I'm not sure I really *understood*, but I think I get the picture now. I don't understand why this is different from a recurring decimal though. Like say 1/7 = 0.142857142857.... Surely it would have been more intriguing if it were an irrational phenomenon, like π (Pi) or √2 (square root)?
Well done! I am a professional mathematician who has done research in homotopy theory, and I am amazed by the quality of this content. Tackling such an abstract topic in a TH-cam video is a very ambitious task, and I think that you succeeded in making the exposition understandable to non-experts.
Interestingly, this relates (and I don’t understand the particularities) to Clifford algebras of different dimensions and in which ways they can be found in each other; they can be expressed in terms of matrix algebras over well-known algebras ℝ, ℂ, ℍ of real and complex numbers and quaternions, and the pattern is one-to-one with the case of Bott periodicity for homotopy groups: we have (matrix algebras over) ℝ, ℂ, ℍ, ℍ ⊕ ℍ, ℍ, ℂ, ℝ, ℝ ⊕ ℝ for dimensions 0…7 and so on. This sequence even has an additional reflection symmetry. Clifford algebras relate to O(n) groups and vector spaces with an inner product (as do O(n)), and also they relate to spinors from physics.
@@md2perpe IIRC it should be related, yeah! Spinor spaces are related to modules acted upon by Clifford algebras (because they contain so-called spin groups: these are precisely the transformations on spinors that are analogous to SO transformations on vectors).
@user-yb5cn3np5q I think there were results about relating algebras for degenerate forms (with squares giving 0) with algebras over nondegenerate forms. As a particular case, if _everything_ squares to 0, we get just an exterior (Grassmann) algebra. Also. Let Cℓ(m, n, k) be an algebra where m generators square to +1, n of them to −1 and k of them to 0; and let Λ(k) be an exterior algebra over k generators (over a k-dimensional space). It might even be the case that Cℓ(m, n, k) ≅ Cℓ(m, n, 0) ⊗ Λ(k), but somebody better check that.
A more precise formulation is that real Clifford algebras display 8-fold periodicity when you consider things up to Morita equivalence which amounts to the representation theory of the algebras being the same. So if you're interested in spin representations (spin groups live in real Clifford algebras), then this periodicity comes into play. For the complex case (with U(n)), then Bott periodicity is 2-fold. With Sp(n) (quaternionic), you have 8-fold periodicity once again but it's shifted by 4 from real Bott periodicity. Probably the reason why Atiyah expressed amazement at Bott periodicity is for its relevance to the Atiyah-Singer index theorem which relates the index of a Dirac operator on bundle over a spin/spin^c/spin^h manifold to some topological invariants (involves Todd class and Chern character). A Dirac operator's definition makes use of Clifford multiplication and thus, a representation; I think people usually are interested in those that extend a spin representation. One approach to this subject is via topological K-theory which is concerned with stable vector bundles. For example, a real stable vector bundle has structure group in O(∞), gives a class in KO, and that's why Bott periodicity becomes relevant. One interesting phenomenon is that KO and KU can be viewed as ring spectra but KSp cannot.
This... kinda feels like why we have the amount of dimensions we do... what if... that's the lowest amount of dimensions that can create a diversity of solutions?
No one is quite sure, but it doesn't seem like a coincidence, especially when you consider the effects of Bott periodicity on the real Clifford algebras, which seem to be critically important when making sense of spinors in quantum physics.
Stunningly good! Thank you. That Group Theory class I took in college, not knowing why, turns out to have been one of the best classes ever (alongside art history!).
Is there a relationship between homotopy and residues or contour integrals in the complex plane? I think they also have the concept of ‘winding number’ when we integrate, which affects the value of the residues.
Yep! What you are looking for is Poincare Duality in the general case. Also there is a theorem about contour integration being the same in the same homotopy class of contours/paths
😊 I get a bit triggered when people say “second dimension” or “third dimension”. It’s “two dimensions” or “three dimensions” respectively. The number of dimensions is a number, a count. Not an ordinal. But suppose we consider two dimensions. Suppose we fix a basis and impose an order on the components. Each of the basis vectors might then be considered to span a dimension and there would be two of them and they’d have an order. Well then the second dimension would be one of them. The second one. In particular, “the second dimension” would not be a two dimensional space! That’s why the phrase doesn’t make sense at all. Rant over… Love the video even despite this minor nit pick 😊
It makes it sound as if the 3d sphere exists in the "third dimension" when it exists in all three dimensions. It is confusing cardinality vs ordinality for the object. It is not that the use of "dimension" as an ordinal is wrong. You can say "exists in the 3rd dimension" in the same sense that one could say "time exists in the 4th dimension"(in the sense it is the 4th component of the space-time vector). But a 3d sphere literally does not exist in the 3rd dimension. One could argue that it is the 3rd dimension of the sphere that gives rise to the 2d sphere and in that sense it might work but it is awkward IMO. That is, S_1 x S_1 x S_1 = S_3 and you could say that it is the x S_1 as "existing in the 3rd dimension but one could then also say it exists in the 2nd dimension since S_2 x S_1 = S_3 or even in the first dimension since there is no preference for singling out the 3rd.
