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- Hacker News
- META: Pulling this out of its original context because I think more readers would find the code amusing. I am breaking the rules, but hopefully for a good/pardonable reason.
> Most of the time we think of complex numbers as vectors in R2 or as rotation+scaling operators, but rarely do we actually we want them in both roles at the same time.
I can give one counterexample.
I was asked to comment on a piece of code that did 2D geometry in Python. There was one piece that was a tangle of trigonometry to find the angular bisector of an angle subtended at the origin by two points.
Using the fact that points can be represented by complex numbers and that rotation is just multiplication one can make that function into a one liner.
The geometric mean of the two points as represented by complex numbers gives you the bisector. Python has native support for complex numbers so all the computation is handled by the runtime.√(z1 * z2)by srean - This is like how one often wants to distinguish the points of an affine space from the vectors representing displacements in that space (there is no distinguished origin for the physical world, but there is a distinguished concept of zero displacement). One can add a vector to a point to get a point, or a vector to a vector to get a vector, but cannot add a point to a point to get another point. Yet, it is meaningful to treat a linear combination of points in an affine space as yielding another point in the same space when the weights of the linear combination sum to 1.
The exact same thing is happening here, only multiplicatively, where z1^(1/2) * z2^(1/2) is a combination with two weights of 1/2 (thus, summing to 1). It is geometrically meaningful to treat 2d vectors (displacements in a plane) as complex numbers, raise them to exponents summing to 1, and then multiply these together to get another vector in the same plane. But it is not generally geometrically meaningful to just multiply one vector by another vector to get a third vector in the same space (because this would require distinguishing some particular direction and magnitude as "1").
by Chinjut - Personally, I've got a line of mileage out of using GA to express animation rigs.
I don't know about the rest of the article—I'm not a mathematician—but I certainly enjoying using GA a lot more compared to linear algebra, I find it way more intuitive and being able to visualize intermediate products on my rig is like a super power.
by erichocean - My feeling on geometric algebra is that you should look too much into it until you exhaust the exterior algebra. That is (in my opinion): it isn't a good use of it to replace the cross product or specialized representations of 3-d geometric rotations. It is good for when you get a bit sick of the bookkeeping of the exterior algebra. From a computer scientist point of view it is sort of adding a bit of type information beyond just vector dimension and depth of product.by jmount
- With my limited knowledge, I read through it stumbling along, and from what I gather, this GA is not Clifford Algebra, and the argument is that the GA movement itself is misguided, and that combining operators and geometric objects without distinguishing between them is problematic.
From a programmer's perspective, it seems like they're saying it's a flawed abstraction, while the GA stance is different. I'd like to hear the other side of the argument too. I'm sure HN will get a long GA comment thread, so from their standpoint, what would it feel like? I agree that merging objects and operators is problematic, but I'm curious what the GA camp would say
by jdw64 - 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 the thing to work".[1]
Certain kinds of perfect correctness are like pure and shining crystallised bits of refined knowledge created by the greatest wizards. "Parse, don't validate" or "Make invalid states unrepresentable." ought to be familiar to the better programmers here, the ones with decades of experience built on iterative, collaborative foundations with real consequences for error.
Theoretical physics doesn't have those same consequences, because there is no real punishment for their equivalent of "spaghetti code". Perversely, there's cachet to be gained for gaining understanding of its unnecessarily esoteric knowledge, much like how biologists and lawyers spend half a decade or more studying... Latin.[2]
Introducing Geometric Algebra to physics is like that wizard coder who sweeps away reams of spaghetti code and replaces it all with a call to a single standard library function. It's that "cheff's kiss" of cleanup. Meanwhile the juniors are screaming about how the senior "deleted all their hard work!"
Meanwhile, I never understood where Pauli and Dirac matrices came from! It's like they were pulled from fat air.
