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What Is a Manifold?

bastian.rieck.me

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Re: What Is a Manifold?

#101

Post by yiyus »

mlthoughts2018 wrote: post #47
krosaen wrote: post #37

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.

What do you mean “works better”? Symbolically simpler once you know the requisite math behind Lie groups? In terms if computation, why would one representation be any different than the other? Unless you mean there’s a subset of certain computations where that representation is better but also a subset where it is worse? Similar to how there are cases where a Hough transform allows a more efficient calculation, but m…

They are more practical, specially when using a computer.

Besides the gimbal lock problem mentioned in another problem, Euler angles are not easy to compose. It is also not obvious how to calculate the misorientation between two rotations with Euler angles, but it is trivial with quaternions.

Other approaches have their own problems. For example, matrices are more expensive to use in a computer (normalizing a quaternion to avoid rounding errors is much faster than normalizing the equivalent matrix, and quaternions need less memory), and Rodrigues vectors, while usually very useful, present the problem of infinite values for rotations of 180 degrees (which, in my opinion, should not be a problem, but it is in many programming languages).

At the end of the day, quaternions present the best compromise.

Re: What Is a Manifold?

#102

Post by tgb »

improbable22 wrote: post #91
tgb wrote: post #88

Earlier quoted context omitted.

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.)

My background is in riemannian geometry so I never had to worry about lack of a tangent space. Certainly you can define orientability of some topological spaces that aren't even manifolds. I'd forgotten but you can even define a local orientation at a point for a general topological manifold in terms of it's top-dimensional homology. I think that's the most general situation it makes sense in, you need a well defined dimension to consider this.

Re: What Is a Manifold?

#103

Post by joncrane »

tyingq wrote: post #34
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...

That picture looks like an intake for a V8 and each trumpet is still for an individual cylinder; it's just that the intake ports on the heads are paired up for manufacturing and thermal efficiency.

Re: What Is a Manifold?

#104

Post by simonh »

Razengan wrote: post #95
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?”

Yes, the original Old English was Manigfeald which means exactly that.

Re: What Is a Manifold?

#105

Post by tomxor »

adrianratnapala wrote: post #85
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…

> Basically, a manifold means something that takes the flow of gases from a one-to-many or a many-to-one. A perfectly sensible meaning given the construction of the word. There is something about mathematics and linguistics (and to some extent CS) that encourages the creation of confusing, meaningless names like "accusative (case)", "(algebraic) ideal" and "(geometric) manifold".

I don't think "encouragement" is quite right, ambiguities are somewhat the default - it's the opposite of encourage, an absence of something, there must be a competitive force present to reduce their likelihood.

I think there are less ambiguities in common language because they share so much context and compete; where as there is enough separation between certain disciplines for the semantics of esoteric words to evolve and coexist independently without issue until viewed externally where it appears ambiguous - this is even true for mathematical notations.

Re: What Is a Manifold?

#107

Post by ginnungagap »

amai wrote: post #99
hopler wrote: post #89

Earlier quoted context omitted.

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.

A hypersurface is usually a codimension one object, while a manifold can have bigger codimension or even not be embedded in an higher dimensional space at all

Re: What Is a Manifold?

#108

Post by commandlinefan »

Topolomancer wrote: post #40
commandlinefan wrote: post #36

Earlier quoted context omitted.

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 :-)

Not at all - I got more than I bargained for and learned something new!

Re: What Is a Manifold?

#109

Post by crdrost »

sideshowb wrote: post #100
crdrost wrote: post #59

Earlier quoted context omitted.

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 val…

Yes, the event horizon of a black hole is precisely this sort of wall of death. Indeed there is an active debate in the literature about whether outside observers ever see black holes because from the outside perspective infalling mass should become “frozen” on the surface, with Liu and Zhang very recently in 2009 arguing that if you consider a shell of nonzero thickness falling symmetrically into the black hole then as this mass approaches the event horizon the event horizon actually expands to gobble up some of the matter—this paper is then cited and refuted in a 2011 paper in the same journal by Penna, who argues that they got one little detail very wrong and that once you correct for this you do indeed see the matter “frozen” on the wall-of-death; see [1] for a PDF of this paper.

The z = -c² / A relation does hold even when you transfer from the first-order-in-β transform to the full Lorentz transform; for an outworking from the inimitable John Baez, see [2].

I did mean that we can’t see beneath our feet, but I see your point: at g = 10 N/kg this term c² / g is something like 10^16 meters away, way way outside of the Solar System, which is only in the billions of km large. I was purely thinking about our Schwarzschild radius, which is millimeters away from the center of the Earth: therefore we cannot see the Schwarzschild event horizon because it is nonexistent; it is “underground” but so far that then most of the mass is outside of that, so you have to recalculate and get an even smaller amount, but that then needs another recalculation... and so on. It vanishes to zero because the mass is not located in a condensed enough space.

[1] http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.758...

[2] http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/r...

Re: What Is a Manifold?

#110

Post by JadeNB »

adrianratnapala wrote: post #85
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…

> Basically, a manifold means something that takes the flow of gases from a one-to-many or a many-to-one. A perfectly sensible meaning given the construction of the word. There is something about mathematics and linguistics (and to some extent CS) that encourages the creation of confusing, meaningless names like "accusative (case)", "(algebraic) ideal" and "(geometric) manifold".

> A perfectly sensible meaning given the construction of the word. There is something about mathematics and linguistics (and to some extent CS) that encourages the creation of confusing, meaningless names like "accusative (case)", "(algebraic) ideal" and "(geometric) manifold".

Confusing is hard to argue, but meaningless is, I think, hard to defend. These all have meanings; I could speak to the latter two, and I'm sure a linguist could speak to the former. They may not be obvious meanings, but that's not the same as saying they're meaningless. (I regard much business jargon, for example, as literally meaningless, defined only in terms of other words that also seem meaningless to me; but I'm sure an MBA would take issue with that characterisation.) I don't know who coined 'manifold', but 'ideal', for example, was literally Kummer's word coined to describe things that behaved like, but weren't quite, numbers in the ordinary sense (https://en.wikipedia.org/wiki/Ideal_number )—much like the "ideal points" of hyperbolic geometry (https://en.wikipedia.org/wiki/Ideal_point).

(EDIT: Fortunately no_identd knows more about the history of 'manifold' than I do (https://news.ycombinator.com/item?id=19659571).)

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