@@MDNQ-ud1ty Thanks for chiming in =) I see a couple of potential misconceptions in your post. Let me see if I can say something about them. Let me clarify a few things. The notion of dimension is defined for vector spaces. It’s the minimal number of elements in that space, that form a basis. Those elements are typically unordered. The set of elements is also not unique. We can get a different basis by linear transformations. There is a simple very common class of vector spaces defined on the n times Cartesian product of R(the real numbers), called R^n. Those vector spaces come with a natural basis {(1,0, …,0),(0,1,0, …,0), …,(0,0, …,1)}. And that basis comes with a clear order. That order can be used to define “components” of a vector. But, as vector spaces two copies of R^n are isomorphic as vector spaces if they are related by a bijective(invertible) linear transformation. R^n can be equipped with several different basis sets. The circle, the sphere and higher spheres are not vector spaces though. We can think of them as for example sets, topological spaces, manifolds or differential manifolds. It depends on how much structure we want to equip. Manifolds are roughly speaking spaces that locally in a neighborhood around each point look like some R^n (Omitting details). Since they locally look like R^n, we can define the dimension of the manifold to be n. Anyway. The 2-sphere is 2-dimensional in the manifold sense. But it is not a vector space. The the 2-sphere doesn’t inherit the order of the coordinates that we might have appealed to in R^3. We might try to equip the sphere with a coordinate system. If we could do that maybe we can call the first coordinate the first dimension and the second coordinate the second dimension and be happy. But there is no naturally preferred coordinates like in R^n. We have to make an arbitrarily choice. Worse still, one can show that it is impossible to construct a single coordinate chart that covers the entire 2-sphere (at least on point will be left out). Ok. Let’s say we ignore this complication, i.e. we remove a point from the sphere, introduce coordinates covering the rest, pick an order for those coordinates, then use those coordinates to construct some notion that we call a dimension. Even so, how is this notion of dimension related to “the dimensions” of R^3? Not in any way really! We’ve just made up a new object and called it the same name as another object. And there is no obvious way to match them up. So the better way to do it is to say that the notion of “the n:th dimension” doesn’t make sense and we should avoid it. So? Does it make sense for spacetime then? Is time the 4th dimension? Well, one can make a better case for it since there is structure that do impose an order. But it’s not as simple as saying the time dimension is the first component of a basis. As a vector space we can use any bases. But we can use the extra structure (like the metric if you know what that is) to transform an arbitrary basis into a basis with one time like vector and three space like vectors (don’t worry about the details to much if the words are unfamiliar). That separation isn’t unique though. But we can choose to only use basis sets that are separated like that. Then we must make sure that we don’t use general linear transformations since they would destroy that separation, but instead we restrict to some smaller set of transformations. But time still isn’t a single unique direction other than in a specific coordinate system. And there is no reason to really prefer a specific coordinate system. Anyway, in general relativity spacetimes are manifolds and not vector spaces. Dimension is just a number and you’re free to pick a basis for tangent vectors at each point. While each such local basis has a time-like element the choice is not unique. The notion of the n:th dimension just isn’t a good one there either. Finally. S^1 x S^1 is not isomorphic to S^2. (!) And S^2 x S^1 is not isomorphic to S^3. I think that’s just a mistake. Thanks for taking the time if you got this far.
@@HyperFocusMarshmallow You are arguing over nomenclature and being pedantic. Dimension has been used before vector spaces were invented and they are not just used for vector spaces. Many times people sacrifice exactness for expediency when things are obvious. It's quite common even by professional mathematicians. By S^1 x ... I do not mean the topological product but the Cartesian product of sets (S^1 being parameterized by r in [0,1)). It was a shorthand. You seem not to be ok with shorthand for some reason... you will learn one day. Nothing in life is perfect.
