Shaping a Universe (manifolds, and some conditions for embedding) 

Hey flammy you too watch his channel..you all are maths' birds.....hope all mathematician ever come together and project a nice discussion

You got a customer right here

@@zachstar lmao

better than 3b1b? I highly doubt that.

@@SolidSiren Absolutely. 3b1b is great, but his memes are weak.

Mr Blue is a bit misleading, if I was to preserve his direction, I needed to "distort" him as he approaches a boundary (as all boundary points are identified). Regarding the flipping, I had originally done that, but it was just too difficult to see the difference between a vertical flip and a rotation, but a horizontal flip looks more apparent. I'm alright with that, because it's really the same innate effect on the object/being (IE, they differ by a rotation within his space).

Would that projection of the cylinder not make it non-continuous? Or require overlapping at some point?

@@No-uu7wm Yes. In fact, the proper embedding would be like cutting the cylinder into a bunch of rings, then shrinking the rings as you go down the cylinder and growing them as you go up, so you can place them as concentric rings in the plane. Or in other words, deform the cylinder into a funnel shape and flatten it out onto the plane. This will leave the some point in the plane unused, but that's okay, an embedding doesn't need to be surjective.

@@alxjones Interesting! I'm a 2nd year physics student and I don't know much topology yet, as such I was just trying to intuit my way to an answer. It's good to know there is a proper way of doing the embedding. Hopefully I'll learn more topology in the future.

For the cylinder to be embedded you must maintain its essential property of being cyclic, so perhaps if you let the angle around its circumference be the polar angle and your height on the cylinder be the distance, you could convert a cylinder into polar coordinates and embed it that way, but then you would gain the property that the bottom circle forms a point, among other failures

Yeah, I was like "is that Reflection?"... And then the bass kicked in, and I was like "yaaaas"

Nah. Dopamine and its counterpart behave so wildly differently (among other chemicals) that such a person would have a horrible toxic reaction to most conventional foods

What's situs ambiguus (both sides are mirror images)?

Thanks bro!

Yes, I'm Red Gianting like crazy. I'll eventually get tired of it, but for now, consider it a new toy.

​@@EpicMathTime I love your style, play with it as long as you want :) yeah, Red Giant is awesome. Cheers

We are a super secret elite special club.

The Klein bottle contains a 3-dimensional volume, but the ambient space is 4-dimensional, so Cliff Stoll's description isn't in conflict.

Epic Math Time so it’s like a plane that has no 3D volume, but has a 2d volume(area)?

@@brunojambeiro6776 Yes, we can say that a square "contains a two dimensional volume", but if it is embedded in R^3, it is not dividing R^3 into an inside and an outside.

My statement that we could not percieve a 2d being was inspired by a conversation I just previously had. There is this popular idea that a 4d being could observe us without knowing, just as we could observe a 2d being from outside of its space without it knowing. The thing is, it is not apparent to me at all why we would have the ability to perceive a 2 dimensional object of any kind, so I don't know why this is said so frequently.

I thought it was a nice idea that Star Trek The Next Generation explored encountering a 2D creature in the episode "The Loss."

There's an invariant that has to do with the group structure of closed paths along the surface of the manifold in question. The torus can be considered as S^1×S^1 (there are two ways you can make a circle which are not "homotopy equivalent," i.e. there does not exist a continuous mapping from one to the other) which forms the free group with two generators, while the sphere can't be represented this way. In fact, any closed path on the sphere can be transformed continuously to a point (we say that these paths are "homotopy equivalent to a point"), which means that the sphere is what we call "simply connected." Intuitively, it seems like we can represent a sphere as a square with opposite sides "glued" together, as we know that going in one direction on the square should make us come out on the other side of the square facing the same direction. But if we simply identify opposite edges, we don't actually get this property in all directions - consider a slanted path: if we start somewhere near the bottom of the left edge of the square and move toward somewhere near the top of the right edge, we would not appear back where we started. Instead, we would appear somewhere near the top of the left edge. If the square represents the whole sphere, this cannot be what happens, as we can choose parameters such that this path will fill the whole space before reaching the starting point, and that's definitely not a possibility when moving in a straight line on a sphere. We can't even just identify all opposite points on the square, as this makes the surface non-orientable (this is a surface known as the real projective plane). Instead, the common approach is to identify ALL points on the edge of the square as the SAME point. This approach doesn't handle direction in an intuitive way, but being careful allows it to work

@@JPK314 Right, the Mr. Blue animation is misleading in that regard. We can't consider the square the "whole" picture. I considered having him "distort" near the edges, but I just took the square visual as being a little "window" of his manifold.

