@@lo1bo2 Mine keeps rolling into the corners of the room, where the Hounds of Tindalos get ahold of it, and when they've finished I have to repair it. Do you have any idea how difficult 7-dimensional sewing is?
There's nothing different about "dimension 4". It's indistinguishable from the other dimensions. But working in 4 dimensions is different from working in 3 dimension. Which 3 or 4 you use, makes no difference.
@@peterwhitey4992 did you watch the video? The Professor says "dimension 4" multiple times so the author of the comment did use a term that everybody in the comments section should be familiar with. If you arent familiar with it just ask: "What exactly do you mean with dimension 4?"
@@sihingvonfelix4251 Peter is clarifying a misconception one might have about what is called “dimension 4” or “the 4th dimension”. Attaching a special significance to one of the dimensions isn’t something you do when studying pure Euclidean spaces. In the video, he briefly mentioned “it could be time”, which would probably cause misconceptions. If you study 4D spacetime (3 space dimensions + 1 time dimension), that’s different from what topology is usually concerned with, and distance is defined differently. In pure Euclidean space, a 4D distance can be found with the Euclidean formula: sqrt(dx^2 + dy^2 + dz^2 + dw^2). In 4D space time, distance is defined as sqrt((c-dt)^2 - (dx^2 + dy^2 + dz^2)).
I didn't even think dimension 4 was strange, I thought the only reason it tends to get more attention than other higher dimensional spaces is because it happens to be the first dimensional space that we didn't evolve to comprehend intuitively. The fact that it stands out topologically from all others blows my mind.
@@nickpatella1525 But we are clearly talking about 4D spaces as a whole here, not some arbitrarily assigned "4th dimension" of any given space with N dimensions. It does seem that Peter hadn't watched the video.
Engineers use manifolds (in N-dimensional complex space) to analyze multiple-antenna communications systems, such as MIMO WiFi. The shape of the manifold is determined principally by the arrangement of the antennas in 3D real space. Each point on the manifold corresponds to a direction (bearing) in 3D real space. The local properties of the manifold about that point tell us how well the communications system can detect, resolve and estimate the parameters of a remote signal source emitting from that direction.
FYI, the disks starting around 12:00 should be filled, not empty. "Disk" means the inside of an n-sphere and "sphere" means the outside. That's why it's okay to contract them, no holes.
If TH-cam allowed us to save comments, that would be on my list. It's times like this I'm reminded that some of the stuff we write online _will_ be read thousands of years in the future, if only by machines. Yikes.
No, only the 2d sphere inside of 3d space (as in that famous video), the 6d sphere inside of 7d space, and (if you want to count it) the 0d sphere (just two points) in 1d space
@ChannelZero Yes, there's a proof. You can see mathematicians agreeing about this in MathOverflow question 115110 "Eversion of the 6-sphere in 7-space". (This parallels other topology facts about spheres and dimensions related to the quaternions and octonions.) For the 0-sphere, it is all solutions of x^2=1, so the two points -1 and 1 on the number line. It's not a smooth object, but passing the points through eachother doesn't pinch it in any way, so it can topologically be turned inside out.
Very humble and knowledgeable professor, nice work trying to help us normal sphere people get a glimpse at those exotics... like a fancy car video for numberphiles
It always amazes me when the debate of the “shape” of the universe comes up. I mean, why does it HAVE to have a shape? It seems rational to believe that it is just never ending. Perhaps, infinity in every direction. Astonishing? Absolutely.
best part of the video: "number phile videos are important too" absolutely agree we need to propagate maths to the whole world.. more people working on maths more probability to solve problems
I want to learn about the different spheres in different dimensions. I'm all about the specialness that is 4D space, but I'm curious about what exactly are the 992 spheres in the 11th dimension
I'd love to see a video on division by invariant multiplication. It's incredibly important in modern computing, but almost unknown to most programmers let alone non-programmers.
well, according to Einstein, if you let go of a doughnut it is the coffee cup which accelerates towards the doughnut, so the coffee cup does indeed dunk into the doughnut
@user-ho3ng2oq2y In Newtonian physics, gravity is a force, and the surface of Earth is not accelerating in general relativity, gravity is curved space-time, and it is more accurate to think of the grounds as accelerating up into you, pushing you off your inertial path (free fall)
Its really weird how many spheres in the numbers have big number gaps but i actually see a pattern in sphere dimensions 7,11 and 15. They all have a relation to mersenne primes. (2^3-1)x2^2=28 (2^5-1)x2^5=992 and (2^7-1)x2^7=16256. 28 doesnt hold for this pattern, but im guessing that the next one might be in the 27th dimension might be: (2^13-1)x2^13 = 67100672. Just an observation though
@@Alex_Deam I'm not saying that it's incorrect but lots of Numberphile videos have taught us that you can't assume a theory is correct just because the first n numbers match.
Prof Manolescu is awesome! He disproved the triangulation conjecture in high dimension. You should ask him if he could do a video in that direction. This video turned out great.
Hahaha, I completely agree! I've even built a 3D shadow of a shadow of a 5D hypercube, but I haven't begun to understand why there are more than 1 type of sphere in any number of dimensions. I did however like the explanation of why 4D topology is more complicated than topology in lower and higher dimensions.
@@evanglickstein8001 Damn, that's awesome. I have a friend doing a maths degree and he is interested in researching how to arrange spheres in the most optimal way possible in higher dimensions. This kind of thing just boggles my mind.
