What Is a Manifold? • Hacker News Board

Re: What Is a Manifold?

#31

Post by falcrist » Sun, Apr 14, 2019, 1:26 PM UTC

wcoenen wrote: post #19

_v7gu wrote: post #12

All is well until the poor bug finds out it's on a Klein Bottle, or worse yet RP2. Is there any way to distinguish between orientable or nonorientable surfaces when you're simply walking on it? If the bug was also 2d, I might have had some ideas hut I'm totally blanking for 3d.

For a Mobiüs strip, the bug would find the world had been mirrored along the width of the strip if it made a round trip and came back to the same point. Other bugs that stayed behind would claim that the traveller itself had become mirrored. (If you do the experiment with paper, the bug would be on the other side of the paper from where it started, but I don't think this concept of "side" exists when the strip is des…

I'm pretty sure the property of sidedness comes up when you embed the manifold in euclidian space. The property seems to be related to orientability.

Re: What Is a Manifold?

#32

Post by amelius » Sun, Apr 14, 2019, 1:27 PM UTC

Topolomancer wrote: post #22

JulianWasTaken wrote: post #11

Earlier quoted context omitted.

Besides the other answers, things also start to get interesting when you add some qualifiers. E.g., asking for a "differentiable manifold" now asks for manifolds with enough additional structure to do calculus on top of.

Good point; I originally wanted to talk about the Poincaré conjecture as well, but then I realized that this would make the post even longer. Do you have some ideas about other interesting topics?

> Good point

Can a point be a manifold?

Re: What Is a Manifold?

#33

Post by patrickthebold » Sun, Apr 14, 2019, 1:33 PM UTC

lordnacho wrote: post #8

Is the conclusion that a manifold is a space that looks like a lower dimensional space that it really is?

Yes and no. The definition doesn't require that what you have is embedded in a higher dimensional space. Though that's a good source of examples.

https://en.m.wikipedia.org/wiki/Whitney_embedding_theorem

Basically says that many manifolds can be described as a lower dimensional submanifold of some euclidean space.

Re: What Is a Manifold?

#34

Post by tyingq » Sun, Apr 14, 2019, 1:35 PM UTC

joncrane wrote: post #25

It's funny, because in my much younger days I dropped out of college and worked as a mechanic for 3 years. In the automotive world and internal combustion World, manifold has a much different meaning, though related. Basically, a manifold means something that takes the flow of gases from a one-to-many or a many-to-one. An intake manifold takes one single entry point for air feeding the engine and splits up into a sep…

That's what manifold means to me as well.

Though, there are exceptions to the "one" part, like intake manifolds with two inlets (dual plane is common), exhaust manifolds with two outlets (common on inline 6 engines), and also some really odd setups: https://speedmaster79.com/media/catalog/product/cache/1/thum...

Re: What Is a Manifold?

#35

Post by mlevental » Sun, Apr 14, 2019, 1:42 PM UTC

xelxebar wrote: post #28

If anyone has questions (especially technical ones), the ##math channel on freenode is pretty phenomenal. It's quite active, with quite a few grad students, across a range of fields. The best part is how friendly and helpful they are to learners. Anyway, highly recommended.

this in general accurate but I would warn people to have a thick skin in there because there are some seriously toxic people in there (TRW3W or whatever his/her name that was fairly knowledgeable but hung out in there I think purely to assert his/her superiority).

Re: What Is a Manifold?

#36

Post by commandlinefan » Sun, Apr 14, 2019, 1:59 PM UTC

joncrane wrote: post #25

It's funny, because in my much younger days I dropped out of college and worked as a mechanic for 3 years. In the automotive world and internal combustion World, manifold has a much different meaning, though related. Basically, a manifold means something that takes the flow of gases from a one-to-many or a many-to-one. An intake manifold takes one single entry point for air feeding the engine and splits up into a sep…

I actually clicked the link expecting exactly that explanation.

Re: What Is a Manifold?

#37

Post by krosaen » Sun, Apr 14, 2019, 2:00 PM UTC

Another connection to make: quaternions can be represented as a Lie group, which is a (smooth differentiable) manifold. This means representing 3d orientation using quaternions works better for optimization than e.g Euler angles (roll, pitch, heading) in things like SLAM.

Re: What Is a Manifold?

#38

Post by [deleted] » Sun, Apr 14, 2019, 2:07 PM UTC

hansen wrote: post #26

math_and_stuff wrote: post #24

Earlier quoted context omitted.

Both of those are typically referred to as manifolds with boundary. The setting of Stokes Theorem is often manifolds with boundary.

Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).

[deleted]

Re: What Is a Manifold?

#39

Post by Topolomancer » Sun, Apr 14, 2019, 2:17 PM UTC

joncrane wrote: post #25

It's funny, because in my much younger days I dropped out of college and worked as a mechanic for 3 years. In the automotive world and internal combustion World, manifold has a much different meaning, though related. Basically, a manifold means something that takes the flow of gases from a one-to-many or a many-to-one. An intake manifold takes one single entry point for air feeding the engine and splits up into a sep…

Sorry, I was not aware of that! It was not my intention to mislead you. Hope you enjoyed the article nonetheless :-)

Re: What Is a Manifold?

#40

Post by Topolomancer » Sun, Apr 14, 2019, 2:17 PM UTC

commandlinefan wrote: post #36

joncrane wrote: post #25

It's funny, because in my much younger days I dropped out of college and worked as a mechanic for 3 years. In the automotive world and internal combustion World, manifold has a much different meaning, though related. Basically, a manifold means something that takes the flow of gases from a one-to-many or a many-to-one. An intake manifold takes one single entry point for air feeding the engine and splits up into a sep…

I actually clicked the link expecting exactly that explanation.

Sorry! It just goes to show that I am living in a filter bubble of mathematics. My intention was not to deceive you :-)