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What Is a Manifold?

bastian.rieck.me

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Re: What Is a Manifold?

#111

Post by JadeNB »

hansen wrote: post #26
math_and_stuff wrote: post #24

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).

> Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).

It depends on which mathematicians! Plenty of differential geometers allow manifolds to have boundary, and say "closed manifold" (https://en.wikipedia.org/wiki/Closed_manifold) to emphasise when they are dealing with a (compact) manifold without boundary (or, as you point out, really a manifold whose boundary is empty).

Re: What Is a Manifold?

#112

Post by xelxebar »

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...

Okay. That's pretty darn cool. I ended up trawling your blog a bit too. Really interesting stuff! Wish there were an RSS feed I could subscribe to!

Re: What Is a Manifold?

#113

Post by amai »

ginnungagap wrote: post #107
amai wrote: post #99

Earlier quoted context omitted.

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

Ok, a line in 3D space can also be called a manifold? Maybe a better question is then to ask, what is not a manifold?

Re: What Is a Manifold?

#114

Post by Sinbe »

termiefoo wrote: post #2

what ain't a manifold at this point

Simplest example of what not a manifold is the shape of the letter “Y”. Technically for a topological space to be a manifold, every small neighborhood around each point should look the same. But “Y” has a special point at the center.

Re: What Is a Manifold?

#115

Post by tyingq »

joncrane wrote: post #103
tyingq wrote: post #34

Earlier quoted context omitted.

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.

My point was that manifolds aren't all many to one. This one is clearly many to many.
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