You are correct. The reason for that is that the effect becomes much less recognizable (it is not immediately apparent that the creature is not just "upside down" rather than reversed). I later realized that maybe keeping the horizontal reflection but moving up would have been best, but as you said, they are really the same innate effect on the creature (since they only differ by a rotation in his own space).

Yes (it's a matter of just going through the orientation reversing loop), but that's not nearly as cool.

Yes, he could. As long as he lives on a Riemannian manifold he could preform experiments such as parallel transport to determine the Riemann curvature tensor and differentiate which manifold he lives on up to homeomorphism. All of these terms are worth reading into if you're interested :)

A vertical or horizontal reflection really results in the same intrinsic effect on the object in its space. They only differ by a rotation within the space (and this is true of reflection with respect to any line). The reason that I chose to flip horizontally instead is that it's much more visually apparent what happens. It's a lot harder to tell the difference between a vertical reflection and simply being rotated upside down. Imagine that you are reflected, as a (presumably) 3-dimensional being. No matter how you are reflected (horizontally, vertically, depthwise, etc) you will always be your mirror image afterwards, and you only have one mirror image. The only thing that differs is your orientation within your space after being reflected.

@@EpicMathTime thank you for your answer ! I saw it upsidedown doing a Mobius strip and drawing on it by my self :)

speaking of tori people are saying problem 1 is a 2-torus but to me it seems like a torus. If you glue the top and bottom you get a cylinder, and then you glue both ends of the cylinder and get a torus.

People who write "2-torus" are just specifying the dimension, so it's not clear to me what your objection is here. A torus does not have the feature of all geodesics being closed. Moving vertically gives a closed geodesic, moving horizontally gives a closed geodesic, but almost any other direction is not a closed geodesic. It is true that Mr. Blue could live on other manifolds, though, but none of them have a straightforward representation as a surface in Euclidean geometry. In total, he could live on: (i)The 2-sphere, (ii) the complex projective plane, (iii) the quaternion projective plane (iv) the octonion projective plane. None of these embed in R^2, and they are rather exotic for this video. The sphere is the "obvious" answer, but generally speaking, the number of closed geodesics won't determine a single topology.

@@EpicMathTime for some reason I thought a 2 torus was a genus, that's what I get for not having taken topology yet. also yeah I know realized my mistake with the torus example oops

Where can i find that explanation?

Check out "PBS Space Time" then, and I think that "Mathologer" does them too.

Similarly, if our 3d universe is spherical, then it does not embed in 3-dimensional Euclidean space, it embeds in 4-dimensional Euclidean space. EDIT: when I said that we couldn't put that "2d manifold into three dimensions," I really meant two dimensions. I'm amazed that I never noticed that I said that. We can certainly embed a 2-sphere into three dimensions, we do it all the time. 😂

@@EpicMathTime The thing is that I heard somewhere that there was a possibility to have a closed '' spherical '' universe without an extra dimension. I may just misunderstood. Thanks for the patience. I'm not a native speaker hahaha

Keep going with the content, I really enjoy it

@@marcovillalobos5177 Whether or not there "is" another dimension is a bit nebulous, what we can say is that it cannot fit into three dimensional _Euclidean_ space, but can fit into four dimensional _Euclidean_ space. This doesn't mean that there is any ambient Euclidean space that contains our universe in some physical, concrete way.

@@EpicMathTime I get it, so is possible to have the blue creature closure effect if the 2D space is not strictly Euclidean??

I agree, it was louder this time than usual, I'll tone it down.

Sure, I can do that!