@@MDNQ-ud1ty Sure. Pedantry was kind of in the premise of my first post, so I assume we were playing that game. It’s not to serious or anything, it’s all for good fun. Trying to get at pedantic (in the good sense) versions of these concepts can be useful for learning or staying sharp. Once we can be pedantic and we know how to really do it, that’s when we can afford to be sloppy. But sometimes we just can’t be bothered and that’s fine too. If I misinterpreted what you were saying that’s another thing entirely though. I wasn’t trying to twist your words or anything. I just interpreted them as best I could. I assume you’re right that dimension has been used before vectors paces. I do know the concept crops up in some other places in mathematics but I don’t know a detailed history of it or versions of it before modern math. Maybe I have my head in the mathematical sand but I do think vector spaces or manifolds is the relevant notion here though. I’m perfectly fine with using loose sloppy language. Of course the potential issue is that we are actually saying things that are nonsense and that we don’t actually know the pitfalls involved or how to rephrase it in the ways that does make sense. Or just that there might be miscommunication. Sometimes we take that risk. Saying that “I’m triggered…” is kind of just my way of phrasing that there is such fudging going on. About the S^1 x … Sorry if I misinterpreted you. I have nothing against using shorthand I just didn’t understand which one you used. It happened to be ambiguous in this case apparently and I read it the way I thoight the shorthand was supposed to be read and answered accordingly. I’m still not quite sure what you intend actually. I think the topology is quite central for distinguishing a circle from an interval or even from just a point set. I’m not sure in what way it would be an n-sphere without such information. Why not just a box or again a point set. Maybe I’m misreading you again. If so, I apologize. You don’t have to answer if you don’t want to, I’m just here to enjoy some math talk and have fun. Maybe learn something. Since I not getting your intention apparently, maybe there is an opportunity to learn something =) Hope you’re having a good day otherwise! Cheers!
I just started watching your some1 submission and instantly found a connection to this video too. Namely the spin 1/2 phenomenon, I believe, is captured precisely in the \theta/2 within q_\theta as flashed on the screen at 10:48 since this covering S^3 \to SO(3) is the spin double cover of SO(3)!
I know that I use math in organic chemistry but my hats off to you guys 😅 I read about, study and have experienced psychedelic experience and I fundamentally understand the concept of infinity in my own way but the way mathematicians deeply analyze and comprehend these complex ideas through numbers and theory is something I strive to understand in more clarity and find absolutely fascinating. I feel like understanding complex mathematics can help anyone who is interested better grasp the deeper concepts about the universe in a more structured way and although my brain is less rigid it's more creative and abstract maybe because of that I find myself curious yet confused. Idk I'm trying to wrap my head around what you are saying and I feel like I kind of get it but it's like you said very hard for someone who exists in the 3rd dimension (arguably 4th) to understand beyond what they can visualize. Idk I nedd to read more 😅 obviously lol
This is making the world pacman you pop out the bottom and pop in the top and it's a repeating pattern pattern every 4 lines has 2 and the two two's are a mirror image ,you need less math to understand the world in only imagined.
What the hell Why is it eight? Why not nine, or seven? Is it because its a power of two? If yes, then why not four or sixteen??? Holy shit this is perplexing.
I dont get any bit of it. But the end sort of reminded me of an octave on a piano. How the notes repeat itself eventually. Guess how many notes there are in an ocatave....8.
Thanks Rad! It's just a page I wrote for the video. You can download it here if you wish: drive.google.com/file/d/1EP_17XTaCR7IVw5oPXNJgGdm8iTiYGnZ/view?usp=sharing
What about the symmetries of a square being generalized to SQ(infinity), is there periodicity there? Is SQ(n) even different from O(n) in the first place so that this question is valid?
not a bad question, but unfortunately it wouldn't work with the square since the symmetries of a square are what we call discrete. This just means that it's finite and it's not like a 'space'. For example, the symmetries of a square is a really nice group called the 4th dihedral group (4 because there are 4 vertices). This just contains 8 symmetries, and hence discrete. So you can't study loops in this space because it's just 8 points - in a sense there isn't enough space to 'move around'.
But, you can define that group, since the symmetry group of the square does embed inside the symmetry group of a cube, and so on for all dimensions. It’s just that it wouldn’t be a connected topological group like the sphere is, for the reasons already pointed out. It would still be a group though
Yes, you can do this. As already stated, you do not have fundamental groups but you do have homology groups(which are the discrete form of the fundamental group). So there will be some discrete analog but is it periodic? I don't know. It's very likely that there will be some analog periodic structure since the hypercube is homeomorphic to the hypersphere. E.g., any embedded manifold in R^n should take on these symmetries of the sphere in some way. E.g., think of the hypercube. Effectively it will sample any "symmetry loops" since some symmetries of the hypersphere will be valid symmetries for the hypercube(just inscribe the hypercube in to the hypersphere). Hurewicz's theorem states that there is a homomorphism between the two so what will happen is that the periodicity that you get from O(infinity) will be collapsed down in some way to the periodicity that you get from the "SQ(infinity)". It may destroy the periodicity or retain it completely. You'd have to investigate to see. In the case of a hypercube it might maintain it completely or modify it slightly. For other shapes it might completely change it. But there definitely is some relationship. You'd have to work it out to actually know.
