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Poor Foundations in Geometric Algebra

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Re: Poor Foundations in Geometric Algebra

#121

Post by elengyel »

ajkjk wrote: post #74
mgarym wrote: post #55

Earlier quoted context omitted.

Lengyel's post doesn't discuss the whole plane-based vs. point-based war at all, and he makes it clear in his book that both approaches are equivalent. He actually says both are happening at the same time whichever way you look at it, but I'm still trying to wrap my head around that. Gunn is not one of the authors of the material Lengyel is talking about in his post.

I am not sure I understand the plane-based GA model, but I imagine it's that in exterior algebra there is another product that's totally dual to the wedge product (people call it the "antiwedge product" or "regressive product"), and the plane-based and point-based models just swap the two symbols.

Hi Alex -- In PGA, every operation comes in pairs. There are two exterior products, two inner products, and two geometric products (and the list goes on). If points are represented by vectors, then the quaternion-like sandwich qpq* with the geometric product, where q is now a more general operator in the algebra, always fixes the origin. Thus, it cannot perform Euclidean isometries in regular space because those (in general) move the origin. However, a fixed origin in regular space means that the horizon is fixed in antispace, so if you were to reinterpret vectors as planes instead, then you do get the set of Euclidean isometries that you want. If you had no knowledge of the geometric antiproduct, then you would just say "vectors are planes" and call it a day. That's where plane-based GA comes from. Just use the geometric product and interpret all geometries in antispace instead of regular space. But this throws out the geometric intuition shown in Figures 2.4, 2.5, and 2.7, where vectors, bivectors, and trivectors are simply projected into the w=1 subspace to de-homogenize points, lines, and planes. Furthermore, we need the general notion of product-antiproduct pairs to get things like norms working, anyway, so we might as well use them to avoid dualizing all the geometries.

The space/antispace duality is discussed in Section 2.6, and the fact that the geometric product fixes the origin is discussed in Section 3.5.1. (In case anyone else is wondering, I know ajkjk has a copy of my book.)

Re: Poor Foundations in Geometric Algebra

#122

Post by elengyel »

ajkjk wrote: post #116
gugagore wrote: post #110

Earlier quoted context omitted.

By the way, I love your blog post https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm... I didn't understand this part: > I strongly believe that if GA would make this distinction they would lose a lot fewer people. It is a completely interesting and useful thing to talk about “a representation of a particular class of operations that makes composition and inversion easy”, and completely offputting when you…

So, I've amended the article some since posting it because the main objection I got from a few GA enthusiasts was this idea of the GP as being used to compose operators. Which I had barely noticed the importance of because it's so hard to identify in the texts! Although once they mentioned it I began to appreciate that, when the GP works and is useful, this is why. So I changed things to address that point more direc…

My position is that the geometric product and antiproduct are good for one thing, performing transformations with sandwich products q ⟑ p ⟑ q̃ or q ⟇ p ⟇ q̰ and composing those transformations. Literally everything else (join, meet, contraction, expansion, projection, inner product, norm, ...) can and should be done in the exterior algebra without any geometric products.

Re: Poor Foundations in Geometric Algebra

#123

Post by altairprime »

obsoletehippo wrote: post #50
altairprime wrote: post #33

Earlier quoted context omitted.

> Yet, I don't think Gunn should be criticized for it at all. The article dedicates itself to critiquing the techniques, but spends no time that I can remember talking about the human beings and their feelings. That’s how I write professionally, and I’ve found it can really upset people who infer that critique of some thing is therefore critique of some person — even when no such inference is intended by the author.…

Well, he describes some folks' work as "crackpot-level bullshit" and repeatedly states that other authors "have no idea what they're talking about". These seem to me to be personal critiques. Generally, I'm not sure the distinction you draw is quite so clear cut -- if I say somebody is great but all their work is hot garbage, that sounds fairly personal.

Mmm. Yeah, I’ll concede that. Agreed that it would be better without — I clearly glossed past it but that doesn’t excuse it being present.

Re: Poor Foundations in Geometric Algebra

#124

Post by altairprime »

andrewflnr wrote: post #52
altairprime wrote: post #33

Earlier quoted context omitted.

> Yet, I don't think Gunn should be criticized for it at all. The article dedicates itself to critiquing the techniques, but spends no time that I can remember talking about the human beings and their feelings. That’s how I write professionally, and I’ve found it can really upset people who infer that critique of some thing is therefore critique of some person — even when no such inference is intended by the author.…

He repeatedly makes assertions that "the authors" have "no idea what they're talking about". The difference between flaming an individual person and "the authors" is thin at best, arguably only different in that it flames more than one person at once. And honestly, to the extent he's correct, I think that's fine. But let's not kid ourselves about what we're reading.

Agreed, I didn’t catch that nuance, thanks for highlighting it.

Re: Poor Foundations in Geometric Algebra

#125

Post by auggierose »

elengyel wrote: post #120
auggierose wrote: post #90

Earlier quoted context omitted.

