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What Is a Manifold?

bastian.rieck.me

41–50 of 115 posts

Re: What Is a Manifold?

#41
post #32

Earlier quoted context omitted.

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?

Yes, but it's a simple one of dimension zero. There's not much to do here---every 'neighbourhood' of the point is exactly that: just the point.

(not that there are different 'classes' of manifolds out there; I am not sure if the point would qualify as a Riemannian manifold, for example)

Re: What Is a Manifold?

#42
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.

Are you familiar with the construction of a torus by taking a square and "gluing the edges together"? It's effectively the game Astroids, where going off one edge of the screen wraps around to the opposite edge. If, instead, you twist the edges 180 degrees before gluing, we then get the Klein Bottle. I.e. when you warp from, say, the left edge to the right edge, you also get flipped vertically. I'm guessing you're al…

I took that idea to its extreme and released "Geodesic Asteroids" on iOS (free) a couple of years ago. You can play on the 2D map or an embedded torus. There is also an old blog post I wrote at the time: http://nbodyphysics.com/blog/2015/03/06/asteroids-on-a-torus...

Re: What Is a Manifold?

#43
post #26

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).

I thought a manifold with a boundary would still be a manifold, but its boundary has to satisfy a dimensionality condition. For example, the 2D disk is a 2-manifold with a 1-dimensional boundary. Strictly speaking, this is a _topological manifold with a boundary_, though.

Re: What Is a Manifold?

#44
"Quick" etymological fact about what mathematicians in English call Manifolds:

If one would translate Riemann's original German word for them, "Mannigfaltigkeit", it would translate to "manifoldyhead", or, more understandable to the speaker of Modern English:

Manyfoldyhood.

(Think of -head as in "Godhead", not as in "Brotherhood", in the same sense that the "ring" in "algebraic ring" refers to "ring" in the sense of "smuggler ring", not in the sense of "gold ring". The two suffixes merged in English, and have very different and rather complex etymological origins.)

While this word sounds somewhat ridiculous and perhaps a bit infantile in English, it, in my opinion, conveys what we mean by them significantly better.

Clifford tried to accommodate Riemann's highly specific word choice in his translation, as Jost notes:

"The English of Clifford may appear somewhat old-fashioned for a modern reader. For instance, he writes “manifoldness” instead of the simpler modern translation “manifold” of Riemann’s term “Mannigfaltigkeit”. But Riemann’s German sounds likewise somewhat old-fashioned, and for that matter, “manifoldness” is the more accurate translation of Riemann’s term. In any case, for historical reasons, I have selected that translation here."

Unfortunately, neither Jost (a native German speaker!) nor Clifford realized that English can and does accommodate Riemann's exact meaning directly.

To make a comparison which might require quite a bit of German knowledge beyond high school education to describe in exact linguistic terms, but which native speakers should hopefully find intuitive to distinguish (I at least, do):

Translating "Mannigfaltigkeit" as "Manifoldness" seems equal to mistranslating "Geheimnis" as "Secrethood" and "Geheimheit" as "Secretness", whereas the opposite pairing would yield an accurate (albeit not necessarily immediately apparent—in terms of the differences between the two—to the English native speaker, unless explicitly pointed out) translation. So much for the last suffix, but that still leaves insertion of the one before it (' * foldy * ' instead of ' * fold[( * )]' unclear.)

One may observe the difference involved there by a converse example, also involving "Geheim", as well as the root word of it, "Heim", by dragging forth a rather rare and archaic—but none the less highly likely intuitive to the native speaker—word:"Geheimig". Whose meaning starkly differs from both "Geheimnis" and "Geheimheit".

For further language related hijinks related to /Manyfoldyhoods/, see:

A) The Dutch word for them, which would back translate to "Variety"; and

B) This quote by Poincaré:

"I prefer the translation of Mannigfaltigkeit by multiplicity, because the two words have the same etymological meaning. The word set is more adapted to the Mannigfaltigkeiten considered by Mr. Cantor and which are discrete. It would be less adapted to those which I consider and which are discontinuous."

(As I don't speak French, I can't make much of any statements about the accuracy about the etymological claim by Poincaré, so I'll close with another Poincaré quote instead:

" Mathematics is the art of giving the same name to different things."

  —Henri Poincaré)

Re: What Is a Manifold?

#45

"Quick" etymological fact about what mathematicians in English call Manifolds: If one would translate Riemann's original German word for them, "Mannigfaltigkeit", it would translate to "manifoldyhead", or, more understandable to the speaker of Modern English: Manyfoldyhood. (Think of -head as in "Godhead", not as in "Brotherhood", in the same sense that the "ring" in "algebraic ring" refers to "ring" in the sense of…

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_50781

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).

regardless in english manifold already exists as an adjective and probably a good translation of the original german (if i'm to understand you correctly) would be to simply describe a space as manifold rather than a manifold.

Re: What Is a Manifold?

#46

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 :-)

No need to apologize. Words have different meanings.

Re: What Is a Manifold?

#47
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 many cases where you can’t efficiently extract other calculations out of the Hough transform with effectively inverting it entirely.

Re: What Is a Manifold?

#48

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 took apart a furnace once to replace an igniter.

I was thinking I was going to find a series of pipes or tubes fitted together.

Instead I found just two pieces of pretty thick sheet metal, close to plate thickness. They just went through a metal bender to form the pipes, and bolted together with a gasket in between.

Maybe that's how it got started. A plane, wrapped around something.

Re: What Is a Manifold?

#49
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…

Representing rotations via Euler angles can lead to gimbal lock[0], whereas using quaternions doesn't. The best explanation I could find quickly is this one:

https://mathoverflow.net/a/95908

[0] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathema...

Re: What Is a Manifold?

#50

I think everything said here can applies to just topologies.

That isn’t really true. Dimension isn’t so well defined in topology but is reasonably straightforward with a manifold. The article also touched on geometry (sum of angles of a triangle). To be topological a bunch of things must change. The bug walking example doesn’t work and is demoted to an analogy. All notion of distance and direction is lost. If the article were about topology then it would probably be talking ab…

If I'm not mistaken, dimension can be defined easily for a topological manifold, which is actually a more fundamental structure than a differential manifold. (The former requires that chart overlaps be homeomorphisms, i.e. continuous bijections), while the latter requires that they be diffeomorphisms, i.e. smooth bijections). Smooth manifolds don't form a nice category for technical reasons, but one can think of a forgetful functor from differential manifolds to topological manifolds, and dimension being defined in the latter.
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