Our spatial universe is a 3-dimensional manifold, and there is nothing "banned" about only considering spatial dimensions in some particular investigation or conversation. I don't know what you mean by "that manifold is not embedded in higher dimensional Euclidean space". Manifolds are not _innately_ embedded into anything.

@@EpicMathTime As I understand it, we cannot separate time from space in GR. The curvature and anything is always considering all 4 dimensions. Outside of GR it still is interesting to describe our universe as potentially being homeomorphic to a 3-sphere or such as you did in your video. I meant that the first manifold one normal y learns about are submanifolds in R^n (M= f^-1({0}) for some f where at any point the df/dx^i are linearly independent), and the universe in GR is one we would not normally think of as embedded (in this sense) in R^n. Of course the Klein bottle and Möbius strip are as such also just general manifolds, but I, at least, first thought of them as submanifolds in R^3/ R^4. However, as any d-dimensional (smooth) manifold can be embedded into R^2d (I wasn't sure about that), there is not really a difference between general (smooth) manifolds and submanifolds.

@@johannesh7610 Yes, every n-manifold embeds in R^2n, that's always true, but it is a rather "blanket statement" of an upper bound. That is, even without restrictions of properties of the manifold, the bound may be lower for a given dimension. Every 3-manifold embeds in R^5, for example. As for GR's model, I don't think looking at the spatial dimensions alone separates space from time, it is more analogous to a cross-section of the spacetime geometry, we don't have to destroy the association between those four dimensions in order to consider a smaller number of them.

Pedro Leao Is it a projective plane? Don't really know how to justify the answers, I just remember it exists and that can be obtained as a quotient space from a square lol

@@pleaseenteraname4824 I also don't know, it was a genuine question

I do think _a space_ homeomorphic to the real projective plane does this. (Namely, I am picturing a half-sphere which glues antipodal points of the cut side together). One thing that I glossed over (regrettably), is that these closed paths' connection to the topology is very tenuous at best ("straight paths" aren't preserved under a homeomorphism, for one). Even if the topology can be determined, it may only hold for a specific _geometry_ as well. I heavily implied that these paths determine the topology, but that is not always true. I think it's better to think of these results as answers of the form "given this number of straight paths doing this, what is a possible topology that is _compatible_ with this?" And yes, I think the real projective plane is _compatible_ with that.

No hassler whitney?? 😢

@@ivanjorromedina4010 Sorry ;(

@@mikhailmikhailov8781 no place for him in the olympus of differential topology??

@@ivanjorromedina4010 Well, many many people are there. Might as well bring up Milnor, Pontrjagin, Chern, Smale, Thom, etc. The Olympus of differential topology is a well populated pantheon to say the least.

I thought it was clear from the context, but it's Ronnie James Dio.

That's right, thank you for the correction.

He would never notice anything has happened to him, to him, the rest of his space has flipped, not him.

@@EpicMathTime but how does it look like when its partially flipped? kind of sounds like stuff far from him have their parts compressed, even farther from him they would look like multiple things are at the same place. im just trying to visualize it and failing miserably

If we gave Mr. Yellow unlimited vision, when he looks down this path, he will see infinitely many copies of his universe. In his vision, he will see his universe continously shrink in size to a point, and then grow back in the reverse orientation at the next copy. When he moves down this path, he never feels that he is shrinking or flipping, he always sees shrinking and flipping in the distance, it is a relative perception of that space from where he is.

@@EpicMathTime wait so...theres a place in the distance where he sees everything shrinking into a singular point, and then, everything after that is getting bigger... but will it look bigger than life or just big as in we know its far, but it grew back to original size..?

Just back to its original size.

why is a cylinder closed?

I think a non-compact cylinder would better describe the second case, so it wouldn't be closed. Also both the compact and non-compact cylinders are homeomorphic to a circular crown (is that how it's called in English?), so they can be embedded in the plane

@@pleaseenteraname4824 I think you mean an annulus or washer, I thought about it as a "punctured plane" since the map is supposed to go forever but I guess that doesn't really matter.

@@chemistro9440 it's an infinite cylinder

@@sashabell9997 Doesn't that mean that it _isn't_ closed?

The universe has three spatial dimensions.