She described an algebra called "Geometric Algebra" that has both standard vector spaces, complex numbers and quaternions as sub-algebras. Supposedly it simplifies a lot of the abstraction surrounding these concepts.
So I dug a little bit deeper and ended up watching these videos on YouTube that really helped me understand what she was talking about:
And WOW, the whole concept just made so much more sense. In 2D if you consider the area created from two vectors, it can have a "positive spin" or a "negative spin" based on the relative direction of the vectors (Like if you turn a handle clockwise vs. counter-clockwise). It turns out this property makes the square of this area negative (like the complex numbers). The same analysis can be done in 3D and the quaternions pop out. Pretty awesome stuff! I have been buying and reading books on the topic ever since, I feel like I need to re-educate myself on all the advanced math I learned in gradschool because this makes it all so much more compact and elegant.
Clifford algebra is exactly how I figured it out as well.
Here's the clearest explanation I can make:
Things have a "sided-ness" to them. In most mathematics, the definition of a plane ignores this sided-ness. Three points and you have a plane, right? But you can't describe the orientation of a plane with the points, because you can't tell which side is "up". A piece of paper has two sides, why doesn't a plane? This is important when describing rotation!
Anyone who has used a look_at function can understand this. You give it a point for the eye position, and a point for the eye to "look at". What is not described is how that eye is oriented. Is it upside down? Sideways? Right side up? Even vectors have a "side"! This is the "twist" that is often mentioned.
If you imagine yourself as the eye, you can imagine your eyes being in the same position, looking at the exact same point, but in many different perspectives. Lying on your side, doing a headstand, standing upright.
I'm not an expert and I'm probably wrong, but intuitively this is what I think that fourth dimension on the quaternion corresponds to.
https://www.youtube.com/watch?v=ikCIUzX9myY
She described an algebra called "Geometric Algebra" that has both standard vector spaces, complex numbers and quaternions as sub-algebras. Supposedly it simplifies a lot of the abstraction surrounding these concepts.
So I dug a little bit deeper and ended up watching these videos on YouTube that really helped me understand what she was talking about:
https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ....
And WOW, the whole concept just made so much more sense. In 2D if you consider the area created from two vectors, it can have a "positive spin" or a "negative spin" based on the relative direction of the vectors (Like if you turn a handle clockwise vs. counter-clockwise). It turns out this property makes the square of this area negative (like the complex numbers). The same analysis can be done in 3D and the quaternions pop out. Pretty awesome stuff! I have been buying and reading books on the topic ever since, I feel like I need to re-educate myself on all the advanced math I learned in gradschool because this makes it all so much more compact and elegant.