Isn't this just a random nonsense pattern that emerges when you compound error upon error? For an example, just because you can represent a 2D sphere and a 3D sphere and their symmetries mathematically, it doesn't follow that you can do it with a "4D sphere" - a completely nonsense object which isn't real but which you can represent in a neat mathematical equation anyway. Multidimensionality nonsense from computer nerds who like their cool geometric screensavers is the weak spot in this "periodicity observation". Perhaps the periodicity of symmetries appears because there aren't more than 3 dimensions.
You can't see imaginary numbers, or quaternions yet those exist. Mathematics is a subject that given a set of axioms, what can you logically conclude. High dimensions "spaces" exist in many data sets.
@@jmd448 Dimensions are just subdirectories within an array, e.g. you've got an array made up of n elements, and each element is a sub-array so that e.g. n1[0...99], and each element of that can be an array, etc. This is not really dimensionality - it's just computer science. What's actually happening is that data is being written to addresses in a computer system that exists in a purely 3D world.
@@jmd448 I think with maths there is a lot of scope to get lost in some esotericism which has no basis in reality. Before maths, it was basic reasoning logic, resulting in wrong conclusions with the right method. Example: man 1 says the Qur'an is true. Man 2 says no it isn't. Man 1 "proves" it's true by saying you can see the sun and the stars, and the Qur'an says there are sun and stars, and by inference everything it says is absolutely true. He wins the argument because it's good logic but bad postulates. In this case what we're doing is interacting in a 3D world which has computers in it, and saying there's a 4D world. This is exactly the same game neoplatonists played, to prove there were 7 layers of higher worlds above this one. Yet they're sitting here in this one, talking about the others which supposedly exist, and being "right". There are many other examples - the process always results in a bad cosmology or ontology. These people are all lost in some esoterism and they become the bishops of their day - social status thing for survival and reproduction of their ape genes, nothing more.
I was reading this comment and then my neighbor came and we read it together. He said this comment really changed his life and it touched my heart. My village people are so grateful. Am proud to say cool comment wow thanks for sharing.
@@loop2427 forgive me but that seems like a cop out to me, these objects are supposed to be geometries in space (I'm obviously kidding about the previous statement lol, but you were the one who said they were equivalent haha), more likely this is an artifact of the framework itself... I've got no clue though, am punching far above my weight.
Like perhaps the fact that we define a dimension as 90 degrees apart from each other and in our space that means there are three dimensions with six total antipodes which divide space into... Octants?
I'm guessing that time may not exist in certain dimensions, and yet may take up one or more dimensions of the number of dimensions under consideration..
Eric Weinstein brought me here
Same 👌🏼
@@ultravidz Chris Williamson podcast??
Yeet
Me too
Same
Nice video, currently learning for my K-theory exam and it was a nice distraction.
Love it!!!!!!!!!
5:08 really difficult to explain this without mentioning homeomorphisms , gjgg
I'm not sure I really *understood*, but I think I get the picture now.
I don't understand why this is different from a recurring decimal though. Like say 1/7 = 0.142857142857....
Surely it would have been more intriguing if it were an irrational phenomenon, like π (Pi) or √2 (square root)?
aren't your animation at 9:50 and 9:57 already loops in O(3)? are these loops individually nulhomotopic?
Nope ... it didn't reach my neurons in charge of understanding.
Does anyone else have Eric Weinstein paused in another tab, having just mentioned Bott?
Sure do!
I have waaaay too many tabs open from both of his chats with Chris Williamson
Yep! How did you guess? : )
Haha! Yep!!!
Yep
Well done! I am a professional mathematician who has done research in homotopy theory, and I am amazed by the quality of this content.
Tackling such an abstract topic in a TH-cam video is a very ambitious task, and I think that you succeeded in making the exposition understandable to non-experts.
I’m a landscaper and I heard this on a podcast thinking it can’t be that hard to imagine.
Turns out it’s pretty hard.
This is excellent content, shame youtube doesn't care about education.
Does not the challenge to explore and the niche of the content not excite and intrege the discovery/study
Interestingly, this relates (and I don’t understand the particularities) to Clifford algebras of different dimensions and in which ways they can be found in each other; they can be expressed in terms of matrix algebras over well-known algebras ℝ, ℂ, ℍ of real and complex numbers and quaternions, and the pattern is one-to-one with the case of Bott periodicity for homotopy groups: we have (matrix algebras over) ℝ, ℂ, ℍ, ℍ ⊕ ℍ, ℍ, ℂ, ℝ, ℝ ⊕ ℝ for dimensions 0…7 and so on. This sequence even has an additional reflection symmetry. Clifford algebras relate to O(n) groups and vector spaces with an inner product (as do O(n)), and also they relate to spinors from physics.
