[jiggawatts]

Worse, they're snidely dismissive.

I've been interested in GA for years now because it helps me visualise and understand otherwise inscrutable mathematics.

Nobody, literally nobody mired in the traditional mathematics of theoretical physics can explain why the Universe is best represented using matrices of complex numbers with constraints on them.

"Shut up and calculate" or some variant is the common response to such probing questions.

More often, it's some variant of "Well, I can understand it, you need to study more.". This is usually stated just politely enough not to be outright insulting. But if you keep asking probing questions, it turns out that they don't really understand either, the "study" didn't help them either. They only got better at pushing the symbols around on paper They're dismissive of such questions because they're too proud to admit their own ignorance.

Geometric Algebra (GA) was my "lightbulb" moment where I finally understood where Dirac matrices, Pauli matrices, and the like come from and why they have the structure that they do.

My logical conclusion was that GA is the far more elegant, clear, understandable mathematical structure that brings a wide range of Physical phenomena under a unified formulation. So clearly, it should be used for pedagogy.

Nobody agrees with that. The attitude is "well, that's nice, but it's mathematically equivalent so there's no benefit." which is just the stupidest thing I've ever heard.

Imagine if you saw a function called "add_num(a,b)" that computed the sum of two integers using the full bit-by-bit adder digital logic circuit simulated in software using boolean logic. Absolutely bonkers, insane code, right? Clearly this ought to be scrubbed from the codebase and replaced with a simple "+" operator, because we're not maniacs. Physicists would argue "no", it's equivalent, it's "working", so shut up, leave it and just move on.

Drives me batty.

[erichocean]

If you haven't used it already, Versor[0] is nice to play with. GA is simple enough that even normal-ish teenagers can understand it and produce useful results (my sons are using it in a game they're building). Math isn't even close to my strong suit, but Dual numbers and GA make sense to me, and have made it a lot easier for me to do (seemingly, to me anyway) advanced stuff. :-)

[0] http://versor.mat.ucsb.edu/

[K0SM0S]

I 100% agree with you in all respects — I don't come from a physics background but I hear you loud and clear. I think the 'why' is deep and psycho-historical in nature:

- we're exiting the "industrial" mindset where everyone is the same making the same products, to a wider topology of knowledge and skills (more and wider horizontals, more and bigger verticals, 'average' profiles become 'scattered'). This clearly drives a need to "learn a little bit of a lot of things" even at expert level.

- The walls and denial you expose here is to me but a symptom of the disease that current academia will either have to heal or die of. Seeing how Khan (and thousands of Udemy's after them, indies) changed the landscape, my money is on a major paradigm shift incoming for academia (it's already done, they just don't seem to know it yet as institutions, most of them). Lots and lots of great teachers around the world almost freely sharing incredible hands-on knowledge and insight.

- Some applied domains with dramatic tension of the demand side (lots of positions to fill) don't have the luxury of elitism and massively adopt "pragmatic" approaches especially in learning. Software dev, programming and tech in general is much like that — the "one liner" installs and 1-page "getting started", all the intelligence solely put into making things intelligible and usable is, frankly, quite humbling and inspiring in that field. A very good side of the SV/Cali culture. So, examples of how to proceed next really do exist.

Now when I think back of topics that I hurt my head against for months or years, that a simple 20-minute video could 'unlock'... Why, why do we not make it a staple of "teaching" to at least consider 2-3 angles to make sure everyone's got a fair chance at getting at least 1?

- On the topic of hubris and laziness, this is where physics went astray, imho. Too much hubris and not enough laziness. That was back in the 1980s and it took 40 years to realize, probably 10-20 more to "fix", if ever before we build a new system (see above).

That being said,

> Geometric Algebra (GA) was my "lightbulb" moment where I finally understood where Dirac matrices, Pauli matrices, and the like come from and why they have the structure that they do.

YES, please! Geometric algebra seems like the thing that could blow my mind too. I am very visual, to a fault maybe.

Would you have a 'favorite' resource to share? (book, course, youtube, whatever?)

[jiggawatts]

GA is just "strongly typed" vector algebra. It recognises and embraces the inalienable fact that areas and volumes are fundamentally different to vectors and scalars.

The reason Physics "went wrong" is that in 3D space (only!) the mathematics of areas and vectors is coincidentally isomorphic, so it's possible to cheat and use only vectors and scalars and then everything "works". Similarly, volumes and scalars are easily confused as well, and appear to work fine.

GA has no such restrictions and the same formulas work in all dimensions, including high-dimensional or with degenerate metrics. Problems from classical geometry such as finding tangent lines to circles can be trivially extended to finding tangent hyperplanes to hyperspheres, even for very complex problems.

The formalities of GA force you to include things like the square of the unit pseudoscalar in some physics formulas that were accidentally dropped in the traditional form because in 3D this is just "1" and hence easily overlooked. This makes some formulas weirdly difficult to extend to become relativistic, when in fact the problem was just the "weak typing" of vector algebra.

Vector calculus also inherently requires a basis, which is an easy way to get bogged down in the weeds and get confused by issues with the algebra itself instead of the truly "hard" aspects of the problem.

