Wikipedia has a thorough intro article https://en.wikipedia.org/wiki/Manifold
What is a manifold?
91–100 of 138 posts
Re: What is a manifold?
#92This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition. It’s actually much more well written than the majority or articles we usually come across.
Is that really a good article? I thought it was average. It had some big flaws but was probably still informative for readers with no mathematical knowledge in the domain. For instance, consider the only concrete example in the article: the space of all possible configurations of a double pendulum is a manifold. The author claims it's useful to see it in a manifold, but why? Precisely, why more as a manifold than as…
> Precisely, why more as a manifold than as a square [O,2π[²?
Because, as the article explains, it's a torus (loop crossed with a loop), not a square (segment crossed with a segment).
Re: What is a manifold?
#93Earlier quoted context omitted.
Quanta’s greatest strength is that it doesn’t pretend to be clever. Many tech publications write as if they’re showing off, and you just end up feeling tired after reading them.
> Many tech publications write as if they’re showing off, and you just end up feeling tired after reading them. I like this honestly because this shows that I learned something intelligent. On the other hand, if I don't feel exhausted after reading, it is a strong sign that the article was below my intellectual capacity, i.e. I would have loved it if I could have learned more.
It's OK to keep going deeper into the material if you aren't tired yet.
Re: What is a manifold?
#94Earlier quoted context omitted.
I'm always surprised more people don't know about Quanta. Seems like it's currently the best science journalism out there, and IMO a very strong candidate for the single best place on the internet that's not crowd-sourced. The mixture of original art and technical diagrams is outstanding. Podcast is pretty good too, but I do wish they'd expand it to have someone with a good voice reading all the articles. Besides not…
It's because of their Simons Foundation support, but not only because of that. I mean, I invite anyone to name another billionaire pet project of comparable quality.
Mathematica?
Re: What is a manifold?
#95Earlier quoted context omitted.
I'm always surprised more people don't know about Quanta. Seems like it's currently the best science journalism out there, and IMO a very strong candidate for the single best place on the internet that's not crowd-sourced. The mixture of original art and technical diagrams is outstanding. Podcast is pretty good too, but I do wish they'd expand it to have someone with a good voice reading all the articles. Besides not…
It's because of their Simons Foundation support, but not only because of that. I mean, I invite anyone to name another billionaire pet project of comparable quality.
Good game and a hard question, especially if you make "comparable" more explicit. I'd add "noncommercial, open-access", and "modern" in the sense that it happened under the current norms with respect to legacy and the social contract.
Re: What is a manifold?
#96So what is the "Not a manifold." part? The actually interesting part.
Re: What is a manifold?
#97This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition. It’s actually much more well written than the majority or articles we usually come across.
Is that really a good article? I thought it was average. It had some big flaws but was probably still informative for readers with no mathematical knowledge in the domain. For instance, consider the only concrete example in the article: the space of all possible configurations of a double pendulum is a manifold. The author claims it's useful to see it in a manifold, but why? Precisely, why more as a manifold than as…
Spacetime is a four-dimensional manifold (at least theoretically - who knows what it is in reality). Technically it's a pseudo-Riemannian manifold since the metric is not positive definite: it can be negative or zero for non-zero vectors. A Riemannian manifold proper has a positive definite metric, but in popularizations like this I wouldn't really expect them to get into these kinds of distinctions.
Re: What is a manifold?
#98Earlier quoted context omitted.
The problem is that this global coordinate system isn't a continuous mapping (see the discontinuity of both angular coordinates between 2*pi and 0). Manifolds are required to have an "atlas"[0]: a collection of coordinate systems ("charts") that cover the space and are continuous mappings from open subsets of the underlying topological space to open subsets of Euclidean space, with the overlaps between charts inducin…
This part I don't grasp: > this global coordinate system isn't a continuous mapping (see the discontinuity of both angular coordinates between 2*pi and 0). I'm guessing that the issue is that I don't know your definition of 'continuous'. I believe every point on the planet (sphere, for simplification) has unique corresponding coordinates on the map projection (chart). The only exceptions I can see are, A) surfaces pe…
Continuity is fundamentally a topological property of a mapping. It just means that for a mapping F and a point p, for any neighborhood del of F(p), we can find a neighborhood eps of p such that F(eps) is contained entirely in del. In simpler terms, if you draw a little ball around F(p), I can find a little ball around p whose image under F is contained in the little ball you drew around F(p). If I have coordinates on the sphere that suddenly jump between 0 and 2*pi, I can’t satisfy this property, because points that are arbitrarily close on the sphere will be mapped to opposite sides of the “coordinate square” with sides [0,2*pi).
The Mercator projection is obtained by removing two points from the sphere (both poles) and stretching the hole at each pole until the punctured sphere forms a cylinder, then cutting the cylinder along a line of longitude. So you can see that the 3 discontinuities in the Mercator projection correspond to the top and bottom edges (where we poked a hole at each pole) and the left/right edges (where we cut the cylinder). (Note that stretching the sphere at the poles changes the curvature, but cutting the cylinder does not. The projection would have the same properties on a cylinder.)
It is possible to continuously map the sphere to the entire (infinite) plane if you just remove a single point (the north pole): place the sphere so the south pole is touching the origin of the plane and for any point on the sphere, draw a line from the north pole through that point. Where that line intersects the plane is that point’s image under this mapping (called the Riemann sphere).
Re: What is a manifold?
#99Does the way "manifold" is used when describing subsets of the representational space of neural networks (e.g. "data lies on a low-dimensional manifold within the high-dimensional representation space") actually correspond to this formal definition, or is it just co-opting the name to mean something simpler (just an embedded sub-space)?
If it is the formal definition being used, then why? Do people actually reason about data manifolds using "atlases" and "charts" of locally euclidean parts of the manifold?
Re: What is a manifold?
#100This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition. It’s actually much more well written than the majority or articles we usually come across.
Agreed. I'm not a mathematician - and to me a manifold is more familar in the context of engines. But I found both the text and the diagrams very useful.
When you use the word "engine" on HN, it can be understood as many things that aren't what you think (e.g. game engines).