4-dimension being special and infinite makes me think of time-slicing the universe into 3d spaces. Pretty wild if the reason we live in 3+1 dimensions is that 4 dimensions holds more potential (is infinitely more likely to occur) than the others. Assuming, of course, that spatial 4d is comparable to 3+1d
Absolutely fascinating. I didn't like functional analysis in university (which kinda relates to topology) and I always thought, that there is nothing special about higher dimensions because you can't properly visualize their objects (I tried hard especially in 4D), but now.. I completely changed my mind
When the video said “Featuring Ciprian Manolescu”, I was delighted. Manolescu is pretty famous in math competition circles because he’s the only person ever to write three perfect papers at the IMO (International Math Olympiad).
This is one of those “unknown unknown” videos where I was like: I knew that I didn’t really understand topology… what I didn’t know was that I don’t really understand topology. Side note, I totally use hyper-spheres and bisectional searches to resolve a convex optimization problem “when have I collected enough data about subject X?”, so it’s not that I wouldn’t love to have a better method by exploiting the manifold or by loosening the requirements of the shape of that manifold… but, I’ve yet to find the way to ask the question in a way that brings me closer to a better implementation. So, great video. Love that numberphile will dive into subjects like this.
Dimension 4 being uniquely special like this feels like it could potentially have implications on why, if the higher dimensional models of e.g. string theories are correct, only three dimensions are extended spatially, the fourth is uniquely temporal, and higher dimensions are curled up and only manifest as phenomena within space over time.
@Mike Foster That ant-on-a-hose analogy is exactly what it meant by the higher dimensions being "curled up"; they're like the dimension around the hose, while the three we're used to are line the dimension along the length of the hose. We're so big compared to the curled-up dimensions that they basically don't exist to us, in the same way that we're so big relative to a hose that it seems almost like a one-dimensional object. But for tiny subatomic particles, the curled-up dimensions are big enough to give them room to do interesting things, which is a proposed explanation for various phenomena that we observe of those particles; what looks like a property of "charge" to us, of an apparently static particle in 3D space, is actually its velocity along a curled-up higher dimension, but since it's just looping around that dimension that's too small for us to see, we don't perceive it as "motion" but as some static property of a motionless particle.
Seriously, I can't even fathom most of this stuff. All those people arguing about 4 dimensional space when I got lost all the way back at "smoothly different."
There's an interesting show called "Dangerous Knowledge" about how some mathematicians/physicists/etc have lost their minds to math. Wrapping your head around infinity isn't as quick & easy as learning to tie your shoes. Pretty interesting watch, although it's a bit dated at this point. I'd like to think that being a mathematician helps to process the deep thoughts they have... it must be nice to have an outlet, a way to express the thoughts, a language (math) to convert ideas to, so that others can interpret/peer review/etc. Knowing how complex the world/universe is, is almost like a blessing and a curse. That said, I'd rather ask the big questions and drive myself a bit mad trying to figure it out, than to never ask them at all and just bumble around taking everything for granted for my short time here.
I've actually ran across a book series where people can use math to do magic, but they also have an increasingly High chance of going insane from doing so. There are ways to reduce the chances, like don't actually do the math all in your head, but the chances never zero.
It isn't really much of a mystery. You can map the 4th dimension, for example. You just need more than 4 dimensions to act as an overarching structure within which to map the lower dimension. Without 4 dimensions we could not map the lower 3. And so on. (Edit, the missing exotic sphere is the exception. Yes, it is considered a quandary and I think of it as a blind spot. To me it is more of an ontological mystery than a topological one.)
They routinely measure the curvature of the universe. So far, nobody has been able to measure any. It is at least flatter than we can currently measure. If it is totally flat, it is likely infinite in size. If there is a curvature, that would tell us the actual size. Based on the fact that we have not been able to measure any curvature, the actual size is at the very least enormous, WAY bigger than the 93 billion lightyears of the visible universe.
Send this to the Flat Earthers so they can realise that the Earth isn't flat, it's just a 2D manifold in 3D space and they're just too close to it to tell the difference between a plane and a sphere ;)
Your video on Ricci flow always confused me, because it never seemed to have an application. Now that I've heard a little about topology here, I can start to see why Ricci flow might be useful.
To draw a 4D sphere, use time as the 4th dimension. Animate it in other words. So you animate a point that grows into a ball then shrinks back to a point then disappears. It will not grow at an arbitrary rate, it will grow quickly at first then slowly, then shrink slowly then shrink quickly.
@@peterwhitey4992 I know he talks about the thesseract and the effect a 4d spatial ball when pass the 3d spatial universe. But as moves have t as 5 dimension, and the projection have t as 4 dimension, standard in our spacetime universe.
Peter whitey is correct. I meant use time as the 4th dimension. X,Y, Z, t. To do it, use art to draw something that looks 3D on a 2D surface, pretty simple. Then animate it. A hypersphere would be as I described. A hypercube would be blank, then suddenly a 1x1x1 cube for 1 unit of time, then suddenly disappear to be blank again.
In a seminar on word embeddings, we heard about a hyberpolic distance function that improved a specific type of classification problem. And I asked the question if the concept of sphere even makes sense in these extremely high dimensional spaces.
really interesting, three spacial and one time dimensions make up a 4 dimensional space, interesting that 4 is the one we have the most trouble saying things about
Interesting topic. I have run into this problem with 4 dimensions in the study of error correcting codes. Coding theory is related to exotic topology, since the number of dimensions affects how code symbols can be decoded. We know a lot about binary codes or trinary codes with symbols that have 2 states (0, 1) or 3 states (-1, 0, +1). We know the best possible error correcting codes for binary codes with length out to 256 bits, and in many useful cases much farther. Something similar is known for trinary codes. But not for quaternary codes. Finding codes for quaternary symbols is much less obvious. In most cases we simply reduce this to a pair of binary symbols, but that ignores the reality of many useful communications systems.