Generally, the "lightbulb" moment for me was that Geometric Algebra has various subsets that are also closed algebras in their own right. For example, the "even" subset of a 3D GA is isomorphic to Quaternions, and the even subset of a 2D GA is basically the same thing as a Complex number. The various "named matrices" are just other subsets of 3D or 4D GAs. Physicists tend to avoid the full general case and simplify their algebras down to the special subset cases, using the historical names and greek symbols. We have to keep the symbols, you see, because otherwise you wouldn't be able to read 2000-year-old ancient greek texts, or... something.

University Physics is actually a study of the History of Physical Philosophy. The computer science equivalent would be learning about abacuses for the entire first semester, then progressing to mechanical calculators in the second semester, vacuum tubes in the second year, and so forth, only to briefly touch on transistors by the end of the third year. Postgraduate research students would be finally told about modern silicon chips and software development, but by this point they're so used to wiring up breadboards manually that it's too late to teach them how to do anything properly.

Starting with something elegant like pure functional programming in the first year is how I studied Computer Science, but I only found out about Geometric Algebra existing after I graduated Physics. It's nuts.

For background reading:

David Hestenes is one of the few physicists trying to reformulate physics in terms of GA: http://geocalc.clas.asu.edu/html/Overview.html

There's lots of papers around: http://geocalc.clas.asu.edu/html/GAinQM.html

Wikipedia is an okay starting point, but not amazing: https://en.wikipedia.org/wiki/Geometric_algebra

Enkimute's "Ganja.js" online demos are amazing, unfortunately the source was written by one of those crazy maths people who think that terseness helps readability: https://enkimute.github.io/ganja.js/examples/coffeeshop.html...

Real industrial use is few and far between, but at least a few folk have discovered that GA is ideal for robotics. Unfortunately, not everyone got the message, and most robotics software libraries are firmly vector/matrix based and have all the usual issues like numerical instability and gimbal-lock. Fun stuff.

[K0SM0S]

Hey there. I'm not sure you'll ever read this, but for the record. THANK YOU, so much.

So.. I've been dabbling with GA since we talked and it is an incredible framework!! I now understand your post loud and clear. It's a new dawn of math for me, I really mean that; Clifford is my new prophet (and I think this one's a keeper possibly for life, I don't know and can't imagine something better for the problem space). So much had not clicked with linear algebra for me, so much of matrices was obscure and had no representation in my mind... And GA's base objects and concepts are so, so elegant, and exquisitely intuitive.

Looking for a short conceptual intro I stumbled upon this channel/playlist: https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ...

Turns out he's an outstandingly good teacher. Strong recommend.

I'll probably take a more "serious" course/book (with problems!) next — if anyone has a recommendation, please do!

Then make progress by working on actual stuff (I guess Hestenes' reformulations are a great starting point, retracing some of these following his reasonning).

And the penultimate goal would be to reformulate stuff myself, if I could — haha, that would be so great. More realistically use GA for research in designing models and representations.

___

TL;DR: you brought Math back into my life. We were on a break (but kept calling each other..) for the last decade and a half. GA is really, really strong. Remind me again, why don't we teach children like that for a century? /s (sigh)

Much thanks again

[K0SM0S]

Wow, thanks so much for all this. I've yet to digest it fully but it's a terrific intro, I love how you worded some of this. You should consider teaching! :)

I can't elaborate much, so just a few "mind blown" moments for posterity:

> GA is just "strongly typed" vector algebra.

That's one hell of $1B slogan, at least around these parts! :) Shut up and take my money.

> in 3D space (only!) the mathematics of areas and vectors is coincidentally isomorphic, so it's possible to cheat and use only vectors and scalars and then everything "works".

I never realized that... there's indeed a lot of confusion in my mind between those concepts. I fail to see how "different" they're supposed to be, I guess really need to go back to sane basic in that regard.

> Geometric Algebra has various subsets that are also closed algebras in their own right.

Just wow. I love this. I actually need this.

> GA has no such restrictions and the same formulas work in all dimensions, including high-dimensional or with degenerate metrics. Problems from classical geometry such as finding tangent lines to circles can be trivially extended to finding tangent hyperplanes to hyperspheres, even for very complex problems.

So that is the real kicker for me, because it fits my problem space so well. I'm exploring highly-dimensional models (basically letting complexity arise from the dimensionality of rather simple/elementary objects, rather than trying to shoehorn complex functions in low-dimensional space in hope of pretty much randomly finding "better fits" — it's a strong desire to not interpret the data before the fact, to remove bias from modeling itself).

There's interesting research around geometric deep learning as well, which seems largely informed by physics as well, and this is sort of the logical conclusion of that for big datasets.

I think industrial use may rise greatly based on this first take. But it's always a generational thing with culture — it takes ~25 years give or take for those who "grew up with it" to finally become the majority of the workforce and sway things their way. Same with politics — looking at you, academia. As you said, "but by this point they're so used to wiring up breadboards manually that it's too late to teach them how to do anything properly."

> It's nuts.

Yeah, it'll take time, never mind how infuriating in the meantime. But good on you, spreading the word about GA is exactly how we move forward, one post, one topic at a time. Eventually, we get there.