What Is Gaussian Curvature?

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.

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!

This is great, thanks for sharing!

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

You can only go so far without doing/understanding the mathematics.

It is possible to learn the math through self study.

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_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 :-)

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...

[2]https://www.3blue1brown.com/

You can absolutely learn math on your own. Many of the most well-known mathematicians, engineers, and scientists were largely self-taught. Examples include da Vinci, Watt, Edison, the Wright Brothers, Heaviside, Ampere, Boole, Galileo, Pascal, Leibniz, and Ramanujan. The learning materials available to those guys were vastly inferior to what we have today. In fact, you could take yourself very far using nothing more than Wikipedia coupled with a divide-and-conquer approach. If you want to understand a topic, do a Wikipedia search on it. There are usually some good references and links at the bottom of the page that can help with further background and intuition. Never be intimidated by the complexity. Every complex topic can be broken down into simple, easy-to-understand pieces. For every subtopic in the description that you don't understand, click on its link, read, and repeat. Eventually, your subtopic dive should reach a low enough level that you can grasp the concept and use it to understand the previous higher-level topic. For each concept, you should develop an intuitive understanding that goes along with the formalism. Try to visualize every concept in some way. Use mental animations, graphs, sounds, colors, etc. One of the tricks is that you should only focus on learning a small number of new concepts every day (maybe 5 to 7 at most). A night's sleep will help you consolidate those concepts and build on them. For even the most accomplished mathematicians, there are papers that they don't understand at first glance. They must go through their own divide-and-conquer process to grasp unfamiliar concepts.

I believe so yes. The curvature of a surface is independent of which "side" you are measuring from.

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.

Not knowledgable in the topic, but 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.

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

Thanks, I will correct this in a subsequent version of the article.