played Settlers of Catan
nah he’d give me a stick
How about 16 emeralds for 4 planks? Or 29 emeralds for a f***ing rotten potato?
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!
Thanks for putting in the time!
3Blue1Brown i was lost in this video sensei :(
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.
"nearly an hour"
I'm a math major and it would probably take me several DAYS to understand this video.
So it's time to learn to evaluate and steal necklaces
"You're probably a mathematician at heart" Thanks for the vote of confidence but I have my doubts lol
i would have liked this but it would have distorted the equilibrium of this world
I mean your on a 3b1b vid sooo
@@GhostGlitch. *you’re
@@ethannguyen2754 you understood my meaning, so why exactly does it matter?
@@ethannguyen2754 also "sooo" isn't a word and I didn't use a period, you going to point them out too?
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
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?
@@ferb1131 Yes please!
@@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.
@@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.
@@AliceYobbythis only proves a mapping exists, not what it is. Finding the mapping is the hard part
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
Probability series waiting room :) is it coming?
Absolutely recomend listening to the podcast. It went really fluently.
Thank you for revisiting this. I understood more this time
Very cool.
I'm considering n+1. N=5. It's combinatorics and topology. 1,2,3,4,5,6...
This channel is sooo wonderful, the “poetry and literature” made accessible to those of us who struggle with the “grammar”!
good analogy 👍
3:00 "trying to minimize sharting"
Generally a good idea
Lol.
(He actually said "sharding," but your version is funnier.)
god i love the internet
@@cristianeering I don't.
@@fernbear3950 "I don't" -🤓
"Lets color each segment of line instead of jewels".
me colorblind: wait what?
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!
I have been asking for this too YESSSS!!!!!!
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.
Yes please
I would love this!
I'm an architect and I would sploosh so hard
Please continue making shorts. I’ve been following you for years, but the shorts always introduce me to older videos I’ve missed
Good to hear! I was a bit worried it may be bothersome to bombard people with excerpts of old content.
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!
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!
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.
I find the Feynman Lectures to be the 3blue1brown equivalent in physics.
My favorite CZcams channel. Always feel enlightened after every video. This guy is simply amazing and probably sets the benchmark of how math needs to be taught.
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
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!
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!
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!
I think this is my favorite 3blue1brown video yet! It's such a beautiful proof! Who knew higher dimensional spheres could be practical?
I smiled when he said "You and your friends want to split the booty evenly".
Great video btw
Sharing is caring ( ͡° ͜ʖ ͡°)
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.
Grant, this is seriously one of my favorite videos ever. The feeling I get when I see the connection... Wow.
Just finished the new vid, this is definitely an improvement! Understanding this one felt effortless :)
Every day you post is like a surprise Christmas
Yeah. Bewarb of those fake math channels. They're no good.
Thank you for changing the title and tricking me into watching this again. Couldn't leave more satisfied.
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.
What is a sphere? A miserable little pile of coordinates of equal metric. But enough talk!
HA
I love when things translate onto others so gracefully. I'm amazed, thank you Grant
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.
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.)
After watching this, I legit ran to my parents screaming "IT'S ALL CONNECTED"
What was their reaction?
Oh my goodness, I am in awe. Understood it much better this time around. Excellent job, Grant, this is among your best work.
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!
I am a simple man
I see 3blue1brown's video...
I click
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?
@@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 CZcams comments' section. If people like the post, it will be more likely to be shown up at top.
Math is deep
42
This, my friends, is the day when peak awakening was reached.
Wok af
Math = 42... How am I just now hearing about this?!?
@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.
1. I was going to say that, but I didn't feel like it...
2. OMG AWAKENING IS ONE OF MY FAVORITE WORDS EVER...!!!!
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.
Absolutely loved the video! Also greatly liked the title, not to complicated, but also not clickbait. I wanted to click the like button multiple times but unfortunately CZcams does not allow me to super like your video. Keep up the amazing work of explaining complex interesting ideas in steps that are followable even when you do not have a background in mathematics. I love you!
This video was first called:
"Who (else) cares about topology? Stolen Necklace Problem"
No wonder I got confused when looking for it again
Love that intro! It's so satisfying to watch. 0:27 for instant replay
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.
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.
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
Please make an “essence of algebraic geometry”!!! You are the hope of mathematics education!
It would be helpful for those who can't tell their a$$ from two holes in the ground.
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.
Once again, you have managed to make my jaw drop. Well done once more.
This might well be one of my favourite channels on CZcams.
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 😂
t. gobold Do you like 1+
If public schools were like this, the society must have become totally different. Just imagine smart and well educated people everywhere you look.
