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What Is a Manifold?

bastian.rieck.me

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Re: What Is a Manifold?

#71

Post by dnautics »

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…

This is also the general usage for chemists... As a chemistry and math major during my college days it was confusing to say the least.

Re: What Is a Manifold?

#72

Post by blt »

ColinWright wrote: post #49
mlthoughts2018 wrote: post #47

Earlier quoted context omitted.

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...

euler angles are sort of the odd one out, a particularly bad way to represent rotations. Orthonormal matrices, quaternions, skew-symmetric tensors with exponential map -- all represent SO(3) without gimbal lock, with different pros and cons for each.

Re: What Is a Manifold?

#73

Post by drilldrive »

musgravepeter wrote: post #42
xelxebar wrote: post #30

Earlier quoted context omitted.

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...

Maybe it would be interesting to have somewhat of a switch between the 2D map or the torus map, in that while playing 2D you can enter a challenge state where the map switches.

Re: What Is a Manifold?

#74

Post by ErotemeObelus »

Topolomancer wrote: post #20
dan-robertson wrote: post #18

Earlier quoted context omitted.

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…

Thanks for pointing out the fuzziness! I did not want to extend this further so as not to confuse readers who have no familiarity with the subject. Of course, the fact that you cross your own path at some point is _not_ indicative of living on a sphere; there are numerous other manifolds that satisfy this (not to speak of the torus, as you yourself mentioned). I will try to be more precise about this in a subsequent…

Don't. Precision obfuscates more than it increases resolution.

You don't want to be precise. You want to increase the resolution of the key ideas being seen. This is also why parables are so popular for teaching: they're literally not true, but they resolve to an image of something that is true. Please don't introduce homotopy/homology. Lines and angles are perfect.

Re: What Is a Manifold?

#77

Post by alcolade »

If the bug were truly mathematical, he would notice the curvature of S^1 and S^2 without walking around (since curvature is defined at each point), but that is obviously very pedantic of me and shows why I wouldn't write an article nearly as interesting as this one.

Re: What Is a Manifold?

#78

Post by FearNotDaniel »

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…

Möbius. Not Mobiüs.

Re: What Is a Manifold?

#79

Post by improbable22 »

tgb wrote: post #70
a1369209993 wrote: post #66

Earlier quoted context omitted.

I think when they say "circle" they're talking about your path, not your orientation reference. That's probably wrong terminology, but it makes some sense if you're thinking about how the path has to be a closed loop.

Yes, I took "circle" to mean any closed path since the difference is immaterial in generic manifold. You wouldn't normally be talking about "orienting a circle" in the manner the original poster did, but I see now what they mean. I took it to mean "choose an orientation along a closed path" which would be impossible to do for some (but not all) closed paths in a non-orientable surface.

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 orientable, then when you take a copy of this little circle (or letter R) and go for a long walk, when you get home your copy will always match the original. That's all that orientable means. In the standard usage, it's a property of the 2D manifold, not of the particular walks you take. I think this is the point of confusion here.

Re: What Is a Manifold?

#80

Post by improbable22 »

alcolade wrote: post #77

If the bug were truly mathematical, he would notice the curvature of S^1 and S^2 without walking around (since curvature is defined at each point), but that is obviously very pedantic of me and shows why I wouldn't write an article nearly as interesting as this one.

But S^1 isn't curved. At least if we mean intrinsic curvature, which is the kind that belongs to the manifold itself.

S^2 is curved, but locally you can't tell it's not (say) RP^2.

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