I find these higher dimension issues fascinating, although I don't understand much about them. Sometimes it seems whoever created the universe did so after drinking a few bottles of wine, and some imperfections became permanent
Assuming evidence-less things like someone created the universe, that it was created at all, that what you see as imperfections are such, tons of logical fallacies in your thinking.
i first dusted a computer in 1971, a univac 9300, i was a tape librarian, which meant i mowed the lawn and filled the coffee machine as well as dropping punch cards all over the floor. i got into computer graphics at uni though, in 1981 i guess (kingston poly) , and then i got poached by digital pictures and worked for them doing pop videos, and then i moved to cfx associates and i learned computer graphics and animation there, going on to freelance later for all the major visual effects houses in soho, working on tv titles and commercials and even doing some feature films and finally decided to retire after a stint at electronic arts. so i love shapes. and time. i have lots of computer graphics of all sorts on my channel.
If you consider the curve itself then yes, they are different. However, you can also consider a parametrization, in which case the crossover point is already counted twice before the deformation. You could say that the objects (figure 8 and circle) are topologically different, but the parametrizations are topologically the same.
"Dimension 4" is no different from "dimension 3", or any other dimension. 4 dimensional space or objects are different from 3 dimensional space or objects though.
@@peterwhitey4992 did you watch the video? The Professor says "dimension 4" multiple times so the author of the comment did use a term that everybody in the comments section should be familiar with. If you arent familiar with it just ask: "What exactly do you mean with dimension 4?"
We aren't "right beside it" it terms of being 3D. That's how we've evolved to experience the world; that's our perception. We exist in all available dimensions.
I feel like I want to intersect a 4d hyperspace with a 3d plane or a 3d object and show how there is another 4d hyperspace where you get something different. Oh, and the answer to "Would you be a famous mathematician if you solved this?" Is "I AM a famous mathematician."
I’ve been studying Quaternions and 4D mathematics, and hearing that the 4th dimension has properties of *infinite unknowables* is kinda mind blowing. It makes me wonder if math is at its perfect structure in the fourth dimension, and matter can only be in 3, so we’re stuck here witnessing it’s insanity.
It is not 4th dimension that is special. It is 4-dimenisonal space that is special. They don't single out some specific dimension in this video. They talk about spaces with different number of dimensions.
If you liked the video, I strongly recommend you play some 4D video games like 4D Toys or Tetraspace (aka. Brane) 🙂(alas, Miegakure is not finished yet...)
Isn't dimension 4 that special, since we ACTUALLY live in it: x, y, z and time (except that we can move to and from in the first 3, but not in time. There we can only go forward)
"...992, then 1, then 3, then 2, then 16,256, and so on..." I feel like this is one of those "find the next number in the series" questions but where I'm a donkey
Is this why we live in 4 dimensions? Perhaps time is something special that only appears in 4 dimensions used to separate our lower dimensions, or some other second law of therm - maximum entropy principle.
@@feynstein1004 No, topology is much, much broader than you have been led to believe. And anyway, the four dimensions are all "real" here, in the sense that every manifold is locally homeomorphic to R^n. Imaginary time is just a clever way of dealing with the Lorentzian metric.
@@feynstein1004 So, for the first part, topology is a very general field of study. It studies the properties of a space in terms of neighborhoods of its points, allowing ideas like connectedness, separability, density, genus, and more. Some topological spaces are associated with what we would think of as a surface or space in a sense. For instance, all spheres share the same topology (essentially). That's the sense in which you can call a coffee cup the same as a donut, because one can be continuously transformed into the other. But most topological spaces are not like that at all. They can encode other abstract ideas like topological graphs or honestly just about anything given the appropriate structure. The four-dimensional space I am talking about is spacetime, the combination of space and time you see in Einstein's theories of Relativity. You mentioned that time is "imaginary," and therefore it's not the same as a real space. That's sort of true; the spacetime in Special Relativity (Minkowski space) has a different way of defining distance than a Euclidean space that you are used to in geometry. Still though, that doesn't actually mean any one of the dimensions has to be imaginary (though treating it that way is a convenient mathematical trick). And importantly, even though the metric is different, the topologies here are the same. If Euclidean space is a donut, Minkowski space is a coffee cup. They look different, but they are topologically the same.
@@EebstertheGreat Hehe thanks for the explanation. I was familiar with the first one. It's the second one that I wanted a bit of clarification on. So, regarding that, what I meant was that our reality is different from a purely spatial 4D space in the same way the complex plane ( x & iy) is different from the real plane (x & y). Idk if this difference matters in topology but I think it does. You might have similar structures in both spaces, for eg you could just draw a circle/donut in both. However, those circles/donuts would end up having different properties and in a sense, mean different things. That's how I think it is anyway. But I'm not a mathematician. So I could be wrong.
Errmm. A coffee mug *doesn't* have a hole in it - it has a dip/depression/recess and you cannot transform a doughnut into a coffee mug that will hold coffee. A cylinder, yes. Coffee mug no. On second thoughts, I take it back - I forgot the coffee mug has a handle...
Two thoughts: 1. Dimensions beyond four always seemed incredibly silly to me until someone explained to me that this is basically what OkCupid or Bumble, etc., are doing. You answer x questions and your march answers y questions, and the algorithm needs to figure out how close you are consodering that you’re judging each other potentially on hundreds of different parameters. So in a 3D space we have an easy time seeing that- like maybe I like strawberries, bicycles, and acoustic guitar and the algorithm wants to place me closest to people who also like all three of these and furthest away from people who hate them all. So what does this say about those questions? Like is it arbitrarily difficult to create an OkCupid result if someone answers 7, 11, or 15 questions but no big deal for 1-3, 56, or 61? 2. Why does the rubber band figure eight example not count for two dimensions? I see why it doesn’t work for three. Is it because really you can’t actually create that pinch point without there being three dimensions? Like we conceptualize it as two but really one part of the rubber band is on top of the other part, so the figure eight cannot topologically exist in two dimensions? Or does that count as two dimensions?
