In this last episode on non-Euclidean geometry, I present the opposite of the spherical geometry, i.e. hyperbolic geometry. This hopefully will give you additional intuition of straight lines in curved spaces…
Straight lines in curved spaces are highly counter-intuitive, but they are the key to general relativity! Now, many mathematicians and physicists would say that they are distance-minimizing trajectories, which is (almost) true. But, like Gabe on PBS Space Time, I actually don’t like this definition. I think it is neither natural nor insightful… Here’s why:
What do you think? Do you get a feel for straight lines in curved spaces now? Are there things that are troubling you? Tell me!
After two episodes about flat surfaces, finally, Episode 9 introduces curved geometry. But we start easy, with the most familiar of all curved spaces: the sphere. Interestingly, the geometry of the sphere has played a crucial in the History of world conquest by European nations…
So, which is your favorite map? Are you deeply disturbed by the upside-down Mercator map? What about he butterfly-shaped map?
Episode 8 here is quite a detour on the way to general relativity. It’s about a very cool piece of geometry that’s embedded in the well-known shapes of soccer balls.
So, how many panels will the 2018 world cup be made off? What’s your bet?
I’ve long regarded my 7th video as my best video. It has plenty of pretty cool stuffs inside it: concrete simple problem, Historical anecdote, well-known scientists, awesome mathematics, recent developments, stunning images… Most importantly, the video introduces the very basics of curved-space geometry, through Nash’s fantastic embedding theorems, with a particular focus on the duality by intrinsic and extrinsic approaches to geometry.
There’s a somehow unfortunate story behind this video: I uploaded it the day before John Nash, a central mathematical figure of the video, sadly passed away. This has led me to make a follow-up video dedicated to Nash, where I also apologized for the somewhat not-very-respectful way I presented it here. In the end, I truly admire the man. Both for his life struggle with schizophrenia, and his epic mathematical journeys.
To this day, this is the video of the series that has had the most views (1.4k). I must say, though, 1.4k views is not a lot. My best earlier videos have 10k, 20k and even 40k views… So, the question I want to ask you is: How do I get more views? How can I improve the videos? Are there things in the videos you did not like? Are there things you did not understand?