Poor Foundations in Geometric Algebra

https://terathon.com/blog/poor-foundations-ga.html

by ibobev • 2 years ago

229 105 2 years ago

terathon.com

mgarym4• 2 years ago

Oh man, I didn't know there was going to be a special episode of Friday Night Smackdown!

user070223• 2 years ago

Note don't to be confused with Algebraic Geometry they are different;

When I first came across this topic it was eye opening, especially the fact that you could squeeze Maxwell's equations into one and the fact that pseudovector create by cross product from physics is just a bivector which in 3d could be represented like a vector orthogonal to the plane created by the two vectors in the product

Primer on the topic https://www.youtube.com/watch?v=60z_hpEAtD8 (And other videos on his channel) Another greate playlist is https://www.youtube.com/watch?v=0VGMxSUDBH8&list=PLLvlxwbzkr...

BTW the author has the following page https://projectivegeometricalgebra.org/ with great infographics and references

BenoitP• 2 years ago

Hah! I knew Sudgy would be your first link. There's relatively little content about GA, but once you're hooked you begin to consume it all.

It's like you never got to study Relativity, conformal geometry, electron spins, quaternions, etc then someone comes with a simple cheat code which introduces you to these topics gently. It's like what category theory wants to do for mathematics, but simple.

Here are additional resources:

GA playground: https://enkimute.github.io/ganja.js/examples/coffeeshop.html...

A physics engine in 100 lines (Gravity, Hook, and damping laws are just one line each!), and you can go from 2D to 3D to 4D by changing a single parameter: https://enki.ws/ganja.js/examples/pga_dyn.html

Other resources: https://bivector.net/

And anything by David Hestenes: https://worrydream.com/refs/Hestenes_2002_-_Reforming_the_Ma...

gowld3• 2 years ago

The absolute first video to watch should be the (ironically titled) Why can't you multiply vectors?

https://www.youtube.com/watch?v=htYh-Tq7ZBI

Certhas3• 2 years ago

Note though that GA is extremely niche and offers very little beyond exterior algebra. Exterior algebra in the context of differential forms is standard part of the postgrad (and sometimes undergrad) Physics curriculum.

See the applications to physics section here: https://en.m.wikipedia.org/wiki/Differential_form

carterschonwald• 2 years ago

This explains very nicely why despite working through lots of GA resources, I never quite grasped it. It’s logically incoherent!

adrian_b• 2 years ago

This is not a critic of all geometric algebra and especially not of its more basic parts. Therefore it is not an excuse for not grasping e.g. what Hestenes has written about GA, or what Eric Lengyel himself has written, e.g. in his new book that is advertised in this article.

It is a critic of many books about geometric algebra, which have made attempts to expand and further develop some parts of its theory, but those attempts have not been thought carefully and they have produced various inconsistent or useless definitions.

It is also a critic of attempts of presenting geometric algebra as preferable for applications where in fact it is not optimal, by showing misleading "benchmarks". Unfortunately this tactic is not at all specific to geometric algebra, but it is frequently encountered for almost any kind of algorithm known to mankind when it accumulates for one reason or another some kind of fan base.

wudangmonk• 2 years ago

The author has books on Geometric Algebra so it makes no sense to assume that he is going against GA as a whole.

joe_the_user• 2 years ago

Calculus had poor foundation and was thus logically incoherent from Newton/Leibniz' discovery to roughly the middle of 19th Century. None-the-less it was a powerful tool and most of the key theorems were discovered then.

The basic situation, I think, is a set of tools can be consistent in the way mathematicians use them but in the way the mathematicians explain them. And the tools can be very useful despite this.

So my guess is saying "it has poor foundations" isn't saying "I'm against it, it's worthless"

catgary• 2 years ago

Yeah I’m pretty well-versed in free constructions of various algebraic objects and how this would interact with things like a quadratic form, etc, but couldn’t sort out GA (I think the authors of “GA4CS” had a very different sort of computer scientist in mind). When I saw the geometric algebra constructions clash with constructions I was already familiar with, I generally got suspicious and lost interest.