The classic "Inside Out" video on smoothly deforming a sphere into itself but inside-out uses the figure 8 as an example of an object that cannot be smoothly deformed into a circle!
Numberphile is VERY necessary. Thank you professor!
I like how he finished the most random looking sequence ever with "and so on".
Hahaha, it's on you to find the pattern
@@adityakhanna113 extrapolating on this pattern is left as an exercise for the reader
@@RubidiumOxide Your comment made me laugh, thank you haha
@@RubidiumOxide If you manage to extrapolate it to n=4 you get a big party
I think he is just trolling and making up numbers
"We topologist are healthy people, we're not supposed to be eating donuts."
Yeah, that's because of the ceramic.
NO, ITS BECAUSE THE GUYS FROM ODD FUTURE WILL CLAP BACK AT YA
@Mike Donuts and coffee mugs are the same to a topologist (they both have one hole), and that's why they are weary of eating donuts.
@Mike missed the joke
@Mike There are a ton of smart people that are obese, smokers, use drugs, are promiscuous, etc. Human being are still human beings after all
A coffee mug isn't the only real world thing a torus (or donut) is homeomorphic to, y'know.
Only a mathematician could refer to an "ordinary 7-dimensional sphere." You know, the common household 7-dimensional sphere. :D
Yeah, like the one you might have in your 8-dimensional home.
Mine is in my kitchen junk drawer somewhere.
@@lo1bo2 i had 8 but it got bent in the drier, now i only have seven again.
@@lo1bo2 Mine keeps rolling into the corners of the room, where the Hounds of Tindalos get ahold of it, and when they've finished I have to repair it. Do you have any idea how difficult 7-dimensional sewing is?
Well yeah, you have like... vigintillions of them in your home right now. Maybe. I don't know how superstrings (allegedly) work.
I love how at 11:54, he says putting a lasso around a donut and the animation shows the coffee cup. After all, they are the same!
animator is also a topologist. :D
This guy is the only person to have scored perfectly on the IMO 3 separate times !
I always knew dimension 4 was strange, but I didn't realize just how strange it was in regards to the rest of them. That's fascinating
There's nothing different about "dimension 4". It's indistinguishable from the other dimensions. But working in 4 dimensions is different from working in 3 dimension. Which 3 or 4 you use, makes no difference.
@@peterwhitey4992 did you watch the video? The Professor says "dimension 4" multiple times so the author of the comment did use a term that everybody in the comments section should be familiar with.
If you arent familiar with it just ask: "What exactly do you mean with dimension 4?"
@@sihingvonfelix4251 Peter is clarifying a misconception one might have about what is called “dimension 4” or “the 4th dimension”. Attaching a special significance to one of the dimensions isn’t something you do when studying pure Euclidean spaces.
In the video, he briefly mentioned “it could be time”, which would probably cause misconceptions.
If you study 4D spacetime (3 space dimensions + 1 time dimension), that’s different from what topology is usually concerned with, and distance is defined differently.
In pure Euclidean space, a 4D distance can be found with the Euclidean formula: sqrt(dx^2 + dy^2 + dz^2 + dw^2).
In 4D space time, distance is defined as sqrt((c-dt)^2 - (dx^2 + dy^2 + dz^2)).
I didn't even think dimension 4 was strange, I thought the only reason it tends to get more attention than other higher dimensional spaces is because it happens to be the first dimensional space that we didn't evolve to comprehend intuitively. The fact that it stands out topologically from all others blows my mind.
@@nickpatella1525 But we are clearly talking about 4D spaces as a whole here, not some arbitrarily assigned "4th dimension" of any given space with N dimensions. It does seem that Peter hadn't watched the video.
The delivery on "for example, here is a donut and here is a coffee mug." is immaculate
Indeed. Some things are easier to think about when you internalise that a donut is a squashed coffee mug.
When someone commits to the joke, that is art. It was Lee Mack-esque
Ciprian is a genius!!! he is the only person ever to score 3 perfect papers at the IMO!, he was also top 5 in putnam (putnam fellow) 3 years!
I didn't know he was American.
@@anticorncob6he’s romanian
@@ABCDEF-it4mlcan tell from the surname
@@ABCDEF-it4mlyou can tell from the surname
I knew topology was interesting, but this blew my mind. Feels like there's so much to learn about higher dimensions
Engineers use manifolds (in N-dimensional complex space) to analyze multiple-antenna communications systems, such as MIMO WiFi. The shape of the manifold is determined principally by the arrangement of the antennas in 3D real space. Each point on the manifold corresponds to a direction (bearing) in 3D real space. The local properties of the manifold about that point tell us how well the communications system can detect, resolve and estimate the parameters of a remote signal source emitting from that direction.
Wow
Shoutout to whomever cleaned those blackboards. Exquisite work!
FYI, the disks starting around 12:00 should be filled, not empty. "Disk" means the inside of an n-sphere and "sphere" means the outside. That's why it's okay to contract them, no holes.
If TH-cam allowed us to save comments, that would be on my list.
It's times like this I'm reminded that some of the stuff we write online _will_ be read thousands of years in the future, if only by machines. Yikes.
This is the guy that proved that there exist manifolds that cannot be triangulated!