You've seen this in code, I bet. Some junior worked really hard on solving a problem and wrote a solid screen-filling wall of "a && b || c || !d && e && (f || g)..." continuing up to "ba, bc, bd", etc.. as they ran out single letters until they're well into the alphabet in double-character symbols.[3]
That's what those matrices are. Someone's hacky attempt at "making things work".
The problem is that we gave those people Nobel prizes and told everyone they're geniuses.
They are, but they were like that brilliant junior. Brilliant.. but junior.
Geometric Algebra sweeps all of that into one beautiful, consistent, crystal clear abstraction that is widely applicable. The magic matrix constants vanish. Bugs in 100-year-old textbook formulas suddenly come to light. Dozens of formulas, one set for each of the 1D, 2D, 3D, and 4D cases collapse into a single formula valid for any number of dimensions.
It's like watching someone struggle with "catching every possible instance of JavaScript injection".
No son, no. Just no. Stop enumerating badness. Stop. Just stop. Escape everything at the boundary instead, enforced by the type system. You'll thank me later.
I know it might be obvious to you, and you always use properly parameterised SQL queries or whatever. This is not the norm everywhere! I still get arguments, long drawn out arguments from people convinced that this is unnecessary and just one more search & replace is all they need to be safe from the bad hackers.
Physicists (and mathematicians) are still making that argument against GA.
"It's isomorphic!"
"That isn't the point!"
[1] You can't convince someone to climb Everest if they struggled to hike up to the top of one of its foothills.
[2] Let me be crystal clear: They're spending their precious time on this Earth learning a dead language instead of learning about the law or bugs. No amount of arguments will sway me. The bugs don't care what you call them. Criminals are guilty or innocent whether or not you speak funny in court. You've just made a simple thing harder for no good reason, that is all. Please stop.
[3] Yes, I've seen this. Twice, from two different people whom have never met. Aliens are amongst us.
by jiggawatts - > this GA is not Clifford Algebra
What makes you say that?
by eigenspace - GA is Clifford algebra plus a bunch of new terminology plus advocacy that it should replace linear algebra.by QuesnayJr
- Mathematician here.
> As I see it, GA is not so much a subject as an ideological position, consisting of basically two ideological claims about the world:
> Claim 1: That the concepts of EA (so, wedge products, multivectors, duality, contraction) are incredibly powerful and ought to be used everywhere, starting at a much lower level of math pedagogy—basically rewriting classical linear algebra and vector calculus.
I support this claim, so I suppose I’m a proponent of geometric algebra.
I think it’s more or less been carried out for vector calculus by Spivak’s “classical” Calculus on Manifolds, which is somewhat widely taught.
> Claim 2: That the Geometric Product (henceforth: GP) should be added to that list as the most fundamental operation, where by “fundamental” I mean that other operations should be constructed in terms of it, and theorems should be stated using it.
Like the author, I also believe this claim is nonsense.
“Rewriting classical linear algebra” is a honored pastime but it’s very difficult to make any headway doing it—the classical texts are classical for a reason, we more or less know how to teach them as an “80% solution” and it’s unclear that the investment in a new pedagogy would get us to an “81% solution.”
Especially with today’s undergrads. If you’re not churning arithmetic, they’re not into it.
- Reading this article, I think there are quite a few interesting points to consider further. C started as a DSL for the Unix kernel. JavaScript is also a DSL, and successful languages are often described as DSLs in certain respects. Then, as they grow and gain broader adoption, they evolve into general purpose languages.
But if you think about it the other way around, since all programs are ultimately about data transformation, you could argue that UIs should essentially be drawn in SQL, but that would sound strange. That's because the tools we use have moved away from that mental model. (Though React's FRP premise does lean in that direction.)
And when I think about why languages split apart, it seems to me that it's because the word 'programming' covers so many different things at once. Languages end up diverging because they serve different purposes. In fact, as a programmer, I see programming languages as a collection of tools that essentially decide what to give up. C gives you safety and low-level hardware access through its ABI. Python gives you expressiveness. They exist because their target goals are fundamentally different.