I’m actually quite interested in checking out Lengyel’s book. It looks rock solid.

mgarym4• 2 years ago

I have Lengyel's book and it's great. Highly recommended.

jdeaton• 2 years ago

very real toxicity within the geometric algebra community. I can’t do much about

I was hoping the article would be about this instead. OP wondering if you have any elaborations for us to hear.

nicf2• 2 years ago

I'm not the OP, I'm not a part of this community, and I don't know if the thing I'm about to complain about is what the author was thinking of with this comment, but as someone who was trained as a mathematician and who has read some of the popularizations of geometric algebra that sometimes get posted to HN, there is a tone that some (though probably a minority) of them take that I find pretty obnoxious.

These pieces are the ones that take the position that geometric algebra is this super secret anti-establishment mathematical samizdat that *they* don't want you to know about. They'll pit themselves against "mainstream mathematics" and say things like, "in differential geometry you do X, but you shouldn't do differential geometry; you should do geometric algebra where we do Y, which is so much better than X."

My reaction is always, "My friend, you are doing differential geometry!" Clifford algebras --- the objects that the geometric algebra people study --- are firmly within the "mainstream" of mathematics; there's simply no conflict here, at least not of the sort that these writers often seem to be imagining. It's great that people are enjoying learning about Clifford algebras. I think Clifford algebras are really fun! But we can all just come together and enjoy them together, and I think this "join me in taking down the cabal of gatekeepers who are suppressing the truth" attitude is unnecessary and turns off a lot of people who might otherwise be fun to engage with.

If you're into this stuff and feel like this doesn't describe you or the people you know, then that's great, keep doing what you're doing! But it does exist and I wish it didn't.

gowld3• 2 years ago

It goes both ways.

Mathematicians will take a moment denigrate Geometric Algebra as "linear algebra with a uselessly nonstandard notation", ignoring that we should prefer a less awkward way of structuring linear algebra than "pseudoscalars" and "pseudovectors".

thechao• 2 years ago

I used GA as a way to bootstrap into 'real' Clifford algebras, and a way to get over a "reader's block" when it came to Lie algebras, tensors, and (finally) algebraic geometry. I'm not sure GA is great math, but it was really great way to learn "advanced math concepts" for "basic..ish math". Personally, I like Alan MacDonald's GA books — they're a great way to learn more complicated concepts, but couched in a very approachable geometry/visual learning style.

MarkusQ• 2 years ago

Ouch.

I love it when people say what they mean and don't beat around the bush.

mncharity2• 2 years ago

So much education content is so very poor. And even the best of it... A first-tier physics professor at a munch was delighted - he regaled believing he had found an error in a highly-regarded introductory textbook! But, upon many day's of thought, and several close reads of the text, he had realized it had been very carefully worded so as to be not incorrect. And so he was so delighted - yay! He thought of this as a good state of affairs, reflecting well on the text, and associated instruction. I... afterwards wished I'd pointed out that the target audience for an intro textbook, was perhaps not well modeled as first-tier experts with a week to wrestle with and closely ponder a paragraph in order to avoid being misled.

I'm unclear on how we get better at this. I've seen OER texts with open errata databases still struggle. Perhaps a github-like fine-grain (Xanadu-like transclusion) wikipedia? Or "nLab all the fields"? Or... ??

selimthegrim• 2 years ago

I tried to report an open calculus textbook from Rice University’s talking about relativistic mass as an error (It’s pretty well-established as a bad concept in physics education at this point as opposed to the momentum energy 4 vector) and they wouldn’t accept my feedback.

mncharity2• 2 years ago

Yeah - I've seen "but it's on lists of most common misconceptions" closed wontfix. Errata are good for "author:oops,tnx", but work much less well for confused authors and bad calls, and not at all for judgment calls and alternate approaches. Some other mechanisms are needed.

meroes• 2 years ago

I don't know the solution either. My stats professor religiously attended and espoused these "teaching stats" conventions. But the end result was him always deferring to how the committee did things. The entire pedagogy including how he answered questions. I really didn't like this solution and it made me hate stats until some reacquaintance with it in discrete math.

