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The case against geometric algebra (2024)

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Re: The case against geometric algebra (2024)

#31

Post by jsLavaGoat »

eigenspace wrote: post #29
jsLavaGoat wrote: post #25

Earlier quoted context omitted.

What interests a mathematician isn't 100% the same as what interests the physicist. All I'm saying is there is some math there that's interesting and people should see it once for the math.

I guess I'd say my point though is that the gauge structure is the mathematically interesting part of Maxwell's equations. (i.e. the fact that `F` is a closed differential form). Without it, I think it'd be of significantly less mathematical interest because it'd lose almost all of its geometric properties.

There isn't just ONE interesting facet of this. There isn't just ONE mathematical formalism of a lot of these things. GA is just one of those approaches and you should see it just once, just like you should see the group structure and all of that as well. For most applications, the standard vector calculus approach is fine. But the math underlying all of this is full of richness and no one approach is the skeleton key.

Same with programming languages. Some people are like RUST RUST RUST and some are like C C C! I'm like, you guys only use one language?

Re: The case against geometric algebra (2024)

#32

Post by Twisol »

aureate wrote: post #24

Tiny nit / check of my understanding: > It was already widely understood that projective geometry allowed one to represent rotations and translations in R^3 with a single linear operator on R^4. I think it's projection operators (in linear algebra) that allow one to do that, not projective geometry [1]. The latter, AIUI, studies projective spaces and projective transformations on them (which differ from vector spaces…

As you say, projections and rotations are easily accounted for in linear algebra. The issue is that translations are not a linear transformation. For instance, consider f(x) = 2 + x. It's certainly not the case that f is linear -- that is, that f(cx + y) = c f(x) + f(y) -- because on the one hand we'd expect 2 + cx + y, and on the other we'd expect (2 + cx) + (2 + y), which is 4 + cx + y.

However, translation is an affine transformation, which is a particular case of a projective transformation [0]. It turns out that we can represent 3D affine (and general projective) transformations using a 4x4 matrix -- that is, as linear transformations in one dimension up, in a similar sense as how we can represent complex numbers as particular 2x2 matrices [1]. So yes, projective geometry is the right theoretical lens, even if we're usually able to forget about it (somewhat) when we use matrix representations.

[0]: https://en.wikipedia.org/wiki/Affine_transformation#Represen...

[1]: https://en.wikipedia.org/wiki/Complex_number#Matrix_represen...

Re: The case against geometric algebra (2024)

#33

Post by groundzeros2015 »

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…

> I could find in there was that the author does not like the people using/doing research in geometric algebra

The start of the article makes a specific technical claims:

> Hestenes’ Geometric Product is not a very good operation and we should not be rewriting all of geometry in terms of it

Later he explains why:

> there is no good general interpretation or usage for the geometric product or mixed-grade multivectors

Re: The case against geometric algebra (2024)

#34

Post by thomasahle »

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…

> After reading the article, the main "case against geometric algebra" I could find in there was that the author does not like the people using/doing research in geometric algebra

Mathematics is a social activity. The research cultures of different branches matter.

Re: The case against geometric algebra (2024)

#35

Post by Sharlin »

aureate wrote: post #24

Tiny nit / check of my understanding: > It was already widely understood that projective geometry allowed one to represent rotations and translations in R^3 with a single linear operator on R^4. I think it's projection operators (in linear algebra) that allow one to do that, not projective geometry [1]. The latter, AIUI, studies projective spaces and projective transformations on them (which differ from vector spaces…

It's one and the same, or rather, one is a special case of the other.

The homogeneous coordinate system used to represent affine transforms in R^n using linear transforms in R^(n+1) is exactly the same as what is used to represent projective transforms in the projective space P(R^n). This is famously exploited in 3D graphics where 4x4 matrices can represent linear and affine transforms and perspective projections (modulo the final w-division normalization step).

Affine transforms are a special case of projective transforms where the last row (or column depending on convention) vector is (0, ..., 0, 1).

Re: The case against geometric algebra (2024)

#36

Post by aureate »

Twisol wrote: post #32
aureate wrote: post #24

Tiny nit / check of my understanding: > It was already widely understood that projective geometry allowed one to represent rotations and translations in R^3 with a single linear operator on R^4. I think it's projection operators (in linear algebra) that allow one to do that, not projective geometry [1]. The latter, AIUI, studies projective spaces and projective transformations on them (which differ from vector spaces…

As you say, projections and rotations are easily accounted for in linear algebra. The issue is that translations are not a linear transformation. For instance, consider f(x) = 2 + x. It's certainly not the case that f is linear -- that is, that f(cx + y) = c f(x) + f(y) -- because on the one hand we'd expect 2 + cx + y, and on the other we'd expect (2 + cx) + (2 + y), which is 4 + cx + y. However, translation is an a…

Ah, interesting. I see "homogeneous coordinates" are covered later in the book I've just started reading (Projective Geometry, Coxeter) as a way of representing projective space. I think that's the link I couldn't see.

Thanks!

Re: The case against geometric algebra (2024)

#37

Post by blurbleblurble »

thomasahle wrote: post #34
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…

> After reading the article, the main "case against geometric algebra" I could find in there was that the author does not like the people using/doing research in geometric algebra Mathematics is a social activity. The research cultures of different branches matter.

Which is why articles like this are obnoxious

Re: The case against geometric algebra (2024)

#38

Post by leoc »

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…

It does starts to sound a bit like chortling about what a weird asshole Semmelweis is. ISTR to recall that US students of linguistics were slow to adopt the International Phonetic Alphabet because it North America it had become associated with elocutionists, and no proper academic linguist wanted to look like an elocutionist grubby.

Re: The case against geometric algebra (2024)

#39

Post by aureate »

Sharlin wrote: post #35
aureate wrote: post #24

Tiny nit / check of my understanding: > It was already widely understood that projective geometry allowed one to represent rotations and translations in R^3 with a single linear operator on R^4. I think it's projection operators (in linear algebra) that allow one to do that, not projective geometry [1]. The latter, AIUI, studies projective spaces and projective transformations on them (which differ from vector spaces…

It's one and the same, or rather, one is a special case of the other. The homogeneous coordinate system used to represent affine transforms in R^n using linear transforms in R^(n+1) is exactly the same as what is used to represent projective transforms in the projective space P(R^n). This is famously exploited in 3D graphics where 4x4 matrices can represent linear and affine transforms and perspective projections (mo…

Yes, I think I understand it (or am at least on the way to understanding it) now. Thanks!

Re: The case against geometric algebra (2024)

#40

Post by cygx »

jsLavaGoat wrote: post #5

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. I guess the people pushing this are a little pushy, but this reminds me of the whole pie fight over th…

Note that by introducing the co-differential δ, you can write the Maxwell equations as a single expression (δ + d)F = J in the differential forms approach.

However, from the perspective of Yang-Mills theory, that's rather questionable as you're stitching together the Bianchi identity and the Yang-Mills equation for no particular reason.

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