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What is a manifold?

quantamagazine.org

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Re: What is a manifold?

#121

Post by frutiger »

idoubtit wrote: post #89
tetrisgm wrote: post #5

This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition. It’s actually much more well written than the majority or articles we usually come across.

Is that really a good article? I thought it was average. It had some big flaws but was probably still informative for readers with no mathematical knowledge in the domain. For instance, consider the only concrete example in the article: the space of all possible configurations of a double pendulum is a manifold. The author claims it's useful to see it in a manifold, but why? Precisely, why more as a manifold than as…

> BTW, I know just enough about relativity

Unfortunately this is one of those things where that knowledge is not enough.

The GR model of spacetime is that it is locally Minkowski but globally a manifold of Minkowski patches.

Re: What is a manifold?

#122

Post by mathgradthrow »

kristopolous wrote: post #75
mathgradthrow wrote: post #66

What a terrible article. Can anyone who is not a mathematician tell me one thing they learned from this? The naked term "manifold" in its modern usage, refers to a topological manifold, loosely a locally euclidean hausdorff topological space, which has no geometry intrinsic to it at all. The hyperbolic plane and the euclidean plane are different geometries you can put on the same topological manifold, and even does n…

Well as a non mathematician all I saw in your description was opaque jargon. "locally euclidean hausdorff topological space" means nothing to me. It'd be like if I asked what the Spanish word "¡hola!" meant and the answer was in evocative Spanish poetry. Extremely unlikely to be helpful to that person who doesn't know basic greetings. This article breaks that loop and it's refreshing to see a large topic not explaine…

If this article broke some loop, you could answer my question and tell me one thing you got out of it. What's a manifold?

Re: What is a manifold?

#123

Post by BobaFloutist »

getnormality wrote: post #21
robot-wrangler wrote: post #12

Earlier quoted context omitted.

I'm always surprised more people don't know about Quanta. Seems like it's currently the best science journalism out there, and IMO a very strong candidate for the single best place on the internet that's not crowd-sourced. The mixture of original art and technical diagrams is outstanding. Podcast is pretty good too, but I do wish they'd expand it to have someone with a good voice reading all the articles. Besides not…

It's because of their Simons Foundation support, but not only because of that. I mean, I invite anyone to name another billionaire pet project of comparable quality.

I like what MacKenzie Scott is doing in the space of "What it takes to actually spend down billions of dollars"

Re: What is a manifold?

#124

Post by [deleted] »

mmooss wrote: post #88
senderista wrote: post #69

Earlier quoted context omitted.

The problem is that this global coordinate system isn't a continuous mapping (see the discontinuity of both angular coordinates between 2*pi and 0). Manifolds are required to have an "atlas"[0]: a collection of coordinate systems ("charts") that cover the space and are continuous mappings from open subsets of the underlying topological space to open subsets of Euclidean space, with the overlaps between charts inducin…

This part I don't grasp: > this global coordinate system isn't a continuous mapping (see the discontinuity of both angular coordinates between 2*pi and 0). I'm guessing that the issue is that I don't know your definition of 'continuous'. I believe every point on the planet (sphere, for simplification) has unique corresponding coordinates on the map projection (chart). The only exceptions I can see are, A) surfaces pe…

[deleted]

Re: What is a manifold?

#125

Post by britzkopf »

Every time I try to get some handle on the essence of this topic I fail. No different here. In the second paragraph it defines manifolds as "... shapes that look flat to an ant living on them, even though they might have a more complicated global structure"

So manifolds are complicated shapes that are at large enough a scale that an ant (which species?) will think they're flat....ok

Re: What is a manifold?

#126

Post by seanhunter »

lisper wrote: post #111
seanhunter wrote: post #84

Earlier quoted context omitted.

> But I would go with something more like: tensors are a way to represent vectors so that the representation of a given vector is the same no matter what basis (or coordinate system) you choose for your vector space. That's just incorrect though for a couple of reasons. Firstly, a vector in the sense in which it is used in physics is a rank 1 tensor so it has this transformation behaviour just like other higher order…

> That's just incorrect though Quite possible. But that's in no small measure because I have yet to find an actual cogent definition of "tensor" that distinguishes a tensor from an array. (I have a similar problem with monads.) > So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe OK, but then "the reality of a tensor" not depending on the…

No it’s very important for physics to have a mathematical object that doesn’t change so that you can represent these characteristics of the universe that don’t change. For every observer in every reference frame even though they will use different basis vectors and different components, the combination of basis vectors and components will be the same. That’s extremely powerful. Try the video I linked a few posts above for what I think is a really excellent explanation of what a tensor is (using practical household objects to illustrate everything practically). I think you’ll get it.

Re: What is a manifold?

#127

Post by kristopolous »

mathgradthrow wrote: post #122
kristopolous wrote: post #75

Earlier quoted context omitted.

Well as a non mathematician all I saw in your description was opaque jargon. "locally euclidean hausdorff topological space" means nothing to me. It'd be like if I asked what the Spanish word "¡hola!" meant and the answer was in evocative Spanish poetry. Extremely unlikely to be helpful to that person who doesn't know basic greetings. This article breaks that loop and it's refreshing to see a large topic not explaine…

If this article broke some loop, you could answer my question and tell me one thing you got out of it. What's a manifold?

I'm pretty confident I'm not going to give a definition that's suitable for you and I don't wish to engage with diminutive minutiae or arcane jargon in a hostile combative atmosphere.

I don't think I can improve this silence.

Have a good day.

Re: What is a manifold?

#129

Post by d4rkn0d3z »

Manifold: Any m dimensional hyperplane embedded in an n dimensional Euclidean space, where m is less than or equal to n. More simply put, a manifold is any set that can be continuously parameterized, with the number of parameters being the dimension of the manifold.

A continuous manifold will have a line element that allows you to compute distances between its points using its parameters. The simplest line element was first written down by Pythagorus I think, it allows you to compute the distance between two points in a flat manifold. In physics we do away with gravitational forces by realizing that masses move along geodesics (shortest paths) of a manifold, hence the saying,"matter tells spacetime how to curve and spacetime tells matter how to move". We stich together large curvy manifolds like a patch quilt from the locally Euclidean tangent spaces that we erect at any point.

Re: What is a manifold?

#130

Post by ChadNauseam »

lisper wrote: post #39
elashri wrote: post #6

This reminds me of how physicists will define a tensor. So a second rank tensor is the object that transforms according as second rank tensor when the basis (or coordinates) changes. You might find it circular reasoning but it is not, This transformation property is what distinguishes tensors (of any rank) from mere arrays of numbers. Looking at things from abstract view does allow us not to worry about how we visual…

> You might find it circular reasoning but it is not Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means .

Of course, there are some definitions that reference themselves that are not circular, or alternatively, some circular definitions that are useful. For example, the definition of "ancestors" might be "your parents and your parents' ancestors" (with the implication that if you don't exist, you don't have any ancestors). But I agree that this is not a useful definition for "tensor".
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