But then if you're at the mercy of a professor who does things their own way, you can have cases like you give.

One thing that helped was getting syllabi from future potential classes and comparing which textbooks they used. My advisor helped me do this and I credit it with making my senior year more tolerable.

ganzuul• 2 years ago

If you don't like their geometric algebra you can try mine:

https://news.ycombinator.com/item?id=41344163

Then you can try stuff like folding these spaces to make your own multiplication and division with numbers you don't have to explain to anyone!

gjm11• 2 years ago

So far as I can see, this has absolutely nothing whatsoever to do with geometric algebra in the sense being discussed here.

rhelz3• 2 years ago

I'm a big fan of Eric Lengyel, and I never would have ever gotten my arms around Geometric Algebra if it hadn't been for his books and articles. How many people can do theoretical math AND code a state-of-the-art game engine? The guy is a walking, coding miracle.

So if I'm a little critical of the tone of the article, it comes from a place of love. There has been a very toxic, clickish vibe in Geometric Algebra circles, which have lead to some pseudo-disputes among those who should be natural allies.

One such is that Gunn, et al, prefer to represent a 3D vector using a dual basis (e.g. [a1, a2, a3]^T = a1*e32 + a2*e31+ a3*e12) whereas Lengyel prefers to just represent them as a1*e1 + a2*e2 +a3*e3. Some really unfortunately hostile back and forth arguing about which one is "the right way"--when in reality, it's a big-endian vs little-endian thing. One of the best parts of projective geometric algebra is that you can flip back and forth to the dual representation whenever you want to, according to what makes sense to you--and what makes the problem at hand easier to solve. Moreover, if you look at the actual calculations doing it one way vs doing it the other it's the same damn exact numbers being multiplied/added in the same damn way. It's not quite as silly as arguing about what font numbers should be printed with, but it's pretty close.

The tone of the article reflects wounds which are still pretty sore from these sorts of battles. So I understand. But sometimes the invective goes a bit to far.

An example of this is his critique of Gunn's initial cut of dualizaiton for PGA. The fact that e0 is not invertible is a big, fat wart, for sure. And frankly, I spent weeks trying to understand Gunn's workaround and its wierd lingo. J-map? What in the world is a J-map? I finally understood the concept, but I've never found out what "J" stands for :-) And Lengyel's treatment is much smoother and more coherent.

Yet, I don't think Gunn should be criticized for it at all. It was an act of courage for Gunn to come up with his janky J-map, and not let its janky-ness stop him and the rest of the field from moving forward. Sometimes that is exactly what is required in mathematics. For example, infinitesimals. For centuries, mathematicians from Archimedes to Newton found them indispensible, even though they had absolutely no coherent mathematical foundation. In point of fact, if you listen to, say, a Feynman lecture, you'll find that they are still indispensible today. But they didn't have any kind of mathematical foundation until the 1960's, when Robinson found a way to coherently axiomitize them.

I saw a video of Freeman Dyson once, where he was talking about how he was able to prove something which was a longstanding open problem. He described his proof as "very ugly" and then went on to say (with tongue partly in cheek) that you can judge how great a mathematician is by how many ugly proofs he creates :-) Because the first time something is proved, the proof is almost always very ugly. It's not until other mathematicians come in and find connections with other branches of math, and start being able to come up with more elegant proofs.

So let's celebrate the ugly, messy, janky-ness which is the reality of how mathematics is actually created, and the courage of the mathematicians to not rat-hole and bike-shed.

