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.
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.
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?"
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.
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?
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.
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 have been semi-diligently (4-5 hours/week as adult life allows) following this approach coupled with working my way through math texts. Although I zero problem grasping concepts and parsing propositions and examples, I find it very difficult to solve end-of-chapter problems. No more than 4 or 5 problems a week, and hence my advancement is agonizingly slow.
I started with Spivak & Apostol but reverted to Riley and Hobson's Foundations text because I was struggling too much with the A&S's problems. Given that I did very well on symbolic-logic proofs in an undergrad course, I figured it would be easy enough to get into math if I had diligence and sincere interest. After having been a 'D' student, my recognition of arithmetic and algebraic expressions and manipulations is hopelessly sub-par. Even many of the R&H problems are beyond my grasp.
My approach has been informed by a sincere desire to engage with mathematics (one I unfortunately did not have through my formal education) as well as several "How do I self-teach maths" threads on HN, Reddit, and /sci/. Previously I had chalked up my poor grades to an undisciplined, unmotivated youth. Now I'm beginning to suspect I may simply not have the requisite intelligence, or am at least outside the age where I had enough time and neuroplasticity to pick math up in earnest.
Given that 1) you have zero problem grasping the concepts and understanding the examples, and 2) your difficulty is in solving the end-of-chapter problems; I think it is almost certain that your issue is one of making small mistakes that throw off your solutions (for example, didn't change a plus to a minus sign, left off a term, made an incorrect assumption, etc.). You just need to learn how to double-check your work and figure out where you went wrong. This is a skill that everyone doing nontrivial math has to learn. In a formal learning environment, you constantly get feedback on your mistakes through the grading process, which you obviously don't get when self-learning. My advice is to work the example problems before looking at the solution. After you have worked it yourself, you can then easily see where you went wrong. You should also get books of solved problems that let you practice this more extensively. There are problem books for pretty much every undergrad math topic as well as some grad-level ones. A lot of textbooks even have solution manuals that will show you how to solve the end-of-chapter problems in the textbook. An Amazon search with the word "solutions" and whatever math topic you want should turn up many examples. With practice, you learn how to detect errors and rectify them even when there is no solution manual.
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)?
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.
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.
> 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.
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.
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
26 comments
[ 4.8 ms ] story [ 74.0 ms ] threadhttps://www.youtube.com/watch?v=gi-TBlh44gY
https://en.wikipedia.org/wiki/The_Cuckoo's_Egg
Good video though!
It is possible to learn the math through self study.
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 :-)
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/
I started with Spivak & Apostol but reverted to Riley and Hobson's Foundations text because I was struggling too much with the A&S's problems. Given that I did very well on symbolic-logic proofs in an undergrad course, I figured it would be easy enough to get into math if I had diligence and sincere interest. After having been a 'D' student, my recognition of arithmetic and algebraic expressions and manipulations is hopelessly sub-par. Even many of the R&H problems are beyond my grasp.
My approach has been informed by a sincere desire to engage with mathematics (one I unfortunately did not have through my formal education) as well as several "How do I self-teach maths" threads on HN, Reddit, and /sci/. Previously I had chalked up my poor grades to an undisciplined, unmotivated youth. Now I'm beginning to suspect I may simply not have the requisite intelligence, or am at least outside the age where I had enough time and neuroplasticity to pick math up in earnest.
Nitpick, but the upper angle of triangle displayed is not right, it's 72 degree (1/5 of 360)
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.
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.
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