What Is a Manifold? • Hacker News Board

Re: What Is a Manifold?

#61

Post by inamberclad » Sun, Apr 14, 2019, 4:33 PM UTC

TeMPOraL wrote: post #58

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…

Isn't this like a multiplexer/demultiplexer in electronics/signal processing?

A multiplexer/demux takes one of many inputs and routes them to a single output, or vice versa. A manifold has no valving, so it's a mixer/adder instead.

Re: What Is a Manifold?

#62

Post by colllectorof » Sun, Apr 14, 2019, 4:44 PM UTC

If you want more, there is an excellent course on "introductory" topology on YouTube:

https://www.youtube.com/watch?v=CEXSSz0gZI4

It is made by a physicist, not mathematician, and it's a great combination of informal explanations and carefully covering the actual math definitions.

I feel that his approach works way better than making analogies.

There is another course from Wildberger. A bit more hardcore, but also has some interesting perspectives:

https://www.youtube.com/watch?v=Ap2c1dPyIVo&list=PL6763F57A6...

Re: What Is a Manifold?

#63

Post by dan-robertson » Sun, Apr 14, 2019, 4:52 PM UTC

senderista wrote: post #50

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…

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 fo…

I don’t disagree that one can define dimension for a manifold (topological or differentiable). I was replying to the parent comment and so I was pointing out that dimension isn’t really well defined for a general topological space.

Re: What Is a Manifold?

#64

Post by ColinWright » Sun, Apr 14, 2019, 5:19 PM UTC

tgb wrote: post #56

ColinWright wrote: post #16

Earlier quoted context omitted.

> 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.

This depends upon the circle you chose. Nonorientable manifolds still have many orientable circles (any sufficiently small loop is orientable).

You confuse me.

If the surface is non-orientable then I can draw a very small circle, put an orientation on it, then there's a path I can take that brings me back to find the orientation has changed.

Let's be more explicit.

Take a non-self-intersecting embedding of RP^2 in R^4, a point P in our RP^2, and a sufficiently small epsilon e. Then take three points on the circle of size e centred on P, and think of them as going in order, thus defining an orientation. Now take an appropriate walk around RP^2 and return to the circle. For an appropriate walk I will now find that the order of the points on my circle has reversed.

That's a more precise way of saying what I originally intended, and in that context your comment doesn't make sense to me. Can you expand on it?

Re: What Is a Manifold?

#65

Post by TeMPOraL » Sun, Apr 14, 2019, 5:34 PM UTC

inamberclad wrote: post #61

TeMPOraL wrote: post #58

Earlier quoted context omitted.

Isn't this like a multiplexer/demultiplexer in electronics/signal processing?

A multiplexer/demux takes one of many inputs and routes them to a single output, or vice versa. A manifold has no valving, so it's a mixer/adder instead.

PWM-switched mux/demux with signal input then :).

Re: What Is a Manifold?

#66

Post by a1369209993 » Sun, Apr 14, 2019, 5:45 PM UTC

ColinWright wrote: post #64

tgb wrote: post #56

Earlier quoted context omitted.

This depends upon the circle you chose. Nonorientable manifolds still have many orientable circles (any sufficiently small loop is orientable).

You confuse me. If the surface is non-orientable then I can draw a very small circle, put an orientation on it, then there's a path I can take that brings me back to find the orientation has changed. Let's be more explicit. Take a non-self-intersecting embedding of RP^2 in R^4, a point P in our RP^2, and a sufficiently small epsilon e . Then take three points on the circle of size e centred on P, and think of them as…

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.

Re: What Is a Manifold?

#67

Post by amai » Sun, Apr 14, 2019, 5:57 PM UTC

So a manifold is a surface?

Re: What Is a Manifold?

#68

Post by [deleted] » Sun, Apr 14, 2019, 6:00 PM UTC

braxxox wrote: post #53

What a great read! I've never thought about how the way we experience the surface of the earth as 2D manifold of a 3D space, and how up until recently (relatively speaking) this was unknown. It's interesting to expand on this idea and realize that maybe we are making the same mistake again, and that from our local perspective the universe is 3D, when in reality, it's a 3D manifold of a higher dimensional space. One t…

[deleted]

Re: What Is a Manifold?

#69

Post by 8note » Sun, Apr 14, 2019, 6:09 PM UTC

TeMPOraL wrote: post #65

inamberclad wrote: post #61

Earlier quoted context omitted.

A multiplexer/demux takes one of many inputs and routes them to a single output, or vice versa. A manifold has no valving, so it's a mixer/adder instead.

PWM-switched mux/demux with signal input then :).

still not quite. more like a node where you join parallel circuits back together; the total volume of stuff moving, current, is the addition of everything coming in, rather than blocking and switching things.

Re: What Is a Manifold?

#70

Post by tgb » Sun, Apr 14, 2019, 6:31 PM UTC

a1369209993 wrote: post #66

ColinWright wrote: post #64

Earlier quoted context omitted.

You confuse me. If the surface is non-orientable then I can draw a very small circle, put an orientation on it, then there's a path I can take that brings me back to find the orientation has changed. Let's be more explicit. Take a non-self-intersecting embedding of RP^2 in R^4, a point P in our RP^2, and a sufficiently small epsilon e . Then take three points on the circle of size e centred on P, and think of them as…

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.