6 ms·

What Is Gaussian Curvature?

94 points by Topolomancer 7y ago

yreg 7y ago
It's a nice article and all, but it can never reach the greatness of this overly enthusiastic backtothefutureesque professor explaining curvature with a slice of pizza. https://www.youtube.com/watch?v=gi-TBlh44gY https://www.youtube.com/watch?v=gi-TBlh44gY
- notfashion 7y ago
  It's Cliff Stoll, not just some wacky guy. He is extremely famous among people who were interested in computer security in the 90s. I know that's not the sense of "hacker" that "hacker news" refers to, but still.
  - espeed 7y ago
    Yes. He wrote The Cuckoo's Egg for one. My uncle gave me the book when I was a kid. The first Java program I wrote was a sequence generator for the Morris Number Sequence [2] given to Stoll by Robert Morris Sr at the NSA: "What is the next number in the sequence 1, 11, 21, 1211, 111221?" https://en.wikipedia.org/wiki/The_Cuckoo's_Egg https://en.wikipedia.org/wiki/The_Cuckoo's_Egg
- petters 7y ago
  He did not mention that the curvature is the product of the minimum and maximum directions. Otherwise, a fair question would be that the gaussian curvature would be different depending on how we draw our lines. In the video, he happens to draw one direction along the fold of the paper, which gives the correct answer. Good video though!
- j88439h84 7y ago
  This is great, thanks for sharing!
  - yreg 7y ago
    Consider checking out some other videos on Numberphile/Computerphile channels. They work with many great minds.
dade_ 7y ago
Roger Penrose gets to this topic early in his book 'The Road to Reality': https://www.goodreads.com/book/show/10638.The_Road_to_Reality https://www.goodreads.com/book/show/10638.The_Road_to_Realit...
- hyt7u 7y ago
  Also covered by Greg Egan in Chapter 2 of Diaspora, with some accompanying drawings here. https://www.gregegan.net/DIASPORA/02/02.html https://www.gregegan.net/DIASPORA/02/02.html
pm90 7y ago
I’m fascinated by these things but afraid of the formalism involved in the papers that define concepts and prove things. Am I doomed to only enjoy math through the generosity of people like the author? Or is there a way for me to dive into math without the help of an actual mathematician using only books?
- JJMcJ 7y ago
  You can only go so far without doing/understanding the mathematics. It is possible to learn the math through self study.
- ssivark 7y ago
  For a bunch of topics in math broadly, this is a nice book, with an emphasis on intuition rather than thorough proofs(hence a compact and readable book) : All the Mathematics You Missed by Thomas A. Garrity https://www.goodreads.com/book/show/967329.All_the_Mathematics_You_Missed https://www.goodreads.com/book/show/967329.All_the_Mathemati... There's of course Roger Penrose's "The road to reality" which has a completely different emphasis (more focused on fundamental physics and therefore introduces a lot of the math used there) but is also very interesting to read. This however, is a real time -- will take some effort to go through, but probably worth it :-)
- credit_guy 7y ago
  It depends what you mean by "dive into math". To get to the level where you understand the proof of the theorem mentioned in this blog post (Theorema Egregium)? This, I think, is roughly where non-elementary math begins. My opinion may prove unpopular, but I don't think this is actually doable by self-study. Well, except that I did something like this myself, I spent 3 months one Summer going page by page through the wonderful "Riamannian Geometry" by Manfredo Perdigao do Carmo [1]. So, it's certainly possible, but also kind of not. It depends where you are in your life, and why you want to do that. If you have a job, maybe a family, and think you'd like to invest one-two hours of day for this hobby, I'm not sure you can get there. If you are willing to invest substantially more time and effort, you need to ask yourself why? Do you want to enhance your career? I think there are better areas of math/CS/Machine Learning that you can attack, with a much better return on investment. Do you already have enough money, that career advancement is not a concern, and you simply want to pursue truth and beauty, and you find advanced math to fit your taste? Then you can go solo, and I think you can succeed, but I think it's much more efficient to actually get feedback from other people (via tutoring or attending courses, or even MOOC). Anyway, not sure if you know about 3brown1blue [2]. Check it out, I hope you'll enjoy it. [1] https://www.amazon.com/Riemannian-Geometry-Manfredo-Perdigao-Carmo/dp/0817634908 https://www.amazon.com/Riemannian-Geometry-Manfredo-Perdigao... [2]https://www.3blue1brown.com/ https://www.3blue1brown.com/
kozlovsky 7y ago
> We already encountered a triangle with three right angles on the sphere Nitpick, but the upper angle of triangle displayed is not right, it's 72 degree (1/5 of 360)
- Topolomancer 7y ago
  Thanks, I will correct this in a subsequent version of the article.
jimbo1qaz 7y ago
So the outside of a sphere is convex (positive curvature), a saddle is concave (negative curvature)... and the inside of a sphere (AKA a concave mirror/lens) is also convex (positive curvature)?
- BoiledCabbage 7y ago
  I believe so yes. The curvature of a surface is independent of which "side" you are measuring from.
  - giomasce 7y ago
    No, you're wrong. In general there is not even a concept of "side", but even if there is a positively-curved surface or manifold seen from the other side is still positively-curved. And the same with negative. You can probably convince yourself of this if you think at the triangles mentioned in the article. Their angles do not change even when you change side.
- pdonis 7y ago
  There is no "outside" or "inside" when you're talking about Gaussian (intrinsic) curvature. "Outside" or "inside" are about how the object is embedded in space, but Gaussian curvature is independent of how (or even if) the object is embedded in space.
  - jimbo1qaz 7y ago
    So Gaussian curvature reuses the words "convex" and "concave" to mean things entirely different from the everyday/lens usage?
    - pdonis 7y ago
      > So Gaussian curvature reuses the words "convex" and "concave" to mean things entirely different from the everyday/lens usage? The everyday/lens usage is talking about how the surface is embedded in a higher dimensional space. Gaussian curvature is not. So you should not expect the meanings in the two cases to be the same.
jvanderbot 7y ago
Can someone connect or point to a connection between this and the measure of curvature of the universe? Im curious what interior angles we're referring to (or equivalent) in 4D spacetime.
- giomasce 7y ago
  Not knowledgable in the topic, but https://en.wikipedia.org/wiki/Shape_of_the_universe https://en.wikipedia.org/wiki/Shape_of_the_universe mentions a couple of measures, both of which cannot distinguish our universe from a flat one.
- Topolomancer 7y ago
  In 4D, we cannot use triangles directly (we could use their higher-dimensional counterparts if we want to classify 3-manifolds), so more complex formulations of curvature are required. I am no expert in curvature myself, but I would wager that researchers would use Ricci curvature here. There's also a nice connection to the Poincare conjecture. I was planning on tackling that in another article. See MathOverflow for an interesting discussion on this subject: https://mathoverflow.net/a/9717 https://mathoverflow.net/a/9717