I assume that the 8-periodicity of spin representations is related:
en.wikipedia.org/wiki/Spin_representation#Symmetry_and_the_tensor_square
@@md2perpe IIRC it should be related, yeah! Spinor spaces are related to modules acted upon by Clifford algebras (because they contain so-called spin groups: these are precisely the transformations on spinors that are analogous to SO transformations on vectors).
Wow this is super interesting!
@user-yb5cn3np5q I think there were results about relating algebras for degenerate forms (with squares giving 0) with algebras over nondegenerate forms. As a particular case, if _everything_ squares to 0, we get just an exterior (Grassmann) algebra.
Also. Let Cℓ(m, n, k) be an algebra where m generators square to +1, n of them to −1 and k of them to 0; and let Λ(k) be an exterior algebra over k generators (over a k-dimensional space). It might even be the case that Cℓ(m, n, k) ≅ Cℓ(m, n, 0) ⊗ Λ(k), but somebody better check that.
A more precise formulation is that real Clifford algebras display 8-fold periodicity when you consider things up to Morita equivalence which amounts to the representation theory of the algebras being the same. So if you're interested in spin representations (spin groups live in real Clifford algebras), then this periodicity comes into play. For the complex case (with U(n)), then Bott periodicity is 2-fold. With Sp(n) (quaternionic), you have 8-fold periodicity once again but it's shifted by 4 from real Bott periodicity.
Probably the reason why Atiyah expressed amazement at Bott periodicity is for its relevance to the Atiyah-Singer index theorem which relates the index of a Dirac operator on bundle over a spin/spin^c/spin^h manifold to some topological invariants (involves Todd class and Chern character). A Dirac operator's definition makes use of Clifford multiplication and thus, a representation; I think people usually are interested in those that extend a spin representation. One approach to this subject is via topological K-theory which is concerned with stable vector bundles. For example, a real stable vector bundle has structure group in O(∞), gives a class in KO, and that's why Bott periodicity becomes relevant. One interesting phenomenon is that KO and KU can be viewed as ring spectra but KSp cannot.
This... kinda feels like why we have the amount of dimensions we do... what if... that's the lowest amount of dimensions that can create a diversity of solutions?
Could the fact that there are infinite 3D loops explain why we have 3 spatial dimensions in the Universe?
No one is quite sure, but it doesn't seem like a coincidence, especially when you consider the effects of Bott periodicity on the real Clifford algebras, which seem to be critically important when making sense of spinors in quantum physics.
12:21 Elevendy-Loops is such a wonderful word
Stunningly good! Thank you. That Group Theory class I took in college, not knowing why, turns out to have been one of the best classes ever (alongside art history!).
Is there a relationship between homotopy and residues or contour integrals in the complex plane? I think they also have the concept of ‘winding number’ when we integrate, which affects the value of the residues.
Yep! What you are looking for is Poincare Duality in the general case. Also there is a theorem about contour integration being the same in the same homotopy class of contours/paths
My brain just just went loop Dee loop with this one. And I enjoyed it
At last, a great lucid visual presentation of Bott periodicity and group symmetries
😊 I get a bit triggered when people say “second dimension” or “third dimension”. It’s “two dimensions” or “three dimensions” respectively.
The number of dimensions is a number, a count. Not an ordinal.
But suppose we consider two dimensions.
Suppose we fix a basis and impose an order on the components. Each of the basis vectors might then be considered to span a dimension and there would be two of them and they’d have an order. Well then the second dimension would be one of them. The second one. In particular, “the second dimension” would not be a two dimensional space!
That’s why the phrase doesn’t make sense at all.
Rant over…
Love the video even despite this minor nit pick 😊
It makes it sound as if the 3d sphere exists in the "third dimension" when it exists in all three dimensions. It is confusing cardinality vs ordinality for the object. It is not that the use of "dimension" as an ordinal is wrong. You can say "exists in the 3rd dimension" in the same sense that one could say "time exists in the 4th dimension"(in the sense it is the 4th component of the space-time vector).
But a 3d sphere literally does not exist in the 3rd dimension. One could argue that it is the 3rd dimension of the sphere that gives rise to the 2d sphere and in that sense it might work but it is awkward IMO. That is, S_1 x S_1 x S_1 = S_3 and you could say that it is the x S_1 as "existing in the 3rd dimension but one could then also say it exists in the 2nd dimension since S_2 x S_1 = S_3 or even in the first dimension since there is no preference for singling out the 3rd.