Eric Leyngel's presentation of projective geometric algebra is, IMHO, far more coherent and elegant than any other presentation. His books (and his source code) are a joy to read. For a newb like me, it is far easier and quicker to absorb. Isn't that good enough? Did he really have to go on to flame everybody else to a crisp? sigh like I said, he has been subjected to very unfair and toxic pillorying, and the wounds are still fresh, so like I said, I understand. But its very regrettable nevertheless.

altairprime3• 2 years ago

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.

Did he really have to go on to flame everyone else to a crisp?

No one person is flamed to a crisp here, but some people’s mathematical works are absolutely set on fire. If this had been a critique about the people in geometric algebra, I never would have read it at all, because that’s not what interests me. Critiquing cargo-curled erroneous hearsay as invalid resonates strongly, especially with the focus on the math instead of the people.

As a newb, I learned a great deal about mathematics from reading this, but I still don’t know who any of the people involved are. Isn’t that the holy grail of professional critique: it’s about the work, not the worker?

Or, am I missing something where this article is personally attacking people rather rhan people’s work output?

obsoletehippo• 2 years ago

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.

andrewflnr• 2 years ago

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.

lupire• 2 years ago

I've never found out what "J" stands for
janky J-map

There you go

Infinitesimals

Infinitesimals are completely well-founded. It's just that Archimedes and Newton didn't know a foundation for them.

https://en.m.wikipedia.org/wiki/Surreal_number

sdenton4• 2 years ago

Yes, that's exactly the point...

"But they didn't have any kind of mathematical foundation until the 1960's, when Robinson found a way to coherently axiomitize them."

Arelius2• 2 years ago

So, I've shipped a game with his engine, so have a unique relation ship with him, and he's a complicated individual.

To be clear, he is a very talented, intelligent individual. But he form strong opinions, has a very hard time taking criticism, or understanding other people's differing context, and has a hard time not taking disagreements personally.

This is really nothing new from him. I think the best way to interact with him, is hear out his points, and use them to synthesize your own viewpoint. And be careful to take his views as your own. I think if you do that, you can learn a lot from him, but to be careful as his strong disagreements are often more nuanced then he makes them out to be.

rhelz3• 2 years ago

// I've shipped a game with his engine //

Do you have a blog post or write-up about the experience? I'd love to know more about what working with both the engine and the man is like--and I suspect there's more stories to be told :-)

mgarym4• 2 years ago

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.

ajkjk9• 2 years ago

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.

mgarym4• 2 years ago

I don't think Lengyel is criticizing anything for just being ugly. When he says something is janky, it's because there's something mathematically wrong/incorrect/broken with it.

gowld3• 2 years ago

Where are the mathematicians who could be describing Geometric Algebra correctly?

For a professional working in linear algebra, supervising a graduate student, it should be straightforward to translate these "higher order" linear algebra objects into the geometric algebra model, without publishing technically incoherent mistakes.

nyssos3• 2 years ago

"Geometric algebras" are Clifford algebras over the reals. Differential geometry is probably the field where you're most likely to see mathematicians discuss them, though they also come up in certain (closely related) areas of mathematical physics via spinors.

koolala11• 2 years ago

What about Clifford algebras over the imaginaries? Is that included in your idea of reals?

fouric• 2 years ago

I'm currently very slowly making my way through Geometric Algebra for Physicists by Doran and Lasenby. The book is a delight to read, but I'm not a mathematician, and this article is showing me that my small amount of understanding is...not nearly as deep, and especially not nearly as rigorous, as I would like. I should try to re-read with Eric's criticisms in mind.

Certhas3• 2 years ago

For a physicist it eventually becomes necessary to understand exterior algebra.

