
What Is Gaussian Curvature? - Topolomancer
https://bastian.rieck.me/blog/posts/2019/curvature/
======
yreg
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
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
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)

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

~~~
hyt7u
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)

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pm90
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?

~~~
1e-9
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.

~~~
nootka
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.

~~~
1e-9
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.

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jimbo1qaz
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)?

~~~
pdonis
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
So Gaussian curvature reuses the words "convex" and "concave" to mean things
entirely different from the everyday/lens usage?

~~~
pdonis
_> 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.

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jvanderbot
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.

~~~
Topolomancer
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)

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

