Visualizing quaternions (4d numbers) with stereographic projection

How to think about this 4d number system in our 3d space.
An equally valuable form of support is to simply share some of the videos.

Quanta article on quaternions:

The math of Alice in Wonderland:

Timestamps:
0:00 - Intro
4:14 - Linus the linelander
11:03 - Felix the flatlander
17:25 - Mapping 4d to 3d
23:18 - The geometry of quaternion multiplication

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

Music by Vincent Rubinetti:

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Various social media stuffs:

[ignaciodemiguel]

My teacher 5 years ago: i is the only complex unit

My teacher now: jk

[paradox]

The animations are so intricate, the math must have taken you so long to put together and understand for yourself, let alone explain to any random YouTube browser. I express my deep gratitude for you taking the time to put together your videos, especially for free for our viewing.

[sonderevokingbuns]

"What you're looking at right now is something called quaternion multiplication; or rather you're looking at a certain representation of a specific motion happening on a 4 dimensional sphere being represented in our 3 dimensional space. One which you'll understand by the end of this video."

That last sentence is the most boldly incorrect assumption anyone has ever made.

[patrickcoffey]

lol good ol' imaginary, jimaginary and kimaginary numbers

[hamsterwheel]

“It might feel weird to talk about two circles being perpendicular to each other, especially when they have the same center, the same radius, and they don’t touch each other at all…”
I understand each word in that sentence individually, and that’s it.

[gexandthecity]

The pure dread of knowing you barely understand the 3d part of the video and know he's going to start speaking in 4 dimensions has more tension than the shining. I would watch a horror movie made by him

[LittleWhole]

I remember in my 5th grade classroom, there was a short fiction chapter book where the plot is based off of the main characters visiting 4D space. I think it was called "The Boy Who Reversed Himself" or something like that. I remember after reading that book I was absolutely captivated by the notion of higher dimensionality and went on gigantic Wikipedia rabbit holes trying to learn all about it. I never really totally understood it but this brought back memories - one of the chapters in the book talked about how the main characters were eating food in 4-space, but because they were 3D objects the 4D beings could see right through them in the 4th axis and were laughing their butts off at the food plopping down and going through the digestive system. It talked about how 3D retinas were required to see in 4-space and how a 4D object would cast a 3D shadow, and how a 4-sphere passing through 3D space would look like a random sphere popping into existence and expanding and shrinking out of existence similarly to how a 3D sphere passing through 2-space would be a circle doing that to a 2D observer.

[entropyz]

"A line is just a circle that passes through the point at infinity" -- Grant Sanderson


That is really deep, man.

[twigglesoft]

You know it's going to be a bumpy road when you don't understand the 1 dimensional analogy lol

[evanwalker]

As a physics PhD student I thank you for helping me understand these things. Group theory is a whole lot simpler now

[maxwellchiu]

I have watched this video at least 5 times in last couple years. I get better understanding every time. Kudos. Totally worth it. Your own work on producing this presentation is admirable.

[yannisconstantinides]

22:00
"The number -1 is sitting off at the point of infinity, which you can easily find by walking in any direction."
Ah, so simple.

[johnchessant]

I think Lord Kelvin was onto something when he said quaternions are an "unmixed evil".

[jethrolarson]

"A line is just a circle that passes through the point at infinity" *head explodes*

[Red-Brick-Dream]

Once again, this channel is in a league entirely of its own. The best of the best on maths-YouTube.

[arlenn]

As an engineer, I pay all my respect to the people who deeply understand these complex math. These math have revolutionary effect on engineering productions these days

[ahmedal-shabi]

"one you will understand by the end of this video"
Wow, some confident statement right there buddy 😂😂

[AleksanderFimreite]

If I'm not mistaken, this helped me understand how scaling the "Field of View" in 3D applications works.
That is essentially scaling one of the axis which is pointing horizontally in relation to the camera if I understood this right.
It still does not help me one bit to grasp which modifications I have to put into a quaternion to reach the exact point I want.
But luckily, applications tend to provide useful features to dumb it down to a matter of providing relative angles.

[yashsharma]

I really cried after watching such a beautiful explanation out of happiness. Hats off to you & I finally understand how to see the beauty inside mathematics. You did a great great great job. Thank you

[PaulPaulPaulson]

This is going to become THE standard reference for learning quaternions!