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The case against geometric algebra (2024)

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Re: The case against geometric algebra (2024)

#121

Post by jiggawatts »

skybrian wrote: post #54
jiggawatts wrote: post #13

Earlier quoted context omitted.

As someone who studies physics and then went into a long IT career (but kept reading papers casually), my view is that this whole GA saga is very reminiscent of how after decades of experience, I still can't convince juniors of the benefits of what I now consider obvious best practices. No amount of demonstrations of the blindingly obvious improvement of some better technique seems to work on someone who "finally got…

As a programmer I’m wondering how you get a decent graphics library out of it. If it’s conceptually better, shouldn’t it make writing code to do calculations easier?

I gave this a go a couple of years ago. GA multivectors are a direct substitute for both vectors and matrix transformations. The challenge is efficiency: a unified type is too expensive for real time games. The typical approach is to create efficient subsets symbolically and bake those into the code so that the compiler has something it can work with. This works, but the "strong typing" of the mathematics and typical programming languages doesn't exactly mesh, so a clean typesystem/interface design is annoyingly difficult.

The main advantage is the elimination of gimbal lock which allows smooth interpolations of arbitrary rotations and translations. This dramatically simplifies certain codes relating to animation and robotics. In mathematics, it simplifies differentiation and integration over curved surfaces in 3D and higher dimensions.

Most developers working in those fields already use one of the many creatively named “impostors” of GA subsets in isolation, such as the quarternions.

Re: The case against geometric algebra (2024)

#122

Post by hamish_todd »

adrian_b wrote: post #89

In my opinion, the article is very mistitled, because it does not contain even a single valid criticism against the geometric algebra theory, despite containing some perfectly valid criticism against some mistakes frequently made by geometric algebra proponents. The author has completely failed to understand the meaning and the purpose of geometric algebras, though to be fair this is not entirely the author's fault,…

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 :`(

Re: The case against geometric algebra (2024)

#123

Post by ajkjk »

hamish_todd wrote: post #110
cherryteastain wrote: post #30

Not a fan of the article. It resorts to ad hominem attacks like > GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots. Hestenes honestly sounds like one a lot of the time, and I’m not really sure whether he is or isn’t. It makes sense, really. > GA ended up appealing to a lot of fringes: people who only had undergraduate degrees, people who had dropped out of PhD…

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)

I completely agree, incidentally... And I have come to see this more clearly since writing the article, in part due to talking to you. Maybe I should update it accordingly. I've been meaning to write a followup where I explain this viewpoint but have had trouble finding the mental energy for it.

Re: The case against geometric algebra (2024)

#124

Post by ajkjk »

chombier wrote: post #91
eigenspace wrote: post #14

Earlier quoted context omitted.

> From a mathematician's point of view, yes, you should write the Maxwell field equations, at least to see it once, that way because you're showing a very low-level symmetry that even the differential forms approach doesn't get all the way to. Differential forms is a standard approach for general relativity, e.g. MTW. While it's neat to write them all as one equation, I disagree that it's an enlightening perspective…

> pull out Hodge Duals every time you want to do something that involves the metric, but I'm also unconvinced that geometric algebra is the answer here. I don't know, I recently tried to work out how the metric on vectors/1-forms induces a metric on higher-degree forms, and if the geometric product magically gives this for free I'd say it's a win (same for the Hodge star).

That comes from exterior algebra on its own, it's the k'th exterior power of the metric. Best not to conflate that with GA (unless I'm misunderstanding what you're talking about).

Re: The case against geometric algebra (2024)

#125

Post by jiggawatts »

SonOfLilit wrote: post #120
jiggawatts wrote: post #116

Earlier quoted context omitted.

To be honest I was struggling to phrase my argument in a cohesive narrative without it turning into a ten page blog post. The point I’m trying to make is that there are necessarily complexities inherent in all areas of study, and there are incidental complexities because of historical reasons, “culture” within certain fields, or juniors putting out their fields’ equivalent of spaghetti code. Geometric Algebra sweeps…

I tried to make it clear that I wasn't arguing against your main point, that was made very clearly, just against a comparison you used that I think was a bit slanderous (tongue in cheek). Yes, obviously Tau is correct, and that's a better comparison to use. Having dived deeper into the essay, author claims that some of the new notation is obviously better (clifford algebras) and the rest is overzealous unification th…

I'm a bit pressed for time, but one annoyance I've had with the classic "greek" physics notation is that they represent things from "both ends" of a graded vector space. So for example, they start with a scalar, then a vector, then ... pseudoscalar-1, and finally the pseudoscalar.

It's a shortcut useful only if you need to scribble on paper and your wrists hurt from writing too much, but it obscures the underlying physics.