HE WHAT
nooo mannnn my simplicial homology :(
I don’t know what it means to triangulate a manifold :|
@@bencressman6110 That you can't rebuild the manifold with a triangle mesh?
What's the lowest dimension such a non-triangulatable manifold exists?
Oh no!
Does it mean I can't use my GPS in a N-Dimentional forest?
*But can you turn a 3-dimensional sphere inside out?*
i know this reference, and i was thinking of it the entire time he was discussing how to turn a figure 8 into a circle.
That was the first thing I thought of when this started!
Yes
No, only the 2d sphere inside of 3d space (as in that famous video), the 6d sphere inside of 7d space, and (if you want to count it) the 0d sphere (just two points) in 1d space
@ChannelZero Yes, there's a proof. You can see mathematicians agreeing about this in MathOverflow question 115110 "Eversion of the 6-sphere in 7-space". (This parallels other topology facts about spheres and dimensions related to the quaternions and octonions.)
For the 0-sphere, it is all solutions of x^2=1, so the two points -1 and 1 on the number line. It's not a smooth object, but passing the points through eachother doesn't pinch it in any way, so it can topologically be turned inside out.
Very humble and knowledgeable professor, nice work trying to help us normal sphere people get a glimpse at those exotics... like a fancy car video for numberphiles
Numberphile never disappoints
False.
True.
@@peterwhitey4992 your assertion is false
Brady's interruptions are becoming more and more annoying. Even Ciprian seemed annoyed a couple of times during the presentation.
@AL W No. Bradys comments are always appropriate and welcome.
yooooo this guy was my professor for a discrete math class at UCLA in 2018, cool to see him on the channel :D
I am not sure whether to feel jealousy or pity. Guy knows his stuff, you would learn a lot and still feel like you are missing something.
IS THAT A MATH CLASS YOU TAKE SECRETLY, SO THEY DONT KNOW YA BOY IS WICKED SMAHT?
Mr Manolescu has an excellent explaining style and a good voice as well. Joy to hear him speak.
What a great voice. Would be a treat to hear him lecture.
It always amazes me when the debate of the “shape” of the universe comes up. I mean, why does it HAVE to have a shape? It seems rational to believe that it is just never ending. Perhaps, infinity in every direction. Astonishing? Absolutely.
One of the best numberphile videos I have ever seen! Please invite pr. Manolescu very often!
Thanks for visiting us from dimension 7 and droppin some knowledge doc. 👍
I dont remember how many times i rewinded 10 seconds back to re hear that new piece of information. This is episode is dense.
best part of the video: "number phile videos are important too" absolutely agree we need to propagate maths to the whole world.. more people working on maths more probability to solve problems
I want to learn about the different spheres in different dimensions. I'm all about the specialness that is 4D space, but I'm curious about what exactly are the 992 spheres in the 11th dimension
Why don't you just try to make them and then you'll see
@@MushookieMan you got me there!
I'd love to see a video on division by invariant multiplication. It's incredibly important in modern computing, but almost unknown to most programmers let alone non-programmers.
Topologists aren't so weird. They dunk their coffee cup into their doughnut just like everyone else.
well, according to Einstein, if you let go of a doughnut it is the coffee cup which accelerates towards the doughnut, so the coffee cup does indeed dunk into the doughnut
Nice.
@user-ho3ng2oq2y In Newtonian physics, gravity is a force, and the surface of Earth is not accelerating
in general relativity, gravity is curved space-time, and it is more accurate to think of the grounds as accelerating up into you, pushing you off your inertial path (free fall)
wrg
Its really weird how many spheres in the numbers have big number gaps but i actually see a pattern in sphere dimensions 7,11 and 15. They all have a relation to mersenne primes. (2^3-1)x2^2=28 (2^5-1)x2^5=992 and (2^7-1)x2^7=16256. 28 doesnt hold for this pattern, but im guessing that the next one might be in the 27th dimension might be: (2^13-1)x2^13 = 67100672. Just an observation though
Nice, this is OEIS sequence A001676 and 27 is indeed 67100672, so looks like your theory is correct!
@@Alex_Deam I'm not saying that it's incorrect but lots of Numberphile videos have taught us that you can't assume a theory is correct just because the first n numbers match.
@@d5uncr I meant the idea that 27 would be 67100672 is correct, not that this proved Mersenne primes are definitely involved
Sounds like you're talking about medicare.
It is false in dimension 35(=2*17+1) in which case it is 2^17-1 2^19 43867
I love this!
I am so confused and do not understand.
But it's wonderful the host understands more than me.
Prof Manolescu is awesome! He disproved the triangulation conjecture in high dimension. You should ask him if he could do a video in that direction. This video turned out great.
What was the triangulation conjecture
I feel like my brain is a smooth object watching this video.
Best comment I've seen all week.
Hahaha, I completely agree! I've even built a 3D shadow of a shadow of a 5D hypercube, but I haven't begun to understand why there are more than 1 type of sphere in any number of dimensions. I did however like the explanation of why 4D topology is more complicated than topology in lower and higher dimensions.
@@evanglickstein8001 Damn, that's awesome. I have a friend doing a maths degree and he is interested in researching how to arrange spheres in the most optimal way possible in higher dimensions. This kind of thing just boggles my mind.
4-dimension being special and infinite makes me think of time-slicing the universe into 3d spaces. Pretty wild if the reason we live in 3+1 dimensions is that 4 dimensions holds more potential (is infinitely more likely to occur) than the others. Assuming, of course, that spatial 4d is comparable to 3+1d
Related question: does the "wavefunction collapse" pick one result (copenhagen), or does it pick every result (many worlds)?