It is for me!
2:30 - you do not need 2nd cut (from the left), 1st sapphire could go down, 2nd and 3rd - up
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!
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.
[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
interlinked.
Shame that it has to be inconsistent. en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems
+totaltotalmonkey
That's not what it's saying...
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.*
Please upload the EE paper link again. The present MIT link is broken. Amazing explanations as always :)
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.
You are really making math easy to understand. Excellent job. Thank you
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.
Thanks! Ben is great. Anyone who doesn't know his content needs to pop over there right now.
0:26 Math is deep -> I would love a T-shirt with that!
My friend has a T-shirt saying MATH: Mental Abuse To Humans
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)
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!
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
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.
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
No, that probably wouldn't work but you're encouraged to dry run it
Meghanto you can fold the flattened necklace, so there is something here
Would flattening the necklace still work for three or more jewels?
Poisonous Hallucinations perhaps not, but there is still a connection
Poisonous Hallucinations perhaps not, but there is still a connection for two jewel types
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!
Absolutely astounding, fascinating! I loved the proof but of course, also the incredibly beautiful symmetry.
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
ALON AMIT INSPIRED 3B1B!!!!
My life is hence complete
I shall now die in peace
Sounds like the biggest crossover in history
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.
@@tesset8828 umm I'm sorry but who??
Tes Set
Or, he's talking about Alon Amit, and not Noga Alon.
@@NoriMori1992 correct
Good lerd your visualizations and explanations help so much in trying to understand abstract concepts and why they're worth pondering at all! 🥰
By far the best math channel!!! I love your videos
5:21, Vsauce flashbacks
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.
if it helps, there's not much to miss on the string section. it's hard to see even with the colors
@@user-vw4xp5nt9f you may want to get checked for color vision.
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.
Thanks Borsaks, Ulams and 3blue1brown!! S-phere sphere SPHERE sounds great when I’m a little giddy!
Beautifully crafted, as always!
Keep up the good work.
I was absolutely smiling like an idiot when you showed the proof
The link to the EE Paper appears to be broken. Edit: He fixed it!
I've seen the previous version of this video before, but man it's still fantastic to watch.
You blowed my mind. I was thinking I am engineer, develooper and math lover. Please don't stop videos.
To get a better understanding of just Borsuk Ulam Theorem watch Vsauce video on Fixed Points.
that logic was little weird.. Grant's line of logic was clear and capturable
I am a simple man.
I see 3Blue1Brown video.
I click even though I don't understand lol.
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
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.
9:54 Vsauce
@Alex DO that soundtrack is playing in my head now
YES
So many parts of this video gave me a smile, best channel ever.
This made me understand why topology is a part of math at all. To say it blew my mind would be an understatement.
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.
You could only be sharing one type of jewel then.
@@totaltotalmonkey what do you mean?
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.
subtitles at 2:37
I watch these videos to help me fall asleep because your voice is so relaxing, but I also feel bad when I fall asleep watching because I get really interested in the topic and end up rewatching it...
Love your videos in general though, more than just to help me sleep, lol.
Only you could explain this in a way that my brain could grasp, Grant. Great job, thanks. :-)
The link for the EE Paper doesn't works for me :c
Just changed the link. Does that work?
Yes, that one seems to be off. But, if you go to the Quora link directly below that one there is a copy of the necklace problem as a pdf that a commenter posted. In particular the commenter that apparantly inspired this video.
You should see the paper as a link in his response. Its a very interesting read and pretty brief, only about 2-3 pages. Enjoy.
@@3blue1brownYes it does. Thanks a lot !
12:30 That line is very thin and the colors are very similar.
Unbelievably great video, anyways!
Your videos have reignited my love for beautiful math. Thank you
I love your videos. I think at 8:35 you should have said "one of these paths" rather than "one of these points".
Please continue doing this - it's the best math channel on CZcams. I hope to be watching this with my kids at some point 😄
Thief returned back the necklace after watching this.😢
TOPOLOGY !!!
This is the most beautiful piece of math I’ve seen in a long time, good lord
Hi Grant. Thank you as always. Your videos are so inspiring...
So, this is the same video but different?!
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?
@@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.
In fact, the winding number can be _any_ odd number (but, crucially, not zero).
TOPOLOGY!!!!!!!!!!!!!!
This is definitely one of your best videos yet, Grant :D
Thank you. It really was beautiful math. Thank you for the EE article link as well.
Is it as sneaky as me, though? Nobody saw me coming
The doctor did........
I was expecting your name to be "The Spanish Inquisition" and now I'm disappointed
@@jemesmemes9026 That wasn't even the sneakiest thing they have done.