The programming equivalent is putting abbreviations in identifiers where, sure, it's fewer characters, but then the reader needs to a track a mental lookup table to translate back to the intended meaning.

Pushing things like this too far results in meaningful aspects of the equations getting squeezed out entirely. For example, the generality of GA means that you have to (correctly) track negative signs and multiplications by pseudoscalars such that your formulas work in all dimensions. In traditional vector algebra it's all too tempting to eliminate certain products because in "your chosen dimension" they multiply to 1 or -1 or whatever and just... disappear due to traditional algebraic simplification conventions. But then if you need to work in 4D SR or curved spaces, you can't, because you threw away something essential while "optimising for characters on a page".

You have then "start over", typically reaching for a partial and incomplete subset of GA, reinventing that wheel over and over.

Hence the push for unification onto GA, to break this cycle.

Re: The case against geometric algebra (2024)

#126

Post by jdshaffer »

hamish_todd wrote: post #109

I'm a GA researcher. I did this livestream of my reaction to the article https://m.twitch.tv/videos/2282548167 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…

I've become more and more curious about GA after bumping into it when learning a little 3D programming for fun. I'm looking to learn more, wondering if it would help my understanding of physics.

Do you have any resources (books, videos, etc.) you would recommend to someone wanting to learn?

Re: The case against geometric algebra (2024)

#127

Post by Certhas »

Loquebantur wrote: post #57
Certhas wrote: post #10

Earlier quoted context omitted.

The space time approach with E as t wedge x and B as x wedge y is purely linear algebra, not differential forms. As opposed to the weird GA form it actually makes the physically most meaningful symmetry (Lorentz transformations) explicit. That's why it's actually used in Physics. Anti symmetric space time tensors are the absolute standard . Further formulations that reveal other aspects, dualities, symmetries are muc…

One should teach the next generation the best way possible, and not turn them into conformists. "Standards" are things to be overcome when they've outlived their prime. Disparaging new ideas as "niche" and "specialised" when their explicit aspiration is to be better foundations is motivated reasoning.

Maybe I wasn't clear: Other formulations that reveal dualities, etc... at the cost of obscuring the fundamental covariance deserve to be treated as secondary.

Re: The case against geometric algebra (2024)

#128

Post by Certhas »

Steuard wrote: post #105
Certhas wrote: post #6

100 percent agree with the article. Wedge products are fundamental, GA is weird ideology. I had the bad fortune of reviewing some GA research articles once upon a time. It was almost embarrassing. Everything of substance had been published in a conceptually cleaner bivector language previously. The only "contribution" was writing everything in terms of weirder, more convoluted concepts that contributed neither techni…

Do you by chance have references to those conceptually cleaner bivector publications? I've spent a good bit of time looking for other people working in that space, and haven't found much other than an article by Jancewicz from 1980. (I like to imagine that my articles these past few years have been "conceptually clean", but they certainly aren't "previous" to much GA work.)

I am talking about some very specialised results in an application area. Not Electro Magnetism which I believe you are referring to?

The formulation of EM with antisymmetric field tensors (which is the same thing as a bivector) is certainly very old and absolute standard in the physics curriculum. And I certainly was taught the definition in terms of wedge products that is also given here in undergrad:

https://en.wikipedia.org/wiki/Electromagnetic_tensor

(See the section: Relationship to the classic fields).

Not sure what you are looking for beyond that?

Re: The case against geometric algebra (2024)

#129

Post by hamish_todd »

markgall wrote: post #114

Is this really a big debate? I am in the similarly-named (but apparently distant) field of algebraic geometry and have never even heard of geometric algebra. Certainly I know about Clifford and exterior algebras, but this debate has never reached me.

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").

Re: The case against geometric algebra (2024)

#130

Post by hamish_todd »

jdshaffer wrote: post #126
hamish_todd wrote: post #109

I'm a GA researcher. I did this livestream of my reaction to the article https://m.twitch.tv/videos/2282548167 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…

I've become more and more curious about GA after bumping into it when learning a little 3D programming for fun. I'm looking to learn more, wondering if it would help my understanding of physics. Do you have any resources (books, videos, etc.) you would recommend to someone wanting to learn?

If you want to learn physics from a geometric viewpoint, GA is brilliant. Here's a playlist of my stuff, which is aimed at computer graphics folks: https://youtube.com/playlist?list=PL9a8DfUJQcuA9AXqvjxYolq5-...

Leading up to classical mechanics, you have the sibgraphi tutorials: https://youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZ... (I also recommend the bivector discord if you want a community)

And from there, sudgy's videos are good, and the tome "geometric algebra for physicists" packs in a huge amount of stuff.

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