What Is a Manifold? • Hacker News Board

Re: What Is a Manifold?

#11

Post by JulianWasTaken » Sun, Apr 14, 2019, 12:06 PM UTC

termiefoo wrote: post #2

what ain't a manifold at this point

Besides the other answers, things also start to get interesting when you add some qualifiers.

E.g., asking for a "differentiable manifold" now asks for manifolds with enough additional structure to do calculus on top of.

Re: What Is a Manifold?

#12

Post by _v7gu » Sun, Apr 14, 2019, 12:08 PM UTC

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.

Re: What Is a Manifold?

#13

Post by Topolomancer » Sun, Apr 14, 2019, 12:08 PM UTC

lordnacho wrote: post #8

Is the conclusion that a manifold is a space that looks like a lower dimensional space that it really is?

To some extent, yes! It needs to have a 'nice' structure, though---this is why we have the condition that is should resemble, say, an Euclidean space.

Re: What Is a Manifold?

#14

Post by JulianWasTaken » Sun, Apr 14, 2019, 12:10 PM UTC

lordnacho wrote: post #8

Is the conclusion that a manifold is a space that looks like a lower dimensional space that it really is?

You can have a thing in some higher dimensional space that still "looks like" a thing in lower dimensional space (like the surface of the sphere in R^3 being two dimensional), but R^3 itself is a 3 dimensional manifold that "looks" 3 dimensional as well.

Re: What Is a Manifold?

#15

Post by [deleted] » Sun, Apr 14, 2019, 12:12 PM UTC

username90 wrote: post #4

termiefoo wrote: post #2

what ain't a manifold at this point

Things with edges, like disks and finite lines.

[deleted]

Re: What Is a Manifold?

#16

Post by ColinWright » Sun, Apr 14, 2019, 12:15 PM UTC

_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.

> Is there any way to distinguish between orientable or nonorientable surfaces when you're simply walking on it?

Draw a circle on the surface and orient it. If the surface is non-orientable then there's a way to take a long walk, return to the circle, and find that the direction of the orientation has changed.

Re: What Is a Manifold?

#17

Post by ColinWright » Sun, Apr 14, 2019, 12:16 PM UTC

termiefoo wrote: post #2

what ain't a manifold at this point

A Y-shaped "line".

Re: What Is a Manifold?

#18

Post by dan-robertson » Sun, Apr 14, 2019, 12:23 PM UTC

ErotemeObelus wrote: post #9

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 about distinguishing shapes by homotopy/homology rather than lines and walks and angles.

The article also gets some details a bit wrong/fuzzy: it says you can tell if you are on a sphere by walking in a straight line infinitely far and seeing if you ever cross yourself. But this property also follows on the surface of a (rounded at the top) cone. Even if you require this property in all directions you get problems on eg a torus.

Re: What Is a Manifold?

#19

Post by wcoenen » Sun, Apr 14, 2019, 12:34 PM UTC

_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 described as a manifold. The bug lives in the strip, it's not walking on top of it. It helps to imagine the strip to be transparent.)

For a Klein bottle the same would happen, but there the bug could also find the world mirrored along a different direction, depending on how it travelled.

Re: What Is a Manifold?

#20

Post by Topolomancer » Sun, Apr 14, 2019, 12:35 PM UTC

dan-robertson wrote: post #18

ErotemeObelus wrote: post #9

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…

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 article!