@@MDNQ-ud1ty Thanks for chiming in =)
I see a couple of potential misconceptions in your post. Let me see if I can say something about them.
Let me clarify a few things.
The notion of dimension is defined for vector spaces. It’s the minimal number of elements in that space, that form a basis.
Those elements are typically unordered. The set of elements is also not unique. We can get a different basis by linear transformations.
There is a simple very common class of vector spaces defined on the n times Cartesian product of R(the real numbers), called R^n.
Those vector spaces come with a natural basis {(1,0, …,0),(0,1,0, …,0), …,(0,0, …,1)}.
And that basis comes with a clear order. That order can be used to define “components” of a vector.
But, as vector spaces two copies of R^n are isomorphic as vector spaces if they are related by a bijective(invertible) linear transformation.
R^n can be equipped with several different basis sets.
The circle, the sphere and higher spheres are not vector spaces though. We can think of them as for example sets, topological spaces, manifolds or differential manifolds. It depends on how much structure we want to equip.
Manifolds are roughly speaking spaces that locally in a neighborhood around each point look like some R^n (Omitting details).
Since they locally look like R^n, we can define the dimension of the manifold to be n.
Anyway. The 2-sphere is 2-dimensional in the manifold sense.
But it is not a vector space.
The the 2-sphere doesn’t inherit the order of the coordinates that we might have appealed to in R^3.
We might try to equip the sphere with a coordinate system.
If we could do that maybe we can call the first coordinate the first dimension and the second coordinate the second dimension and be happy.
But there is no naturally preferred coordinates like in R^n.
We have to make an arbitrarily choice.
Worse still, one can show that it is impossible to construct a single coordinate chart that covers the entire 2-sphere (at least on point will be left out).
Ok. Let’s say we ignore this complication, i.e. we remove a point from the sphere, introduce coordinates covering the rest, pick an order for those coordinates, then use those coordinates to construct some notion that we call a dimension.
Even so, how is this notion of dimension related to “the dimensions” of R^3?
Not in any way really!
We’ve just made up a new object and called it the same name as another object. And there is no obvious way to match them up.
So the better way to do it is to say that the notion of “the n:th dimension” doesn’t make sense and we should avoid it.
So? Does it make sense for spacetime then? Is time the 4th dimension? Well, one can make a better case for it since there is structure that do impose an order.
But it’s not as simple as saying the time dimension is the first component of a basis.
As a vector space we can use any bases. But we can use the extra structure (like the metric if you know what that is) to transform an arbitrary basis into a basis with one time like vector and three space like vectors (don’t worry about the details to much if the words are unfamiliar).
That separation isn’t unique though. But we can choose to only use basis sets that are separated like that.
Then we must make sure that we don’t use general linear transformations since they would destroy that separation, but instead we restrict to some smaller set of transformations.
But time still isn’t a single unique direction other than in a specific coordinate system. And there is no reason to really prefer a specific coordinate system.
Anyway, in general relativity spacetimes are manifolds and not vector spaces. Dimension is just a number and you’re free to pick a basis for tangent vectors at each point. While each such local basis has a time-like element the choice is not unique.
The notion of the n:th dimension just isn’t a good one there either.
Finally.
S^1 x S^1 is not isomorphic to S^2. (!)
And
S^2 x S^1 is not isomorphic to S^3.
I think that’s just a mistake.
Thanks for taking the time if you got this far.
Its a well established phrase. I also hate when people write f = O(n) instead of f "is and element of" O(n), but all of this is here to stay.
@@HyperFocusMarshmallow You are arguing over nomenclature and being pedantic. Dimension has been used before vector spaces were invented and they are not just used for vector spaces. Many times people sacrifice exactness for expediency when things are obvious. It's quite common even by professional mathematicians.
By S^1 x ... I do not mean the topological product but the Cartesian product of sets (S^1 being parameterized by r in [0,1)).
It was a shorthand. You seem not to be ok with shorthand for some reason... you will learn one day. Nothing in life is perfect.
@@MDNQ-ud1ty Sure. Pedantry was kind of in the premise of my first post, so I assume we were playing that game. It’s not to serious or anything, it’s all for good fun. Trying to get at pedantic (in the good sense) versions of these concepts can be useful for learning or staying sharp. Once we can be pedantic and we know how to really do it, that’s when we can afford to be sloppy.
But sometimes we just can’t be bothered and that’s fine too.
If I misinterpreted what you were saying that’s another thing entirely though. I wasn’t trying to twist your words or anything. I just interpreted them as best I could.
I assume you’re right that dimension has been used before vectors paces. I do know the concept crops up in some other places in mathematics but I don’t know a detailed history of it or versions of it before modern math.
