[howling]

An article that argues against GA: https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...

TLDR:

- GA tends to attract a lot of crackpot. In fact most mathematicians avoid the name Geoemtric algebra and call it Clifford algebra to disassociate with them.

- Most of the usefulness of GA comes from just exterior algebra and exterior product/wedge product is more important than geometric product.

- It shows up in spinor field in physics but this does not concern most people that do not work in quantum physics.

My rudimentary view towards GA:

- It is useful in game physics since rotors can represent n-dimensional rotation in 2^{n-1} numbers instead of n^2 numbers as 2^{n-1} < n^2 when n <= 6. You can use PGA if you want to add translation as well. It is also better in interpolation.

- Outside of this you should just probably just learn exterior algebra instead.

[hamish_todd]

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.

[asplake]

A quaternion is just an even subalgebra of the Clifford algebra Cl3,0(R), what’s the problem?

[howling]

> but the biggest giveaway is how he says we should move away from quaternions

I'm sorry I must have missed that part. Can you point me to where did he say this?

[aap_]

The article reads more like a trolling attempt. Geometric/Clifford algebra is incredibly useful and by throwing away its product you lose a lot of the power of the algebra. It's like saying matrix multiplication is not useful and you really want to be multiplying and adding numbers in various ways. After all GA/CA elements can always be mapped to elements of a matrix algebra. To get rid of the idea of linear transformations and that they should compose just doesn't sound well thought through.

I don't know what sort of crackpots he's talking about, personally i haven't heard of them, only the accusations. If the author can't separate the math from the people who developed and/or popularized it, too bad. Does GA magically give intuitive explanations for all sorts of weird things? no. Can you formulate a lot of stuff much more efficiently and concisely, and does it help gain new perspective on some things? yes, absolutely. It provides a wonderful framework for expressing geometric ideas.

[dist-epoch]

> As for pure math—it seems like research mathematics readily talks about and uses Clifford Algebra, but is uninterested in or specificaly avoids the terms and concepts that are specific to Hestenes’ “Geometric Algebra”. I can speculate as to why: even by the 90s/00s, GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots.

https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...

[howling]

> Can you formulate a lot of stuff much more efficiently and concisely, and does it help gain new perspective on some things? yes, absolutely. It provides a wonderful framework for expressing geometric ideas.

Can you elaborate on what stuff does it help to formulate much more efficiently and concisely?

[rhelz]

// tends to attract a lot of crackpot. //

Come on. You know what else attracted a lot of crackpots? The internet. If you are criticizing math, criticize the math, not the people.

// just learn exterior algebra instead of//

YMMV, but I like to know where the mathematical concepts came from. GA gives a nice origin story, see below:

// Most of the usefulness of GA comes from just exterior algebra //

Dot products come from the geometric product. If e1 & e2 are two basis vectors such that e1*e1 = 1, and e1e2 = -e2*e1, then if you multiply two vectors:

(a1*e1 + a2*e2)(b1*e1 + b2*e2) =

a1*b1*e1*e1 + a1\b2*e1*e2 a2*b1*e2*e1+ a2*b2*e2*e2 =

a1*b1 + a2*b2 + (a1*b2 - b2*a1)*e1*e2 =

(a . b) + (a ^ b)

The first is the dot product. The second is the exterior product that everybody agrees is so useful. Now you know where both concepts came from. They are just from multiplying polynomials. The geometric product is a *a product*, it's the product of two polynomials.

Yes, sometimes you just need the dot product, and sometimes you just need the exterior product. If you are coding, or giving the final form of some formula, you don't have to always put both of them in your code or paper. But neither the dot product nor the wedge product are investable by themselves. Having an investable product on vectors is endlessly useful while you are *deriving* the formulas.*

[howling]

> Yes, sometimes you just need the dot product, and sometimes you just need the exterior product. If you are coding, or giving the final form of some formula, you don't have to always put both of them in your code or paper.

In my experience 99% of the time you just want the dot product or the exterior product. Even when you want both it is rare that you want to combine them linearly except in some niche physics/mathematics.

> But neither the dot product nor the wedge product are investable by themselves. Having an investable product on vectors is endlessly useful while you are deriving the formulas.

Do you mean invertible? Why is invertibility is so useful?

[buescher]

Without reading either article - yet - I can tell you "crackpot" can mean a lot of things, and one of them that is relevant to this context is "an academic that's more interested in ideas than in being a good research bureaucrat".

[adornKey]

Interesting article.

I have been telling people for more than a decade now to use the exterior product (and e.g. get rid of the clutter of strange Minus-Signs and div, grad and rot in Electrodynamics).

And I was really happy to see that people finally start doing that.

But when I saw the Geometric Product, it didn't look like anything I want. If someone says that it looks like a thing that the cat brought in, I'll think about it and will probably agree.

[hamish_todd]

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.