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What Is a Manifold?

bastian.rieck.me

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Re: What Is a Manifold?

#91

Post by improbable22 »

tgb wrote: post #88
improbable22 wrote: post #79

Earlier quoted context omitted.

A closed path is S^1, which is certainly orientable, no matter what it's embedded into. You just draw an arrow on the line. ColinWright's point is that drawing such an arrow on a tiny circle is equivalent to drawing the letter R on the 2D manifold. It gives a local orientation to the 2D surface. (If you wish you may think of this as an arrow into some 3D embedding space, but you don't have to.) If the 2D space is ori…

Yes, you can always choose an orientation of the path but not necessarily of the manifold along the path. I understand what he was saying now and misunderstood since it is not the usual way one talks about specifying an orientation at a point and in fact had imagined he was specifically specifying a "big" circle, like a nontrivial loop on the torus. You'd more typically talk about choosing a basis of the tangent spac…

Cool, I think we're on the same page. I know what you mean by "orientation of the manifold along the path" but this isn't precisely the standard usage. (My vague memory is that you can define an orientation without needing a tangent space, but I can't think of an example, so could be wrong here.)

Re: What Is a Manifold?

#92

Post by mlevental »

no_identd wrote: post #52
mlevental wrote: post #45

Earlier quoted context omitted.

i'm sorry i don't quite follow (even though i'm pretty keen on etymology in general and etymological origins of mathematical objects in particular). you say that >Think of -head as in "Godhead", not as in "Brotherhood" and >Translating "Mannigfaltigkeit" as "Manifoldness" seems equal to mistranslating "Geheimnis" as "Secrethood" and "Geheimheit" as "Secretness", but https://www.etymonline.com/word/-head#etymonline_v_…

>indicates to me that -ness is exactly the correct translation of Mannigfaltigkeit, since in the instance of godhead a accurate synonym would be godliness (and in the instance maidenhead maybe maidenly). Strictly speaking, if one entertains the distinction involved here, "Godhead", as a noun, serves as a hypernym to "God" and "Goddess"; and "Godheads" as a hypernym to "Gods" and "Goddesses", which neither "godness",…

i don't speak german at all (outside of what i've been exposed to in the sciences (ansatz, eigen, etc.) so the nuanced distinction is not clear to me (albeit i can take it on faith visavis your translation to english) but i'll say that i appreciate your bit about rings and varieties (names of objects whose relationship to the objects i've always been curious about - in the case of algebraic ring i always assumed it had something to do with closure). thank you.

Re: What Is a Manifold?

#94

Post by loeg »

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…

Ditto plumbing (liquids), e.g., https://www.supplyhouse.com/Bluefin-CM4-75PXZ-PXV-3-4-PEX-Cr...

Re: What Is a Manifold?

#95

Post by Razengan »

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…

> An exhaust manifold takes the hot exhaust from each cylinder separately and combines them into one big pipe.

Does that have anything to do with “many-fold?”

Re: What Is a Manifold?

#96

Post by loeg »

mindcrime wrote: post #81
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…

A similar notion exists in the fire service, where a manifold is an appliance for combining or splitting water supply lines. A typical example of a fire service manifold would be an appliance with 3 2.5" threaded ports and a single 5" Storz port, which can be used to either merge multiple 2.5" or 3" supply lines into one 5" supply line, or to divide the flow from a 5" line up into multiple smaller lines. Example: htt…

And on a smaller scale, ordinary water plumbing :-).

Re: What Is a Manifold?

#98

Post by Razengan »

braxxox wrote: post #53

What a great read! I've never thought about how the way we experience the surface of the earth as 2D manifold of a 3D space, and how up until recently (relatively speaking) this was unknown. It's interesting to expand on this idea and realize that maybe we are making the same mistake again, and that from our local perspective the universe is 3D, when in reality, it's a 3D manifold of a higher dimensional space. One t…

I'm always interested in this as well.

Are there any experiments to rigorously prove or disprove the existence of further spatial dimensions?

Re: What Is a Manifold?

#99

Post by amai »

hopler wrote: post #89
amai wrote: post #67

So a manifold is a surface?

A generalized surface in arbitrary dimensions. A common English "surface" maps naturally to a 2D manifold embedded in 3D space.

Ok, so a manifold is simply a hypersurface. At least I‘m pretty sure in machine learning it means exactly that.

Re: What Is a Manifold?

#100

Post by sideshowb »

crdrost wrote: post #59
braxxox wrote: post #53

What a great read! I've never thought about how the way we experience the surface of the earth as 2D manifold of a 3D space, and how up until recently (relatively speaking) this was unknown. It's interesting to expand on this idea and realize that maybe we are making the same mistake again, and that from our local perspective the universe is 3D, when in reality, it's a 3D manifold of a higher dimensional space. One t…

Sort of, not quite. The basic background is that Einstein wrote a paper taking Lorentz more seriously than Lorentz took his own work: and that paper suggested that just maybe, when you accelerate in any given direction by an acceleration A , you see all of the clocks ahead of you some distance z tick faster by a factor A z / c², where c is the speed of light in vacuum, and behind you they tick slower with the corresp…

Thanks for this, it's very clearly expressed.

I've not noticed before that special relativity implies a 'wall of death' as you put it but I see now you can get that from the Lorentz transform t'=(t-vx/c^2)/sqrt(1-v^2/c^2) combined with v=at.

Does the z=-c^2/a relation still hold in general relativity or is it modified by other terms?

Also

> we can't see beneath our feet

I presume you mean that we can, but for the values of A and z we experience here on earth, the radius of the Earth is much smaller than z so a point at a distance z "beneath our feet" doesn't exist as it's up and out the other side of the gravitational well

> And now we have a photo of a black hole to prove it!

I'm guessing you mean the event horizon is exactly such a "wall of death"?

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