Visualizing the 4d numbers Quaternions

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] [VIDEO]

> 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: [VIDEO]

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] [VIDEO]

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] ; [VIDEO] The second part of the article about untangling strings after a 720° turn is also covered in: [LINK] ; [LINK] ; [LINK] ; [LINK]

3B1B is also informative, especially in explaining why C was not extended using triplets but 4-tuples were needed: [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.

I found this video explaining quaternions to actually be a great way to explain (and visualize) higher spatial dimensions: [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.

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: [VIDEO] AdS/CFT correspondence [LINK] Plato's Cave: [LINK]