94 points by Topolomancer 7y ago

yreg 7y ago

It's a nice article and all, but it can never reach the greatness of this overly enthusiastic backtothefutureesque professor explaining curvature with a slice of pizza. https://www.youtube.com/watch?v=gi-TBlh44gY https://www.youtube.com/watch?v=gi-TBlh44gY

notfashion 7y ago
It's Cliff Stoll, not just some wacky guy. He is extremely famous among people who were interested in computer security in the 90s. I know that's not the sense of "hacker" that "hacker news" refers to, but still.
- espeed 7y ago
  Yes. He wrote The Cuckoo's Egg for one. My uncle gave me the book when I was a kid. The first Java program I wrote was a sequence generator for the Morris Number Sequence [2] given to Stoll by Robert Morris Sr at the NSA: "What is the next number in the sequence 1, 11, 21, 1211, 111221?" https://en.wikipedia.org/wiki/The_Cuckoo's_Egg https://en.wikipedia.org/wiki/The_Cuckoo's_Egg
petters 7y ago
He did not mention that the curvature is the product of the minimum and maximum directions. Otherwise, a fair question would be that the gaussian curvature would be different depending on how we draw our lines. In the video, he happens to draw one direction along the fold of the paper, which gives the correct answer. Good video though!
j88439h84 7y ago
This is great, thanks for sharing!
- yreg 7y ago
  Consider checking out some other videos on Numberphile/Computerphile channels. They work with many great minds.

dade_ 7y ago

Roger Penrose gets to this topic early in his book 'The Road to Reality': https://www.goodreads.com/book/show/10638.The_Road_to_Reality https://www.goodreads.com/book/show/10638.The_Road_to_Realit...

hyt7u 7y ago
Also covered by Greg Egan in Chapter 2 of Diaspora, with some accompanying drawings here. https://www.gregegan.net/DIASPORA/02/02.html https://www.gregegan.net/DIASPORA/02/02.html

pm90 7y ago

I’m fascinated by these things but afraid of the formalism involved in the papers that define concepts and prove things. Am I doomed to only enjoy math through the generosity of people like the author? Or is there a way for me to dive into math without the help of an actual mathematician using only books?