Maybe I have my head in the mathematical sand but I do think vector spaces or manifolds is the relevant notion here though.
I’m perfectly fine with using loose sloppy language. Of course the potential issue is that we are actually saying things that are nonsense and that we don’t actually know the pitfalls involved or how to rephrase it in the ways that does make sense. Or just that there might be miscommunication. Sometimes we take that risk. Saying that “I’m triggered…” is kind of just my way of phrasing that there is such fudging going on.
About the S^1 x …
Sorry if I misinterpreted you. I have nothing against using shorthand I just didn’t understand which one you used. It happened to be ambiguous in this case apparently and I read it the way I thoight the shorthand was supposed to be read and answered accordingly.
I’m still not quite sure what you intend actually. I think the topology is quite central for distinguishing a circle from an interval or even from just a point set.
I’m not sure in what way it would be an n-sphere without such information. Why not just a box or again a point set.
Maybe I’m misreading you again. If so, I apologize.
You don’t have to answer if you don’t want to, I’m just here to enjoy some math talk and have fun. Maybe learn something.
Since I not getting your intention apparently, maybe there is an opportunity to learn something =)
Hope you’re having a good day otherwise! Cheers!
Great video, thanks for the upload!
Really great video! Should get more views
Thanks!
I just started watching your some1 submission and instantly found a connection to this video too. Namely the spin 1/2 phenomenon, I believe, is captured precisely in the \theta/2 within q_\theta as flashed on the screen at 10:48 since this covering S^3 \to SO(3) is the spin double cover of SO(3)!
@@loop2427 Yes, you are right! I was surprised to find your video for the same reason
unfortunately, I don't have the DNA to understand this.
i blame my parents for not doing embryo selection.
I added this to the list of all SoME1 videos that I personally could find thus far. @
th-cam.com/video/MsNQtj3zVs8/w-d-xo.html
Love this Video!!!
Are you planning on doing other videos?
I know that I use math in organic chemistry but my hats off to you guys 😅 I read about, study and have experienced psychedelic experience and I fundamentally understand the concept of infinity in my own way but the way mathematicians deeply analyze and comprehend these complex ideas through numbers and theory is something I strive to understand in more clarity and find absolutely fascinating. I feel like understanding complex mathematics can help anyone who is interested better grasp the deeper concepts about the universe in a more structured way and although my brain is less rigid it's more creative and abstract maybe because of that I find myself curious yet confused. Idk I'm trying to wrap my head around what you are saying and I feel like I kind of get it but it's like you said very hard for someone who exists in the 3rd dimension (arguably 4th) to understand beyond what they can visualize. Idk I nedd to read more 😅 obviously lol
Anyone else here because they're math nerds and not because it's a fad subject?
Nice! Algebraic topology is fascinating.
This is making the world pacman you pop out the bottom and pop in the top and it's a repeating pattern pattern every 4 lines has 2 and the two two's are a mirror image ,you need less math to understand the world in only imagined.
yad think thered be an infiite number off homotopographic loops in every dimension
What the hell
Why is it eight? Why not nine, or seven? Is it because its a power of two? If yes, then why not four or sixteen???
Holy shit this is perplexing.
Thank you. I think I want to study topology in school. Cheers, keep up the good work. Very well paced
I think broke my brain trying to understand this.
What a fine ass introduction to bott
Very easy to understand! 🎉 thanks mate
Ok, someone call Terrence 😂
I dont get any bit of it. But the end sort of reminded me of an octave on a piano. How the notes repeat itself eventually. Guess how many notes there are in an ocatave....8.
Great video, hope it's first of many!
What is the paper you show at 10:46 ?
Thanks Rad!
It's just a page I wrote for the video. You can download it here if you wish:
drive.google.com/file/d/1EP_17XTaCR7IVw5oPXNJgGdm8iTiYGnZ/view?usp=sharing
This is super cool
like I've always said .. cycles within cycles
What about the symmetries of a square being generalized to SQ(infinity), is there periodicity there? Is SQ(n) even different from O(n) in the first place so that this question is valid?
not a bad question, but unfortunately it wouldn't work with the square since the symmetries of a square are what we call discrete.
This just means that it's finite and it's not like a 'space'. For example, the symmetries of a square is a really nice group called the 4th dihedral group (4 because there are 4 vertices).
This just contains 8 symmetries, and hence discrete.
So you can't study loops in this space because it's just 8 points - in a sense there isn't enough space to 'move around'.
