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

A laid back and informal view of a couple of properties that a "world map" (topological manifold) could possess that require it to embed in a higher-dimensional ambient space.

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[AndrewDotsonvideos]

I'm starting a super store that sells fans as it's sole product. I'm calling it only fans. I'd love for you to be part of it.

[PapaFlammy]

Fkin sick <3 Keep it up you crazy dude!

[pendragon]

I love how it starts as a serious math video and slowly devolves into 4 dimensional fever dream with metalhead grim reaper manipulating a manifold with the force.

[Higgsinophysics]

I'm starting a Gofound me for mr yellow's upcoming psychologist bills after that diss 6:03

[cycklist]

That was superb. The very best maths channel of them all.

[carsonellaruddii]

Tool, math, universe talk, if this video could pour me a beer it would be perfect

[joshvenick]

1a. 2-torus
b. Yes, it is closed and bounded, ie compact
c. No it’s not nonorientable (it is orientable)
d. It cannot be embedded in R^2 as it is compact.
2a. 2-cylinder
b. No, it is not closed
c. It is orientable
d. Yes, it can. Let u and v be in R^2 minus the origin. Let x(u, v)=(u/sqrt(u^2+v^2), v/sqrt(u^2+v^2), ln(sqrt(u^2+v^2)))
This takes each point of R^2 minus the origin and projects it onto the unit circle, where points further from the origin are mapped towards plus infinity in the z direction and points closer to the origin are mapped towards minus infinity. I believe this is in fact a diffeomorphism, but please correct me if I’m wrong.

[timberfinn]

Tool as background music was unexpected yet awesome, great video!

[sleepheartcat]

Just some nitpicks on the animations in case anybody wonders:


The flipping on the Möbius strip and Klein bottle are also a bit misleading: Mr Yellow and Mr Green ought to have been flipped vertically instead of horizontally.

Otherwise, great video! These topics are really perfect examples of why topology is interesting :)

[matron]

Oh boy, you ain’t dead :)

Edit: Loved the video! Was fully worth the time waiting!

[uv]

Having Tool in the background is very appropriate for a math channel

[Assault_Butter_Knife]

Damn this has to be one of your best videos so far. I had no background in topology when I started this video, but understood everything perfectly. Can't wait for part 2 :D

[DannyVass]

For 1 it sounds like a 2-torus to me, which I think is closed and orientable. Since it's closed that means we can't embed it into R^2 based on what we learned in this video.

For 2 we have the surface a cylinder that extends out infinitely in either direction. Since is extends infinitely it is not closed, but the cylinder is orientable. I believe it can be embedded into R^2, one possible way is to lie the cylinder along the y-axis and then project each point from the cylinder to the plane so that the line on the opposite side of the cylinder is at infinity. Hopefully this makes sense! I would attach a sketch if I could to help explain.

I guess another way to think about it is to make an infinite cut along the length of the cylinder, then you uncurl and stretch it in such a way that it covers the whole plane.

[authentic]

Can we all appreciate the beautiful editing and hardwork he has put in to make such a great video. This channel deserves a lot more :)

[natealbatros]

i really like all the effects lol make it feel like im in a movie through its math, keep up the great work

[Tuffadandem]

I'm a Math major and literally just stumbled upon your channel. Your content is AMAZING! Keep up the fantastic work!

[ec]

Maybe people who have their insides flipped (situs inversus; about 1 in 10, 000 have this) have that condition because some 4-dimensional being came along and flipped them through that 4th dimension 🤔

[artunsaday]

You have quickly grown to be my second favorite math channel.

[sanjaycosmos]

i like you explaining the difficult things in a simple way.

[fritzheini]

very nice to see some geometry and topology