This is often done in the context of differential forms, but of course can be brought back to vectors easily. With those well established tools GA doesn't offer much. This blog post seems to point out exactly this fact.

https://en.m.wikipedia.org/wiki/Exterior_algebra

oddthink• 2 years ago

I always come back to Schutz's Geometrical Methods of Mathematical Physics as my reference for notation, but I agree. I came to this by way of General Relativity, so that colors my perceptions. The few treatments of GA that I've looked at (briefly) weren't very clear about the distinction between 1-forms and 1-vectors and seemed to assume Euclidean metric everywhere, so I left thinking that it seemed a little weird and not quite trusting it.

In any case, my experience is that the coordinate-free manipulations only go so far, but that you pretty quickly need to drop to some coordinates to actually get work done. d*F=J is nice and all, but it won't calculate your fields for you.

kevinventullo3• 2 years ago

As a former mathematician from one of the more elitist subfields of math, I have to say this is the first time I’ve heard anyone claim geometric algebra lacks rigor.

hkmaxpro2• 2 years ago

This is the second time I’ve heard geometric algebra lacks rigor. The first time:

https://news.ycombinator.com/item?id=39576214

even by the 90s/00s, GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots
those reasons disproportionately attract people who are not actually capable of rigorous mathematics, or are slightly prone to conspiratorial thinking, or are otherwise slightly deranged
if you look around for papers that explicitly talk GA, they are very disproportionately (a) non-theoretical, (b) poorly-written, (c) trivial, i.e. restating widely-known results as if they’re novel, (d) only citing other GA papers, and of course (e) just plain crackpotty

kevinventullo3• 2 years ago

In the linked blog, the author describes himself: “I am not a mathematician or physicist and am definitely not credible at all”.

generationP2• 2 years ago

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/~lundholm/clifford.pd...

elengyel4• 2 years ago

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.

koolala11• 2 years ago

This person is selling "rigor"...

ziofill• 2 years ago

Virtually all authors define the dual of A […] and there seems to be no consensus as to which one is best. It doesn’t matter because every single one of these definitions is crap.

Love it :)

jmount• 2 years ago

A lot of the (valid) criticism can be summarized by the "No scalar product necessary. Get rid of it" point (early in one of the tables). The next point on inner product is good, and related.

The point is: there is no natural or distinguished isomorphism from a finite vector space to the vector space of linear functionals over it. We know they are isomorphic- but there are many such isomorphisms with little to distinguish between them. The note's "arbitrary multivector, G" fix for inner products is good. The idea is most of us are using the same G that looks a lot like an identity matrix. And as long as you always use the same G you uniquely identify the inner product by equality (as the note does). But if you accidentally combine work that is using different G you run into trouble. You can end up with different Gs if you don't worry enough about naming your basis.

A lot of this keeps getting re-invented as variations of tensors, the exterior algebra, wedge product and so on. I think most of these end up being more bookkeeping than consumers want.

nyssos3• 2 years ago

There's no mathematically natural choice. In physical settings, on the other hand, you almost always want the one induced by the metric.

kkylin• 2 years ago

Not a physicist, but this seems to depend on the physical setting? Seems there are usually metric with physical meaning in continuum mechanics, e.g., elasticity or GR, but not so much if one is working in say geometric mechanics — one can define a Hamiltonian flow on a symplectic manifold without a metric.

forrestthewoods3• 2 years ago

As a fellow game dev this article should be targeted at readers like me. But my eyes can't help but *immediately* glaze over as soon as I read all the math notation.

I'm pretty ok at 3d video game math. I do lots of work with matrices, quaternions, vectors, and friends. It's not particularly difficult.

I can't for the life of me read mathy math. Wikipedia Math is inscrutable hieroglyphics. It's quite frustrating.

I wish someone would write a "foundations" article or book that spoke in language understandable by normal humans. Or at minimum had a bloody legend that explained what all the %&(#%& symbols meant.

runevault2• 2 years ago

Honestly I'm surprised no one has done the programming/CS version of it where all the symbols and formula are explained with code. I'm rusty as hell on math but seems like it should be pretty reasonable to translate a lot of them to functions or similar per symbol.

forrestthewoods3• 2 years ago

Yes please.

