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.
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.)