Visualizing the 4d numbers Quaternions on HackerTube: recommended by 6 Hacker News commenters across 6 separate months, 2018–2023.
explainer · intermediate · evergreen
3Blue1Brown builds visual intuition for quaternions, showing how to think about this four-dimensional number system using stereographic projection and rotations in 3D space. It is part one of a two-part series with an interactive companion site.
HN commenters recommend it as the best visual explanation of quaternions and a good way to visualize higher dimensions.
“He does a great job of explaining quaternions though visual intuition.”
7 Hacker News comments link this video. Each one links back to its thread.
I'm happy to see 3Blue1Brown's most recent video [0] get a mention in this article! He does a great job of explaining quaternions though visual intuition. (I do think simply calling Grant a "math animator" is selling him a bit short. He's a true educator in every sense; animation is just one of the (very effective) learning aides he uses.) [0] ▶ this video
in “Quaternions: the Strange Numbers That Birthed Modern Algebra” · View on HN
I found this video explaining quaternions to actually be a great way to explain (and visualize) higher spatial dimensions: ▶ this video When you think about the nth dimension as a projection of the n+1th dimension, it begins to make a lot more sense. That doesn’t mean you can necessarily visualize the 4th spatial dimension directly, but you can at least visualize how the 3rd dimension can be a projection of the 4th, just like how we can easily see that the 2nd dimension as a projection of the third.
in “It’s hard to give a good intuitive explanation for the volume of a cone” · View on HN
3B1B is also informative, especially in explaining why C was not extended using triplets but 4-tuples were needed: ▶ this video . Here’s a Math SE answer on why: [link] , here’s a more ELI5 discussion: [link] . I especially like this very intuitive comment in that thread : [link] Left as an exercise to reader: prove that no such n-tuples exist for n>4.
in “Quaternions” · View on HN
The first part of the article about describing sphere rotations with a direction and magnitude has some overlap with quaternion rotations, described and visualized nicely here: [link] ; ▶ this video The second part of the article about untangling strings after a 720° turn is also covered in: [link] ; [link] ; [link] ; [link]
in “The Dirac String Trick (2015)” · View on HN
I still remember the first time when I saw videos about non-Euclidian Geometry[1] and Higher Dimensions[2]. I was amazed with such concepts, but I still havent read books about them. They are related somehow with topology[3] and curvature (and tangentially, to differential equations). [1] [link] (5 sided square) [2] [link] (The road to 4D part 1) [link] (Carl Sagan's explanation) [3] [link] (Zach Star's intro) Also, a nice exercise to visualize 4D is to study quaternions and their properties. Vectors are somehow a nice simplification of them, see Maxwell laws. 3blue1brown has a video[4]. [4] ▶ this video
in “What is the Fourth Dimension? (1884)” · View on HN
> rotors operate fully in 3D By this, the author means: you can visualize a rotor FULLY in 3D, no extra dimension neccessary. I suspect you never have tried to think about how quaternions work mechanically, I suggest this neat video: ▶ this video
in “Let's remove quaternions from every 3D engine (2018)” · View on HN
Perhaps this montage offers a clue about the possible existence of Higher Dimensions in Physics... ...And then again, perhaps not! :-) Related: Professor Leonard Susskind - The World As Hologram: [link] 3Blue1Brown - Visualizing quaternions (4d numbers) with stereographic projection: ▶ this video AdS/CFT correspondence [link] Plato's Cave: [link]
in “Cool Examples of Shadow Art (2016) [video]” · View on HN