You are correct and this is another failing of the article. Hestenes has very little relevance to the field today - the Atiyah/Penrose school, via Jon Selig, is much more influential. To be fair the article would have been correct ten years ago, but not now (sorry Alex!)
Very sorry to hear about the brain fog, that's rough. Best of luck getting through it.
> in some cases
What cases do you have in mind?
> why are these your operators in the first place?
This is a perfectly reasonable question but I want to point out that it's a bit philosophical and not the kind of thing physics undergrads tend to enjoy hearing about. For example Clifford algebras like matrix algebras are associative algebras over commutative fields (real or complex numbers) - why? Why does the universe like that? I can hazard guesses but it's mostly above my paygrade, possibly above anyone's paygrade. But if it's to be asked of GA it should be asked of matrices too, I'm sure you can agree.
There's something that distinguishes Clifford algebras from matrix algebras. They start from the assumption that vectors anticommute when they're orthogonal. That's easy to explain. It says that if A and B are orthogonal vectors then A->B is the opposite (negation) of B->A.
Aside from those the last thing is that you say your basis vectors have a certain "metric" on them. There's deep philosophical questions about why the universe cares so much about metrics, but they're not at all specific to Clifford Algebra.
Personally I find that very intuitive, much more intuitive than any other tensor algebra I'm aware of.
In physics and especially computer graphics, yes. These are two fields that are culturally very different from algebraic geometry, in that mathematical techniques that appear at all "exotic" are "hard to sell", even if they're useful. Hence having to have a more marketable name for Clifford algebra ("geometric algebra" is just Clifford algebra), and having to have endless screaming from the rooftops for people to learn the slightest thing (like "multiplication is transform composition").
The geometric product is transform composition. It plays the same role as matrix multiplication: if a group is represented by a set of matrices or multivectors then the GP will be group composition.
For example Cl(4,2) can represent the spacetime conformal group and Cl(3,0,1) can represent the Euclidean group. If A and B are elements of those groups represented by multivectors then AB will be their composition.
That is an extremely fundamental operation. Almost as, if not more, fundamental than inner and wedge.
I mention this every time this article is posted on hacker news and nobody wants to talk about it :`(
Can do either! If your multivectors represent linear transformations then their products will also be linear transformations. If they're affine transformations then their products will be affine transformations. Euclidean, conformal, etc!
TLDR it is quite a bad article. One of the closest thing he has to a real argument is "I don't like it when geometric objects are identified with operators, I want those to be separate things". But this is both anti-GA and anti-Lie-Theory. As he says, he is critical of mathematics as conventionally practiced. So be warned that if you find yourself disliking GA for anything like the reasons he dislikes GA, there's a lot of other (mainstream/prestigious) fields you dislike too.
You're making the word "some" do a lot of work in this comment. It's true that eg I would represent the inertia tensor with a matrix. But I would calculate it with GA... and I struggle to think of many other physics problems for which I wouldn't use GA (Poisson bracket in GA is particularly elegant). See my other comment on projections I suppose.
On the other hand, from what you've posted you're clearly experienced with computer graphics - it sounds like you have at some point interpolated with dual quaternions?
This hackernews thread may not be sustainable, so perhaps join the bivector discord? https://discord.gg/bBvkuTrM I would be very interested to find out the precise things you care about, on the off chance that there is some not-yet-known-to-you way that GA can help you.
The geometric product does transform composition. It has all the properties you want for multiplication in an algebra, or indeed a monoid or group.
So it is like matrix multiplication, but for transforms represented as multivectors. Multivectors are nicer than matrices because they are made out of the separate (exterior algebra) objects so you can geometrically interpret them. For example, a rotation-reflection (rotoreflection/improper rotation) will have a grade 1 part and a grade 3 part. One of them is the plane you reflect in, one is the point you rotate around.
There are ten thousand examples I want to give of why you're wrong. We have to start somewhere so here's a favourite, the "universal projection formula":
(A.B)/B
Projects any A onto any B, in any number of dimensions and with any signature (eg hyperbolic/Euclidean/elliptic). A and B can be lines, planes, points, and with a conformal or anti de Sitter metric a sphere or hyperboloid etc ("blades").
It works because A.B is dimension independently the object "orthogonal to A and containing B or vice versa". And division by B will intersect that orthogonal object with B.
Concise, intuitive, and powerful. What's the linear algebra formula you'd consider to be comparable?
A quaternion is some amount of identity averaged with some amount of line reflection. You can visualize the line reflection as a pair of planar reflections at 90 degrees to each other. You can visualize the identity as a pair of planar reflections that are the same. Averaging the two of those will give you a pair of planar reflections that are some angle apart.
You're correct that you can construct a quaternion with the exponential map - but the most common way to make a quaternion is with a pair of vectors. Every game engine will have that function. GA will tell you how that function works - the vectors are planar reflections, you compose those to get a rotation by twice the angle, and you add (average with) the identity rotation to get a rotation by the precise angle.
> how to detach the two notions from each other
Can you say why would you want to do that? A plane always defines a planar reflection, a point always defines a point reflection, a line always defines a line reflection (assuming we're in euclidean space, which engineering is). To me this doesn't seem to be "happen to have" territory, this seems fundamental.
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!
I did two streams where I went through this article and explained the many places it is wrong. The second part of the article has more maths in it, so most of the content is there, you can watch it here: https://www.twitch.tv/videos/2282548167
(it's very long so I plan to edit the two streams into a digestible 10-15m or something. His fault not mine I'd say!)
Probably other commenters have already said, but the biggest giveaway is how he says we should move away from quaternions, and then demonstrates little to no awareness of why quaternions are used in engineering (vital in gamedev for example, your animations will look awful without quaternions). Yes, quaternions are hard if you are completely married to the idea that everything in geometry is ""vectors"". But the games industry put on its big-boy pants and learned to use them - they wouldn't do that if the things weren't useful for something, so it's bit silly to write an article like this if you haven't figured out why that happened.
They include a visual code editor I worked on - it's very janky (no proper errors) but you can play with it here https://hamishtodd1.github.io/ga/ed - ALT+ENTER to compile
From that page: "When Euler Angles interpolation is used, Unity internally bakes the curves into the Quaternion representation used internally. This is similar to what happens when importing animation into Unity from external programs"