In that sense, though I'm not an expert in this field, from my limited perspective this debate feels like it's just the noise that arises when Algebra tries to encompass too much and inevitably splits apart. I imagine these kinds of cases will only increase in the future. As things become more specialized, there will be more situations where existing frameworks don't fit, and new systems will be needed. Is there a term for this phenomenon? At that point, we might say we need to change the old system to fit the new one.
Personally, I wonder if there isn't a general purpose language at the bottom that models the entire world, with other languages layered on top of it.
by jdw64 - > That part is fine. But why, then, does multiplying zzˉ give a “magnitude” that works in a reasonable way?
Because the product of all Galois conjugates is a norm and the determinant of the linear operator defined by general field multiplication of a primitive element when viewing the field extension as a vector space of the extension field over the base field.
Although the geometric interpretation of norms in Galois theory really only works for the complex numbers because only the complex numbers are a field. Quaternions are not a field.
by jordigh - The basic issue with geometric algebra is that geometric vectors generally do not have a distinguished notion of unit magnitude (is unit magnitude 1 meter? 1 mile? 1 inch?), so it is silly to work in a framework that requires pretending they do (since the definition of the geometric product of two vectors is dependent upon this choice). Dimensional analysis (a very handy way of tracking mathematical symmetries and thus sanity checking results) goes out the window when working with mixed grade multivectors.
This is not an issue when working with non-mixed-grade multivectors, for which dimensional analysis works just fine in the ordinary way. As the linked article notes, exterior algebra/the wedge product is great. Thinking about exterior powers of vector spaces is great. It's the further move of forcing everything into a Procrustean bed of Clifford algebra that is misguided for almost any application other than some spinor stuff.
by Chinjut - from a theoretical physicist point of view, i find GA don't add much to the standard tooling ppl use, i.e. Lie algebras, Clifford and (sometimes) differential forms. while it's always nice to have a formalism that "hides indices", in most cases (for (super-)gravitation at least) just writing tensor/clifford/lie indices is just much faster and less error prone.
i used to use differential form for gauge theories, einstein-cartan gravitation and ramond-ramond fields.
also, in a paper, we used O(D,D) clifford algebras/spinors to represent differential forms, which worked quite well in our very specific case (appendix A)
https://arxiv.org/pdf/1304.1472
ps: i had colleagues that worked on GA for ML in robotics but wasn't really impressed by what it accomplished
by immmmmm - 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 technical clarity nor conceptual parsimony.,
by Certhas - 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.)by Steuard
- I tried to solve some engineering problems with PGA few years ago. Seemed to work OK up to a point, and at least for me was easier to approach than say Lie algebra or differential geometry.
TFA denigrates papers and websites that are "non-theoretical" or "trivial". As a user of the formalisms, these kinds of materials are exactly what I need. I don't care about proofs or theoretically problematic corner cases that "real mathematics" seems to be almost exclusively interested in.
I did hit a wall quite soon with GA, and got a feel that it may indeed be overhyped, but at least the scene seems to be interested about applied use.
There seems to be similar debate about nonstandard calculus. For my modest use it has provided some tools that can give me results that I don't know how to get with epsilon-delta etc. I don't really care if I don't "really understand" it because the underlying proofs need some heavy machinery. I don't understand those for standard calculus either, and in applied use you either manipulate infinitesimals without any proper algebra, or just hope what you need is in some table.
I can't comment on deeper theoretical or philosophical questions about these, and I don't really care about them. But to me maths communication often seems analogous to making people learn turing machines and lambda calculus before they are allowed to program in Javascript.
I don't think the author necessarily disagrees with me much, but this is maybe a kinda mini rant from a perspective of someone who is just an "end user" of mathematics.
by jampekka - > I don't care about proofs or theoretically problematic corner cases that "real mathematics" seems to be almost exclusively interested in.
That is a rather strange take for a software engineer.
