Comment by at_compile_time | Hacker News Reader

Comment by at_compile_time | Hacker News Reader

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at_compile_time

>unpedagogical

Does this word mean "not the way things mathematicians teach things"? My experience has been that mathematicians teach things that are useful to mathematicians to students who are not going to be mathematicians and would be better served learning other things. I wasted countless hours of my life finding analytical solutions to toy calculus problems in a universe that will never yield to those methods.

>My stance on quaternions is that they are an opaque representation of what they are trying to do, which makes them unnecessarily-difficult and annoying. Not to mention hard to learn to visualize.

I've seen some remarkably confusing attempts to understand and visualize these things. This has always baffled me because the equivalent objects in geometric algebra aren't that hard to understand. I really think this is a problem with your pedagogy. You've hidden the geometric meaning of the bivector components in these imaginary components, i,j,k, and you have to take it on faith that i*j=k and instead of it being just the product of two bivector blades.

>The notion of an exponential map and exponentiating generators is, IMO, much more "natural" and intuitively straightforward than the alternatives.

That's because you learned it that way. Exponentiating an oriented area to generate a rotation is perfectly intuitive to me, and I'm not sure how somebody could be confused by the fact that a bivector can be an oriented area or the logarithm of a rotation because it's a simple matter of context.

Eh, I disagree.

By "unpedagogical" I mean: very hard to learn, and even when you learn it, often hard to explain. It's information that doesn't compress well, generalize well, or really convey direct understanding of what's going on. Maybe not the best word for this. Really, I just mean "bad". To me it's incomplete knowledge; it needs to be improved upon so that it makes more sense for people who need to know it. In doing so it will also become easier to learn.

I think quaternions are confusing, GA bivector notations are less confusing but still confusing, and the operator version which I endorse is the least confusing of the three. This is just an aesthetic judgment on my part. IMO if things were taught in the way I prefer, more people could learn them, faster, and come away with a more solid understanding afterwards for less work. (You and I agree that GA's version is better than quaternions for the same reason. I just think that there are better versions still.)

If you want to tell someone that exponentiating an area gives a rotation, you need to basically deal with the fact that that sentence sounds like nonsense. An area's an area, why would exponentiating it... do... anything? My preferred explanation of all this stuff avoids that saying things that sound like nonsense.

(I prefer to think of a bivector not as an oriented area per se but as a type of tensor which happens to represent those things, but also represents other things, including the logarithms of rotations, due to the properties that those two things happen to share. That's a perspective that generalizes very well, compared to the GA version, because it separates the tensor representation from the operators; the two sides end up generalizing in different directions as you go to more complicated objects. When I get around to writing to my own exposition on this I'll go through that perspective very methodically.)

>If you want to tell someone that exponentiating an area gives a rotation, you need to basically deal with the fact that that sentence sounds like nonsense.

It's not just an area though, it's an area with orientation and magnitude. It's the complex exponential function extended to 3 dimensions.

>By "unpedagogical" I mean: very hard to learn, and even when you learn it, often hard to explain. It's information that doesn't compress well, generalize well, or really convey direct understanding of what's going on. Maybe not the best word for this. Really, I just mean "bad".

I can see where you're coming from on the learning part. There are as many different ways of learning GA as there are teachers, and this does hold it back.

>To me it's incomplete knowledge; it needs to be improved upon so that it makes more sense for people who need to know it. In doing so it will also become easier to learn.

I wish you luck, and I hope that I find it on here someday.

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