JJMcJ 7y ago
You can only go so far without doing/understanding the mathematics. It is possible to learn the math through self study.
ssivark 7y ago
For a bunch of topics in math broadly, this is a nice book, with an emphasis on intuition rather than thorough proofs(hence a compact and readable book) : All the Mathematics You Missed by Thomas A. Garrity https://www.goodreads.com/book/show/967329.All_the_Mathematics_You_Missed https://www.goodreads.com/book/show/967329.All_the_Mathemati... There's of course Roger Penrose's "The road to reality" which has a completely different emphasis (more focused on fundamental physics and therefore introduces a lot of the math used there) but is also very interesting to read. This however, is a real time -- will take some effort to go through, but probably worth it :-)
credit_guy 7y ago
It depends what you mean by "dive into math". To get to the level where you understand the proof of the theorem mentioned in this blog post (Theorema Egregium)? This, I think, is roughly where non-elementary math begins. My opinion may prove unpopular, but I don't think this is actually doable by self-study. Well, except that I did something like this myself, I spent 3 months one Summer going page by page through the wonderful "Riamannian Geometry" by Manfredo Perdigao do Carmo [1]. So, it's certainly possible, but also kind of not. It depends where you are in your life, and why you want to do that. If you have a job, maybe a family, and think you'd like to invest one-two hours of day for this hobby, I'm not sure you can get there. If you are willing to invest substantially more time and effort, you need to ask yourself why? Do you want to enhance your career? I think there are better areas of math/CS/Machine Learning that you can attack, with a much better return on investment. Do you already have enough money, that career advancement is not a concern, and you simply want to pursue truth and beauty, and you find advanced math to fit your taste? Then you can go solo, and I think you can succeed, but I think it's much more efficient to actually get feedback from other people (via tutoring or attending courses, or even MOOC). Anyway, not sure if you know about 3brown1blue [2]. Check it out, I hope you'll enjoy it. [1] https://www.amazon.com/Riemannian-Geometry-Manfredo-Perdigao-Carmo/dp/0817634908 https://www.amazon.com/Riemannian-Geometry-Manfredo-Perdigao... [2]https://www.3blue1brown.com/ https://www.3blue1brown.com/

kozlovsky 7y ago

> We already encountered a triangle with three right angles on the sphere Nitpick, but the upper angle of triangle displayed is not right, it's 72 degree (1/5 of 360)

Topolomancer 7y ago
Thanks, I will correct this in a subsequent version of the article.

jimbo1qaz 7y ago

So the outside of a sphere is convex (positive curvature), a saddle is concave (negative curvature)... and the inside of a sphere (AKA a concave mirror/lens) is also convex (positive curvature)?

BoiledCabbage 7y ago
I believe so yes. The curvature of a surface is independent of which "side" you are measuring from.
- giomasce 7y ago
  No, you're wrong. In general there is not even a concept of "side", but even if there is a positively-curved surface or manifold seen from the other side is still positively-curved. And the same with negative. You can probably convince yourself of this if you think at the triangles mentioned in the article. Their angles do not change even when you change side.
pdonis 7y ago
There is no "outside" or "inside" when you're talking about Gaussian (intrinsic) curvature. "Outside" or "inside" are about how the object is embedded in space, but Gaussian curvature is independent of how (or even if) the object is embedded in space.
- jimbo1qaz 7y ago
  So Gaussian curvature reuses the words "convex" and "concave" to mean things entirely different from the everyday/lens usage?
  - pdonis 7y ago
    > So Gaussian curvature reuses the words "convex" and "concave" to mean things entirely different from the everyday/lens usage? The everyday/lens usage is talking about how the surface is embedded in a higher dimensional space. Gaussian curvature is not. So you should not expect the meanings in the two cases to be the same.

jvanderbot 7y ago

Can someone connect or point to a connection between this and the measure of curvature of the universe? Im curious what interior angles we're referring to (or equivalent) in 4D spacetime.

giomasce 7y ago
Not knowledgable in the topic, but https://en.wikipedia.org/wiki/Shape_of_the_universe https://en.wikipedia.org/wiki/Shape_of_the_universe mentions a couple of measures, both of which cannot distinguish our universe from a flat one.
Topolomancer 7y ago
In 4D, we cannot use triangles directly (we could use their higher-dimensional counterparts if we want to classify 3-manifolds), so more complex formulations of curvature are required. I am no expert in curvature myself, but I would wager that researchers would use Ricci curvature here. There's also a nice connection to the Poincare conjecture. I was planning on tackling that in another article. See MathOverflow for an interesting discussion on this subject: https://mathoverflow.net/a/9717 https://mathoverflow.net/a/9717