@@RTzarius you clearly do not believe in the heart of the cards, tsk tsk
This is one of the most fascinating numberphiles videos ive ever seen! so cool!
same haha
So amazing! I got an interesting point: on corners, derivatives are not defined, so u can't say anymore where to grow or shrink to deform the objects!
Best explanation of "manifold" I've ever heard. I still don't quite fully understand it but I understand it better than before I saw this video.
28 = 32-4 = 2^5 - 2^2
992 = 1024-32 = 2^10 - 2^5
16256 = 16384-128 = 2^14 - 2^7
Absolutely fascinating. I didn't like functional analysis in university (which kinda relates to topology) and I always thought, that there is nothing special about higher dimensions because you can't properly visualize their objects (I tried hard especially in 4D), but now.. I completely changed my mind
Topology is sometimes called 'soft analysis'. I find it much easier and much more fun.
Strange as functional analysis deals with infinite dimensional spaces, thiugh you typically don't think of them as "space".
We will understand dimension 4 once we understand time t (the 4th dimension). Great explanation for the exotic structure with corners. thank you!
Haha I like this guy, hope to see more of him
When the video said “Featuring Ciprian Manolescu”, I was delighted. Manolescu is pretty famous in math competition circles because he’s the only person ever to write three perfect papers at the IMO (International Math Olympiad).
I would definitely like to see a video about those infinite exotic planes!
would there be time, i mean, really, would there be time? or would you have to fold the video?
Yes Numberphile videos are important too. You may be inspiring the person who discovers the 4h dimensional exotic sphere
This is one of those “unknown unknown” videos where I was like: I knew that I didn’t really understand topology… what I didn’t know was that I don’t really understand topology.
Side note, I totally use hyper-spheres and bisectional searches to resolve a convex optimization problem “when have I collected enough data about subject X?”, so it’s not that I wouldn’t love to have a better method by exploiting the manifold or by loosening the requirements of the shape of that manifold… but, I’ve yet to find the way to ask the question in a way that brings me closer to a better implementation.
So, great video. Love that numberphile will dive into subjects like this.
Dimension 4 being uniquely special like this feels like it could potentially have implications on why, if the higher dimensional models of e.g. string theories are correct, only three dimensions are extended spatially, the fourth is uniquely temporal, and higher dimensions are curled up and only manifest as phenomena within space over time.
@Mike Foster That ant-on-a-hose analogy is exactly what it meant by the higher dimensions being "curled up"; they're like the dimension around the hose, while the three we're used to are line the dimension along the length of the hose. We're so big compared to the curled-up dimensions that they basically don't exist to us, in the same way that we're so big relative to a hose that it seems almost like a one-dimensional object. But for tiny subatomic particles, the curled-up dimensions are big enough to give them room to do interesting things, which is a proposed explanation for various phenomena that we observe of those particles; what looks like a property of "charge" to us, of an apparently static particle in 3D space, is actually its velocity along a curled-up higher dimension, but since it's just looping around that dimension that's too small for us to see, we don't perceive it as "motion" but as some static property of a motionless particle.
It's also curious that dimension 4 happens to be the one we live in.
Kind of. We only have 3 spatial dimensions. Time isn't acceptable as another dimensio to us. Unless you move close to the speed of light, etc.
Absolutely wonderful!!!
How do you become a mathematician and not become insane? This seems to me like looking straight into the abyss.
Seriously, I can't even fathom most of this stuff. All those people arguing about 4 dimensional space when I got lost all the way back at "smoothly different."
Maybe a mathematician's job is to take an abyss and reveal its structure.
There's an interesting show called "Dangerous Knowledge" about how some mathematicians/physicists/etc have lost their minds to math. Wrapping your head around infinity isn't as quick & easy as learning to tie your shoes. Pretty interesting watch, although it's a bit dated at this point.
I'd like to think that being a mathematician helps to process the deep thoughts they have... it must be nice to have an outlet, a way to express the thoughts, a language (math) to convert ideas to, so that others can interpret/peer review/etc. Knowing how complex the world/universe is, is almost like a blessing and a curse. That said, I'd rather ask the big questions and drive myself a bit mad trying to figure it out, than to never ask them at all and just bumble around taking everything for granted for my short time here.
I've actually ran across a book series where people can use math to do magic, but they also have an increasingly High chance of going insane from doing so. There are ways to reduce the chances, like don't actually do the math all in your head, but the chances never zero.
Careful, you're pinching it infinitely tight!
14:10 professor drops a brain-melting uncountable "absolute" infinity on us
I had some intrinsic feeling that 4th is a mystery.
It isn't really much of a mystery. You can map the 4th dimension, for example. You just need more than 4 dimensions to act as an overarching structure within which to map the lower dimension. Without 4 dimensions we could not map the lower 3. And so on. (Edit, the missing exotic sphere is the exception. Yes, it is considered a quandary and I think of it as a blind spot. To me it is more of an ontological mystery than a topological one.)
The sequence shown in this video is A001676. 1 is specified for dimension 4 but it's just a conjecture.
They routinely measure the curvature of the universe. So far, nobody has been able to measure any. It is at least flatter than we can currently measure. If it is totally flat, it is likely infinite in size. If there is a curvature, that would tell us the actual size. Based on the fact that we have not been able to measure any curvature, the actual size is at the very least enormous, WAY bigger than the 93 billion lightyears of the visible universe.
Incredible video. I'm in love with topology now.
Prof Manolescu's face as he desperately tries to keep things simple and not go completely off the rails.
Are there infinitely many dimensions for which there are no exotic spheres? Is this known?