This image makes the rounds semi-regularly: https://twitter.com/FreyaHolmer/status/1436696408506212353

stainforth• 2 years ago

"It just means that there’s a lot of low-quality stuff under the same label, which has made that label questionable, and if you want to sift through it you have to be ready to filter for quality yourself... At some level GA is trying to “democratize” geometry." Hit me just now reading the case against GA linked in this thread, maybe the same way Munger remarks on Costco's counter-intuitive membership fee, the notation and systems of "real" math as we encounter them like you say in Wikipedia is the field's membership fee keeping out the riff-raff from overwhelming the gathering place and making the entire store burn down.

"But the point is to make the existing math more intuitive, not to discover new results. The fact that research mathematics is generally not concerned with making calculation and intuition easier to think about is, I think, a giant failure that it will eventually regret. There’s as much value in making things easy to use as there is in discovering them."

gatane• 2 years ago

Modern vector notation (i, j ,k) came to be thanks to an electrical engineer bored of dealing with quaternions lol (Oliver Heaviside)

koolala11• 2 years ago

Squaring and Cubing should be geometric ideas - not just another way to do 1D multiplication. We need new operators for transforming a number into a higher dimention.

Certhas3• 2 years ago

As mentioned elsewhere, that's the wedge product of exterior algebra. Standard stuff first introduced in 1844 that has been foundational to a ton of modern math.

https://en.m.wikipedia.org/wiki/Exterior_algebra

nkhodyunya• 2 years ago

The claim that geometric algebra is not rigorous made me confused. But the lack of proper references is much more confusing to me. Back in the days when I was doing research in Chevalley groups the definitive and most cited source was (and still is) Geometric algebra by E. Artin. And there are some more recent treatments recognized in math community, e.g. classical groups and geometric algebra by L. Groove and the geometry of the classical groups by D. Taylor. At least this books made everything rigorous and clear. But may be there is some other field with the same title?

I've heard about book by Doran and always thought that it's more about applications.

auggierose2• 2 years ago

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.edu/~macdonal/

elengyel4• 2 years ago

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, at the bottom of page 111, that the Hodge dual can't be defined directly in the exterior algebra, which is not correct.

meindnoch3• 2 years ago

I always had the feeling that the bivector.net guys are using a bit weird definitions for certain operations, which then result in weird formulas further down the line. Nice to see Eric putting in the effort to examine their mistakes.

at_compile_time• 2 years ago

Geometric algebra was, for me, an easier on-ramp to more advanced topics in mathematics, leveraging the geometric intuition built into these objects. I learned vector algebra in university, so extending the idea of a direction with magnitude and orientation into areas and volumes with direction and magnitude was a manageable step. Projective algebras similarly allow the construction of points, lines, and planes with orientation and magnitude. Learning that you can generate transformations by dividing one object by another was also really neat. Then you can take the logarithm of that transformation to perform linear interpolation. Geometric algebra is a tool that I use to manipulate geometric objects as easily as I would manipulate real numbers.

The hard part was getting there amidst a host of confusing and conflicting source materials, as this post highlights, and as its author helps proliferate. I used Eric Lengyel's materials a lot in my journey, but I really dislike his decision to represent points as vectors. Notice how every transformation in his poster [1] uses the antigeometric product and antireverse? These operations are only necessary because he's defined everything in the point-based dual space and has to bring everything back to the plane-based space to perform transformations. But he has a wiki and posters, and I'm just here doing my own thing, so I guess that's that.

Alan MacDonald's two books [2] were great, and I would recommend them as an introduction to geometric algebra. Work through the examples and you will learn the material.

[1] http://projectivegeometricalgebra.org/projgeomalg.pdf

[2] http://www.faculty.luther.edu/~macdonal/index.html#geometric...

Poor Foundations in Geometric Algebra

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