hamish_todd

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!)

You're certainly correct that it would be the done thing in physics (and I have written a draft for my blog).

But note that Geometric Algebra is significantly a tool for graphics/game developers, who tend to prefer videos (even the older generation I think)

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.

Ah! If you do it would probably would need to be a followup rather than addendum

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 :`(

He's distressingly wrong about this, the geometric product is transform composition

(Apologies to folks who have seen me repeat this a million times; but it's very important folks be aware of 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!

The geometric product is transform composition.

TRANSFORM COMPOSITION!

(sorry, it's not your fault or even his that you didn't know this - GA textbooks should have it as the first thing they teach but they don't)

Since Geometric Algebra == Clifford Algebra, Michael Atiyah, Roger Penrose, and Paul Dirac disagree with 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.