Send this to the Flat Earthers so they can realise that the Earth isn't flat, it's just a 2D manifold in 3D space and they're just too close to it to tell the difference between a plane and a sphere ;)
Your video on Ricci flow always confused me, because it never seemed to have an application. Now that I've heard a little about topology here, I can start to see why Ricci flow might be useful.
Ricci flow is a part of Perelman's proof of Poincare's conjecture.
I always thought that higher dimensions where funky, but "2+2
love the animations here ☆
The 28 spheres of dimension 7 are also known to form a cyclic group. I assume that the non exotic one is the identity element.
Cool, under what operation?
@@samthedog8391 connected sum
Really interesting! Would love to learn more about this topic.
OMG finally an ending that makes me sit through the credits!!!! On TH-cam!!! (not the ad though)
Numberphile video, and all the sister channels, are very important.
Back then I knew numberphile through Vsauce ...Never disappoint 😊
Vsauce was the entry drug for educational youtube
Congratulations on reaching the 15TH DIMENSION! Enjoy your reward of SIXTEEN THOUSAND TWO HUNDRED AND FIFTY-SIX SPHERES!
To draw a 4D sphere, use time as the 4th dimension. Animate it in other words. So you animate a point that grows into a ball then shrinks back to a point then disappears.
It will not grow at an arbitrary rate, it will grow quickly at first then slowly, then shrink slowly then shrink quickly.
You mean time as a 5 dimension. If is at 4 dimension, is our time growing a normal 3d ball.
@@RF-fi2pt - No, he meant as a 4th dimension.
@@peterwhitey4992 I know he talks about the thesseract and the effect a 4d spatial ball when pass the 3d spatial universe. But as moves have t as 5 dimension, and the projection have t as 4 dimension, standard in our spacetime universe.
Peter whitey is correct. I meant use time as the 4th dimension. X,Y, Z, t. To do it, use art to draw something that looks 3D on a 2D surface, pretty simple. Then animate it.
A hypersphere would be as I described. A hypercube would be blank, then suddenly a 1x1x1 cube for 1 unit of time, then suddenly disappear to be blank again.
In a seminar on word embeddings, we heard about a hyberpolic distance function that improved a specific type of classification problem. And I asked the question if the concept of sphere even makes sense in these extremely high dimensional spaces.
From now on, when people ask me why I didn't specialize in topology, I'll tell them it's because I like doughnuts.
Those blackboards look amazingly clean!
I know, Selman Akbulut and Simon Donaldson, working on dimension 4
Topology - my favorite piece of Math!
👍
That's like a whole mysterious world at our doors yet to discover !
That's what mathematics is, my guy.
really interesting, three spacial and one time dimensions make up a 4 dimensional space, interesting that 4 is the one we have the most trouble saying things about
11:54 - Let's take a moment to appreciate this illustration of a donut. 😆
I wonder for years why only three physical dimensions were opened in this Universe. This question must be on the list about 4th dimension.
Interesting topic. I have run into this problem with 4 dimensions in the study of error correcting codes. Coding theory is related to exotic topology, since the number of dimensions affects how code symbols can be decoded. We know a lot about binary codes or trinary codes with symbols that have 2 states (0, 1) or 3 states (-1, 0, +1). We know the best possible error correcting codes for binary codes with length out to 256 bits, and in many useful cases much farther. Something similar is known for trinary codes. But not for quaternary codes. Finding codes for quaternary symbols is much less obvious. In most cases we simply reduce this to a pair of binary symbols, but that ignores the reality of many useful communications systems.
another reason why the universe should've been 4-dimensional
What are some other reasons?
I find these higher dimension issues fascinating, although I don't understand much about them. Sometimes it seems whoever created the universe did so after drinking a few bottles of wine, and some imperfections became permanent
Assuming evidence-less things like someone created the universe, that it was created at all, that what you see as imperfections are such, tons of logical fallacies in your thinking.
But even in a different universe these same topological problems would remain meaningful and have the same answers and easy or difficult proofs.
The shrinking the loop on the coffee cup handle reminded me of Ricci Flow for some reason... too much Numberphile!
i first dusted a computer in 1971, a univac 9300, i was a tape librarian, which meant i mowed the lawn and filled the coffee machine as well as dropping punch cards all over the floor. i got into computer graphics at uni though, in 1981 i guess (kingston poly) , and then i got poached by digital pictures and worked for them doing pop videos, and then i moved to cfx associates and i learned computer graphics and animation there, going on to freelance later for all the major visual effects houses in soho, working on tv titles and commercials and even doing some feature films and finally decided to retire after a stint at electronic arts.
so i love shapes. and time.
i have lots of computer graphics of all sorts on my channel.
7:11 But that replaces a single point (the crossover) with two points, so the result is topologically different.
If you consider the curve itself then yes, they are different. However, you can also consider a parametrization, in which case the crossover point is already counted twice before the deformation. You could say that the objects (figure 8 and circle) are topologically different, but the parametrizations are topologically the same.
@Roger. That makes sense.
Dimension 4 being so weird while we’re right beside it feels like it has some sort of meaning…
"Dimension 4" is no different from "dimension 3", or any other dimension. 4 dimensional space or objects are different from 3 dimensional space or objects though.
@@peterwhitey4992 did you watch the video? The Professor says "dimension 4" multiple times so the author of the comment did use a term that everybody in the comments section should be familiar with.
If you arent familiar with it just ask: "What exactly do you mean with dimension 4?"
@@sihingvonfelix4251 See my response under Kris’s comment
Probably not a coincidence that spacetime is a 4-dimensional manifold...
We aren't "right beside it" it terms of being 3D. That's how we've evolved to experience the world; that's our perception. We exist in all available dimensions.