@@blakenator123 unfortunate
But, you can define that group, since the symmetry group of the square does embed inside the symmetry group of a cube, and so on for all dimensions. It’s just that it wouldn’t be a connected topological group like the sphere is, for the reasons already pointed out. It would still be a group though
Yes, you can do this. As already stated, you do not have fundamental groups but you do have homology groups(which are the discrete form of the fundamental group). So there will be some discrete analog but is it periodic? I don't know. It's very likely that there will be some analog periodic structure since the hypercube is homeomorphic to the hypersphere. E.g., any embedded manifold in R^n should take on these symmetries of the sphere in some way.
E.g., think of the hypercube. Effectively it will sample any "symmetry loops" since some symmetries of the hypersphere will be valid symmetries for the hypercube(just inscribe the hypercube in to the hypersphere).
Hurewicz's theorem states that there is a homomorphism between the two so what will happen is that the periodicity that you get from O(infinity) will be collapsed down in some way to the periodicity that you get from the "SQ(infinity)". It may destroy the periodicity or retain it completely. You'd have to investigate to see. In the case of a hypercube it might maintain it completely or modify it slightly. For other shapes it might completely change it.
But there definitely is some relationship. You'd have to work it out to actually know.
@@MDNQ-ud1ty how do you get nontrivial homology groups on a discrete space though?
Isn’t the orthogonal group called the general orthogonal group GO(n)
At least not by mathematicians... We have GL, SL,, O, SO, SP and some other cool groups.
SO mentioned
Sir which software did u use for animations?
I used adobe after effects for pretty much everything.
@@loop2427 and u prolly kno presh twakar myd right?is the videos of that possible from adobe effects?
@@gardenmenuuu Not sure, sorry
Awesome
Isn't this just a random nonsense pattern that emerges when you compound error upon error? For an example, just because you can represent a 2D sphere and a 3D sphere and their symmetries mathematically, it doesn't follow that you can do it with a "4D sphere" - a completely nonsense object which isn't real but which you can represent in a neat mathematical equation anyway. Multidimensionality nonsense from computer nerds who like their cool geometric screensavers is the weak spot in this "periodicity observation". Perhaps the periodicity of symmetries appears because there aren't more than 3 dimensions.
You can't see imaginary numbers, or quaternions yet those exist. Mathematics is a subject that given a set of axioms, what can you logically conclude. High dimensions "spaces" exist in many data sets.
@@jmd448 Dimensions are just subdirectories within an array, e.g. you've got an array made up of n elements, and each element is a sub-array so that e.g. n1[0...99], and each element of that can be an array, etc. This is not really dimensionality - it's just computer science. What's actually happening is that data is being written to addresses in a computer system that exists in a purely 3D world.
So you are a strict materialist, believing that concepts aren't real? You believe math is invented not discovered?
@@jmd448 I think with maths there is a lot of scope to get lost in some esotericism which has no basis in reality. Before maths, it was basic reasoning logic, resulting in wrong conclusions with the right method. Example: man 1 says the Qur'an is true. Man 2 says no it isn't. Man 1 "proves" it's true by saying you can see the sun and the stars, and the Qur'an says there are sun and stars, and by inference everything it says is absolutely true. He wins the argument because it's good logic but bad postulates. In this case what we're doing is interacting in a 3D world which has computers in it, and saying there's a 4D world. This is exactly the same game neoplatonists played, to prove there were 7 layers of higher worlds above this one. Yet they're sitting here in this one, talking about the others which supposedly exist, and being "right". There are many other examples - the process always results in a bad cosmology or ontology. These people are all lost in some esoterism and they become the bishops of their day - social status thing for survival and reproduction of their ape genes, nothing more.
This is such a weird argument. Seems like an AI output
omg this is sooooooo boring
Omg please shut up
I was reading this comment and then my neighbor came and we read it together. He said this comment really changed his life and it touched my heart. My village people are so grateful. Am proud to say cool comment wow thanks for sharing.
So basically we live in a 3D reality with a fourth dimension of time but also 11D reality with a 12th dimension that is time?
I wouldn't say that Bott periodicity has anything to say about the physical world we find ourselves in, outside of the symmetries of spheres.
@@loop2427 forgive me but that seems like a cop out to me, these objects are supposed to be geometries in space (I'm obviously kidding about the previous statement lol, but you were the one who said they were equivalent haha), more likely this is an artifact of the framework itself... I've got no clue though, am punching far above my weight.
Like perhaps the fact that we define a dimension as 90 degrees apart from each other and in our space that means there are three dimensions with six total antipodes which divide space into... Octants?
I'm guessing that time may not exist in certain dimensions, and yet may take up one or more dimensions of the number of dimensions under consideration..
I didn't realise 03 was D1 of the periodicity table...
Is the hole in O3 technically: "the necessity of having to rotate the sphere to achieve a loop"?