When implementing something I do need to know what the corner cases are, whether the runtime can enter such a state. I need to think how to put in checks so that they cannot be reached, or alternatively, how to recover gracefully. That's my job after all, why would anyone pay me if I didn't.
Perhaps a topical example is a gimbal lock. I need to be aware that it can happen and I need to know how to prevent it.
by srean - > I did hit a wall quite soon with GA
can you give an example of what's impossible/hard to do?
by NooneAtAll3 - This article captures so much of what I have felt but been unable to put into words about GA. I come from a computational physics background, and when translating theory into numerical algorithms, the dimensional analysis and units are very important (you have to be able to relate the simulation to something in the real world!). GA dispenses entirely with any notion of meaningful units, making the dimensional analysis and error checking extremely difficult. The geometric product has always seemed like some strange mathematical trick or coincidence that happens to maybe have some useful properties. Almost like how "new math" is perhaps easier to learn or understand at first, but you really just need to sit down and understand algorithmically what is going on with the basic arithmetic operations.by CyLith
- 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 the Rust community. OK, so they're pushy. Nothing to do with the merits or demerits of the language (or of C for that matter).
If you're a baby duck about linear algebra and geometry, there's no need to care about different formalisms. Do whatever works. But it's interesting to see how all of this stuff comes together at different levels, whether it's the geometric product, differential forms, or just linear algebra.
by jsLavaGoat - 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 much more niche and specialized subjects and not how the subject should be taught when first encountering it.
https://en.wikipedia.org/wiki/Covariant_formulation_of_class...
by Certhas - what is MTW?by NooneAtAll3
- 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.
by cygx - > 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 to learn. While it seems like writing Maxwell's equations in one equation instead of two is a step forward with even more symmetry, what is actually going on is that you are obscuring the most important part of Maxwell's equations: the gauge structure. Without this, it actually becomes much more hidden just how geometric electromagnetism is.
When you write Maxwell's equations as the pair `dF = 0`, `d*F = J`, the first of those two equations is exactly what tells you that this is a gauge theory, and thus may write `F = dA` where `A` is a vector potential. This vector potential then becomes the connection which defines a covariant derivative in a fibre bundle, and one then sees that charged particles follow geodesics now in spacetime, but in an enclosing fibre bundle. This is foundationally important to modern physics, and IMO obscured by writing Maxwell's equations as `∇F = J`
____
n.b. I'm not a particularly big fan of differential forms either, I think it leaves a lot to be desired, and it's super awkward to constantly have to 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.
by eigenspace - 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 PhDs, people with PhDs from unrigorous programs, people who had been good at math but were perhaps going a bit senile, random passerbies from engineering or computer programming, run-of-the-mill circle-squarers, people who had a bone to pick with establishment mathematics and felt like all dissenting views were being unfairly suppressed
> It didn’t help that a lot of the texts by the actually-competent GA people, like the Cambridge group, tended to say things that sounded and still sound kind of crackpotty as well.
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, such as the ostensibly failed academics from a Cambridge research group [1] which the article links to.
I was expecting in the "An Actual Case Against GA" section that the author would demonstrate something like "Geometric Product actually does not work if you apply it to xyz domain". Rather, the section just ended up being mostly about the type of bikeshedding you see about naming of variables in programming.
There is I guess merit to the core "there is no good general interpretation or usage for the geometric product or mixed-grade multivectors" thesis of the article but calling other academics crackpots really subtracts from that message.
- > 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.
by thomasahle - > 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
- 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)
by hamish_todd - 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.by leoc
- you might want to read this post by a GA researcher: https://terathon.com/blog/poor-foundations-ga.html
especially the part about duals -- made me feel like I was going crazy when I was trying to figure out degenerate metrics: every source deals with it in a slightly different (often sloppy) way; you're sure it all must be possible to resolve and get something beautiful and consistent, but not while you're trying to apply it to a specific problem you need to solve
by anuramat