I feel like I want to intersect a 4d hyperspace with a 3d plane or a 3d object and show how there is another 4d hyperspace where you get something different.
Oh, and the answer to "Would you be a famous mathematician if you solved this?" Is "I AM a famous mathematician."
I’ve been studying Quaternions and 4D mathematics, and hearing that the 4th dimension has properties of *infinite unknowables* is kinda mind blowing. It makes me wonder if math is at its perfect structure in the fourth dimension, and matter can only be in 3, so we’re stuck here witnessing it’s insanity.
Math is not perfect in any dimension. The incompleteness theorem will always keep mathematics at most incomplete.
It is not 4th dimension that is special. It is 4-dimenisonal space that is special. They don't single out some specific dimension in this video. They talk about spaces with different number of dimensions.
You cannot imagine how much I love your channel. It's like 🎉🎉🎉🎉😆😆
If you liked the video, I strongly recommend you play some 4D video games like 4D Toys or Tetraspace (aka. Brane) 🙂(alas, Miegakure is not finished yet...)
Ditto for 4d golf (alas it's not finished)
Isn't dimension 4 that special, since we ACTUALLY live in it: x, y, z and time (except that we can move to and from in the first 3, but not in time. There we can only go forward)
We definitely need more topology videos
"...992, then 1, then 3, then 2, then 16,256, and so on..."
I feel like this is one of those "find the next number in the series" questions but where I'm a donkey
Thus it make sense we live in 3rd dimension plus time - it lets us move very intriguing in timespace
I love math, but I am happy this dimension 4 is someone else's problem.
Is this why we live in 4 dimensions? Perhaps time is something special that only appears in 4 dimensions used to separate our lower dimensions, or some other second law of therm - maximum entropy principle.
He squared the circle in less than 45 seconds. Genius.
Did he?
@@niamhgirling6000 ‘twas a joke 😂
The fact that spacetime appears to be a four-dimensional Lorentzian manifold seems to make this special case extremely important.
Not really. Afaik topology deals with all dimensions being real, whereas in our universe the 4th dimension is imaginary (time).
@@feynstein1004 No, topology is much, much broader than you have been led to believe. And anyway, the four dimensions are all "real" here, in the sense that every manifold is locally homeomorphic to R^n. Imaginary time is just a clever way of dealing with the Lorentzian metric.
@@EebstertheGreat In English please? 😅
@@feynstein1004 So, for the first part, topology is a very general field of study. It studies the properties of a space in terms of neighborhoods of its points, allowing ideas like connectedness, separability, density, genus, and more. Some topological spaces are associated with what we would think of as a surface or space in a sense. For instance, all spheres share the same topology (essentially). That's the sense in which you can call a coffee cup the same as a donut, because one can be continuously transformed into the other. But most topological spaces are not like that at all. They can encode other abstract ideas like topological graphs or honestly just about anything given the appropriate structure.
The four-dimensional space I am talking about is spacetime, the combination of space and time you see in Einstein's theories of Relativity. You mentioned that time is "imaginary," and therefore it's not the same as a real space. That's sort of true; the spacetime in Special Relativity (Minkowski space) has a different way of defining distance than a Euclidean space that you are used to in geometry. Still though, that doesn't actually mean any one of the dimensions has to be imaginary (though treating it that way is a convenient mathematical trick). And importantly, even though the metric is different, the topologies here are the same. If Euclidean space is a donut, Minkowski space is a coffee cup. They look different, but they are topologically the same.
@@EebstertheGreat Hehe thanks for the explanation. I was familiar with the first one. It's the second one that I wanted a bit of clarification on.
So, regarding that, what I meant was that our reality is different from a purely spatial 4D space in the same way the complex plane ( x & iy) is different from the real plane (x & y). Idk if this difference matters in topology but I think it does. You might have similar structures in both spaces, for eg you could just draw a circle/donut in both. However, those circles/donuts would end up having different properties and in a sense, mean different things. That's how I think it is anyway. But I'm not a mathematician. So I could be wrong.
This is a nice description of what a manifold is, but it would have been super helpful to have a *non*-example here, not just a lot of examples.
One example of this is a figure 8 as in the middle section there is a sharp corner thing that you cannot get rid of
Errmm. A coffee mug *doesn't* have a hole in it - it has a dip/depression/recess and you cannot transform a doughnut into a coffee mug that will hold coffee. A cylinder, yes. Coffee mug no.
On second thoughts, I take it back - I forgot the coffee mug has a handle...
Two thoughts:
1. Dimensions beyond four always seemed incredibly silly to me until someone explained to me that this is basically what OkCupid or Bumble, etc., are doing. You answer x questions and your march answers y questions, and the algorithm needs to figure out how close you are consodering that you’re judging each other potentially on hundreds of different parameters. So in a 3D space we have an easy time seeing that- like maybe I like strawberries, bicycles, and acoustic guitar and the algorithm wants to place me closest to people who also like all three of these and furthest away from people who hate them all. So what does this say about those questions? Like is it arbitrarily difficult to create an OkCupid result if someone answers 7, 11, or 15 questions but no big deal for 1-3, 56, or 61?
2. Why does the rubber band figure eight example not count for two dimensions? I see why it doesn’t work for three. Is it because really you can’t actually create that pinch point without there being three dimensions? Like we conceptualize it as two but really one part of the rubber band is on top of the other part, so the figure eight cannot topologically exist in two dimensions? Or does that count as two dimensions?
The classic "Inside Out" video on smoothly deforming a sphere into itself but inside-out uses the figure 8 as an example of an object that cannot be smoothly deformed into a circle!
These animations are great!
Fantastic animation !!!!!