The second paragraph of the conclusion: "Nor should we be trying to make everything look more like complex numbers and quaternions. Those are already weird and confusing; we should be moving away from them!"
It's also implicit in the thing he says throughout: "bivectors and trivectors are good, but there's no reason to add a scalar to a bivector or a trivector to a 1-vector, nor is there a reason to multiply such objects". A quaternion is a scalar and a bivector added together!
Hi I wrote that article and I would say that I am extremely aware of how quaternions are used in engineering.
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. But GA isn't much less opaque either. The "actual" representation which I find to be most agreeable is the stuff I mentioned about viewing them as operators. Given a bivector B which describes a rotation, you can treat it as an operator on vectors via contraction R(v) = B⋅v. Then exponentiating e^(Rθ)(v) (either the one-sided rotations or two-sided rotors) gives the same rotation formalism as quaternions and GA, but without any of the weird unpedagogical stuff. The notion of an exponential map and exponentiating generators is, IMO, much more "natural" and intuitively straightforward than the alternatives. Perhaps I should update the article to make this more clear.
What irritates me about GA---well, one of the things---is that it treats bivectors/trivectors as both geometric primitives (oriented areas) and operators (rotations, say), and totally conflates the two and never explains to the student how to detach the two notions from each other (and I doubt most of the writers on the subject even know). IMO it is best viewed as a version of representation theory: rotations are operators which happen to have representations on bivectors of the EA; not all operators will have that property, and then you will want other algebraic structures to do algebra with them, if that's a case you're considering.
> I have given a lot of reasons why I think GA is problematic: the Geometric Product is a bad operation for most purposes. It really implements operator composition and is not a very fundamental or intuitive thing. Using a Clifford Algebra to implement geometry is an implementation detail, appropriate for some problems but not for general understandings of vector algebra and all of geometry. Giving it first-class status and then bizarrely acting like that is not weird is weird and alienating to people who can see through this trick.
If I understand him correctly, he means Clifford algebra is "appropriate for some problems" but we should "move away" from "giving it first-class status" as it is not more fundamental and often does not help students understand geometry better. I also readily admitted that it has some use cases in game physics in my comment.
hamish_todd|1 year ago
It's also implicit in the thing he says throughout: "bivectors and trivectors are good, but there's no reason to add a scalar to a bivector or a trivector to a 1-vector, nor is there a reason to multiply such objects". A quaternion is a scalar and a bivector added together!
ajkjk|1 year ago
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. But GA isn't much less opaque either. The "actual" representation which I find to be most agreeable is the stuff I mentioned about viewing them as operators. Given a bivector B which describes a rotation, you can treat it as an operator on vectors via contraction R(v) = B⋅v. Then exponentiating e^(Rθ)(v) (either the one-sided rotations or two-sided rotors) gives the same rotation formalism as quaternions and GA, but without any of the weird unpedagogical stuff. The notion of an exponential map and exponentiating generators is, IMO, much more "natural" and intuitively straightforward than the alternatives. Perhaps I should update the article to make this more clear.
What irritates me about GA---well, one of the things---is that it treats bivectors/trivectors as both geometric primitives (oriented areas) and operators (rotations, say), and totally conflates the two and never explains to the student how to detach the two notions from each other (and I doubt most of the writers on the subject even know). IMO it is best viewed as a version of representation theory: rotations are operators which happen to have representations on bivectors of the EA; not all operators will have that property, and then you will want other algebraic structures to do algebra with them, if that's a case you're considering.
howling|1 year ago
> I have given a lot of reasons why I think GA is problematic: the Geometric Product is a bad operation for most purposes. It really implements operator composition and is not a very fundamental or intuitive thing. Using a Clifford Algebra to implement geometry is an implementation detail, appropriate for some problems but not for general understandings of vector algebra and all of geometry. Giving it first-class status and then bizarrely acting like that is not weird is weird and alienating to people who can see through this trick.
If I understand him correctly, he means Clifford algebra is "appropriate for some problems" but we should "move away" from "giving it first-class status" as it is not more fundamental and often does not help students understand geometry better. I also readily admitted that it has some use cases in game physics in my comment.