Less Weird Quaternions Using Geometric Algebra

Re:Less Weird Quaternions Using Geometric Algebra by klodolph on Monday August 07, 2017 @05:57PM (#14951606)

I aiming that explanation at people who had read and understood the article. The article explains bivectors and how they can be used to represent reflections, and "closure" is a fairly common concept, so once you put those two together you should get a mathematical object which I've called "the even-ordered subalgebra". I haven't explained why it's even-ordered or what a subalgebra is, but I used those terms so you could at least have the vocabulary to talk about it or do a Google search.

Mathematics education is hard. In my experience, you start out with no understanding of a subject and can't understand it when people explain it to you, and at some point it clicks and you can't understand why it was ever difficult. I could be intentionally obtuse and, for example, describe a vector space as an "abelian group, field, and homomorphism from the field to group endomorphisms", but I feel that's the only people who would use that definition already have a good understanding of vector spaces.

The reason that I consider the non-GA approach to quaternions as rotations "hand wavy" is because it's not constructive, or perhaps just because I personally don't understand it. Using GA, I can construct a representation for rotations in any Euclidean space, not just 3D space, but 2D, 4D, 5D, whatever. However, without GA at my disposal, the fact that unit quaternions are a double cover for SO(3) seems like some kind of black magic that came from the void.

I have a few drafts of an introductory article I was writing on geometric algebra sitting on my hard drive, but I've never been able to get the article into a state I'd consider publishable. So instead, I'm trying to inject what I know into HN discussions.

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Re:Less Weird Quaternions Using Geometric Algebra by yequalsx on Tuesday August 08, 2017 @12:51PM (#14959745)

I was a math Ph.D. student at Purdue University and studied commutative algebra. I understand what you were getting at. My comment was mostly tongue in cheek. For someone not versed in mathematics what you wrote could be ironic in a slightly humorous way. I.E. that the hand wavy way explanation is more understandable to a layman than subalgebras, and whatnot. That's the ironic difference between mathematicians and non-mathematicians. What is hand wavy to us is concrete to them and vice versa.

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