They are talking about different stuff. Geometric algebra is here the name used for the popularisation of a brand of Clifford algebra. The popularisation books are often written by physicists and engineers, and their maths is shaky, because they care more about applications than about proofs and rigour. There are exceptions, for example I like the two books by Alan Macdonald about the topic: http://www.faculty.luther…

I think Macdonald's book is very concise and clean compared to others, but it does have some of the same issues that I wrote about in my post. In particular, Definition 6.15 gives one of the problematic definitions of the inner product, and Definition 6.23 gives the same broken definition of dual that has an inconsistent orientation and fails to extend to the degenerate metrics of projective algebras. Has also says,…

Thank you for this analysis, will check it out!

Re: Poor Foundations in Geometric Algebra

#126

Post by meindnoch »

elengyel wrote: post #121
ajkjk wrote: post #74

Earlier quoted context omitted.

I am not sure I understand the plane-based GA model, but I imagine it's that in exterior algebra there is another product that's totally dual to the wedge product (people call it the "antiwedge product" or "regressive product"), and the plane-based and point-based models just swap the two symbols.

Hi Alex -- In PGA, every operation comes in pairs. There are two exterior products, two inner products, and two geometric products (and the list goes on). If points are represented by vectors, then the quaternion-like sandwich qpq* with the geometric product, where q is now a more general operator in the algebra, always fixes the origin. Thus, it cannot perform Euclidean isometries in regular space because those (in…

>In case anyone else is wondering, I know ajkjk has a copy of my book.

Is there only a paper version of your GA book?

Re: Poor Foundations in Geometric Algebra

#127

Post by ajkjk »

elengyel wrote: post #122
ajkjk wrote: post #116

Earlier quoted context omitted.

So, I've amended the article some since posting it because the main objection I got from a few GA enthusiasts was this idea of the GP as being used to compose operators. Which I had barely noticed the importance of because it's so hard to identify in the texts! Although once they mentioned it I began to appreciate that, when the GP works and is useful, this is why. So I changed things to address that point more direc…

My position is that the geometric product and antiproduct are good for one thing, performing transformations with sandwich products q ⟑ p ⟑ q̃ or q ⟇ p ⟇ q̰ and composing those transformations. Literally everything else (join, meet, contraction, expansion, projection, inner product, norm, ...) can and should be done in the exterior algebra without any geometric products.

agree but I am still trying to grok the divine truth as to why exactly that is. What's up with the sandwich products? Why do they work? I guess it is like a change-of-basis for a matrix (PAP^{-1)) but I still don't quite see why, and why it works as a change-of-basis on multivectors, not just vectors.

Re: Poor Foundations in Geometric Algebra

#128

Post by rhelz »

meindnoch wrote: post #126
elengyel wrote: post #121

Earlier quoted context omitted.

Hi Alex -- In PGA, every operation comes in pairs. There are two exterior products, two inner products, and two geometric products (and the list goes on). If points are represented by vectors, then the quaternion-like sandwich qpq* with the geometric product, where q is now a more general operator in the algebra, always fixes the origin. Thus, it cannot perform Euclidean isometries in regular space because those (in…

>In case anyone else is wondering, I know ajkjk has a copy of my book. Is there only a paper version of your GA book?

Just buy his books man. They'd be a bargain at 10x the price.

Re: Poor Foundations in Geometric Algebra

#129

Post by meindnoch »

rhelz wrote: post #128
meindnoch wrote: post #126

Earlier quoted context omitted.

>In case anyone else is wondering, I know ajkjk has a copy of my book. Is there only a paper version of your GA book?

Just buy his books man. They'd be a bargain at 10x the price.

I would buy a digital version. I don't have more space for paper books in my apartment.

Re: Poor Foundations in Geometric Algebra

#130

Post by generationP »

elengyel wrote: post #119
generationP wrote: post #65

Earlier quoted context omitted.

The subject doesn't lack rigor, but all of the popular and quite a few of the mathematically minded introductions to it do. Bourbaki and Chevalley did a good job introducing Clifford algebras properly some 80 years ago, but even this has been forgotten by the modern bloggy expositors. This here seems to be one of the few good modern texts that are elementary yet rigorous: https://www.mathematik.uni-muenchen.de/~lundh…

Unfortunately, the Lundholm and Svensson text you've cited suffers from the same problems I wrote about in my post. Definition 2.7 connects the interior products to the scalar product, not the inner product. Definition 2.8 gives the same broken definition of dual that has an inconsistent orientation and fails to extend to the degenerate metrics of projective algebras.

It fixes the worst problem, which is the well-definedness of the interior product, since it defines the product explicitly on the standard basis rather than through a bunch of not-obviously-consistent axioms. It is too basis-dependent for my tastes (it should depend on the symmetric bilinear form, not on the basis). As for the sign conventions, I think they're ultimately matters of taste, and while I agree with you about the inner/scalar product, I disagree about the derivation-ness of a contraction (I like it to satisfy the Koszul sign rule, which allows signs to be inserted only when some factors move past each other).

My ideal approach is along the lines of Chevalley's "The Algebraic Theory of Spinors and Clifford Algebras", Chapter III, but he doesn't get to much geometric algebra.

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