The case against geometric algebra (2024) • Hacker News Board

Re: The case against geometric algebra (2024)

#61

Post by srean » Sun, Jun 21, 2026, 2:51 PM UTC

skybrian wrote: post #54

jiggawatts wrote: post #13

Earlier quoted context omitted.

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…

As a programmer I’m wondering how you get a decent graphics library out of it. If it’s conceptually better, shouldn’t it make writing code to do calculations easier?

> shouldn’t it make writing code to do calculations easier?

You need an optimizing compiler that would take the high level description (in GA) and compile it to add subtract multiply divide of reals (the assembly language). I don't think we have that yet.

Till we have such a compiler it will be tempting to drop down to assembly. Assembly being a metaphor.

Re: The case against geometric algebra (2024)

#62

Post by immmmmm » Sun, Jun 21, 2026, 2:52 PM UTC

eigenspace wrote: post #14

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…

> 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…

agreed, when you start needing the the hodge star, diff form loose quite a lot of their interest.

i'd add it's quite nice in string theories for RR fields and coupling to D-branes, where writing 10 anti-symmetrized indices quickly gets annoying.. and topological field theories..

Re: The case against geometric algebra (2024)

#63

Post by blauditore » Sun, Jun 21, 2026, 2:53 PM UTC

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…

Only tangenially relevant, but the exagerrated differentiation of universities, and levels of education (e.g. PhD vs. not) has always been bothering me. I only have experience in a different field (CS), and yes, those things can be indicators, but I've experienced so many outliers in both directions to know that degrees need to be taken with more than just a grain of salt.

Re: The case against geometric algebra (2024)

#64

Post by ajkjk » Sun, Jun 21, 2026, 2:53 PM UTC

srean wrote: post #55

Chinjut wrote: post #53

Earlier quoted context omitted.

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…

I agree with you on three dimensional vector products. It's too special, too cute and doesn't generalize to all dimensions, and as you said, you have to keep track of the two types of vectors. On complex multiplications though, I disagree. It's a great way to do Euclidean manipulations on the 2d plane. Rotations, translations and reflections (via conjugates) are simple. You rarely need calls to trigonometric function…

What magnitude are you expecting your angle bisector to have afterwards?

Re: The case against geometric algebra (2024)

#65

Post by ajkjk » Sun, Jun 21, 2026, 2:55 PM UTC

cherryteastain wrote: post #48

ajkjk wrote: post #43

Earlier quoted context omitted.

I meant it more as an assessment of the state of affairs, not as an ad hominem (I have no opinion about the people at all). IMO the crackpottery is impossible to ignore, and if you don't talk about it everyone feels like they're going crazy. It's a very widely-noticed thing that is distinct and bizarre compared to other parts of math.

Crackpot really has connotations like "flat earther" and "aliens built the pyramids". It's one thing to say "I believe GA proponents' claims regarding the usefulness of the geometric product are overstated". It's another to say "GA proponents are crackpots".

Well, I said "sound kind of crackpotty", not "are crackpots", about the reputable writers. My point (ish) is that I would really like it if the non-crackpot GA writers would try a lot harder not to sound like crackpots! It is a real issue. I've never seen writing in any other field of math which so frequently strays into sounding unhinged. (Referring mostly to Hestenes + the Cambridge group here.)

Re: The case against geometric algebra (2024)

#66

Post by srean » Sun, Jun 21, 2026, 2:57 PM UTC

ajkjk wrote: post #64

srean wrote: post #55

Earlier quoted context omitted.

I agree with you on three dimensional vector products. It's too special, too cute and doesn't generalize to all dimensions, and as you said, you have to keep track of the two types of vectors. On complex multiplications though, I disagree. It's a great way to do Euclidean manipulations on the 2d plane. Rotations, translations and reflections (via conjugates) are simple. You rarely need calls to trigonometric function…

What magnitude are you expecting your angle bisector to have afterwards?

There was only the need for the point on the bisector (on the unit circle). There was no need for the magnitude of the angle.

The only thing that needed care was which sign of the sqrt bisects the internal angle as opposed to the external angle.

In general I prefer not to deal with angles when dealing with 2D rotation. Get inputs in angles if need be and from then onwards use the (cos,sin) tuple or, equivalently, use complex numbers. One can get rid of calls to trascendentals as long as you are happy to call sqrt.

In other words angle is a tuple.

Re: The case against geometric algebra (2024)

#67

Post by turtleyacht » Sun, Jun 21, 2026, 3:01 PM UTC

Are these titles then the wrong avenue for learning math?

Projective Geometric Algebra: Illuminated (2024) (Not mentioned directly in the article [1]; including a quote from link [2].)

Algebraic Calculus (2016)

Divine Proportions: Rational Trigonometry to Universal Geometry (2005)

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

[2] "If you want solid foundations, this book is for you."

Re: The case against geometric algebra (2024)

#68

Post by ajkjk » Sun, Jun 21, 2026, 3:05 PM UTC

srean wrote: post #66

ajkjk wrote: post #64

Earlier quoted context omitted.

What magnitude are you expecting your angle bisector to have afterwards?

There was only the need for the point on the bisector (on the unit circle). There was no need for the magnitude of the angle. The only thing that needed care was which sign of the sqrt bisects the internal angle as opposed to the external angle. In general I prefer not to deal with angles when dealing with 2D rotation. Get inputs in angles if need be and from then onwards use the (cos,sin) tuple or, equivalently, use…

if they're unit vectors then yes that makes a lot of sense.

The same calculation works in R^n, incidentally, using the geometric product. This is pretty much the ideal usecase for it, for constructing operators between vectors.

Re: The case against geometric algebra (2024)

#69

Post by srean » Sun, Jun 21, 2026, 3:10 PM UTC

ajkjk wrote: post #68

srean wrote: post #66

Earlier quoted context omitted.

There was only the need for the point on the bisector (on the unit circle). There was no need for the magnitude of the angle. The only thing that needed care was which sign of the sqrt bisects the internal angle as opposed to the external angle. In general I prefer not to deal with angles when dealing with 2D rotation. Get inputs in angles if need be and from then onwards use the (cos,sin) tuple or, equivalently, use…

if they're unit vectors then yes that makes a lot of sense. The same calculation works in R^n, incidentally, using the geometric product. This is pretty much the ideal usecase for it, for constructing operators between vectors.

Maybe that's my missing link for

https://news.ycombinator.com/item?id=48619191

You probably know this, but this is one way to generalize beyond 2D

https://en.wikipedia.org/wiki/Angle_bisector_theorem

Re: The case against geometric algebra (2024)

#70

Post by SonOfLilit » Sun, Jun 21, 2026, 3:22 PM UTC

zarzavat wrote: post #45

groundzeros2015 wrote: post #33

Earlier quoted context omitted.

> 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

How is the geometric product any less motivated than any other notation? Ultimately the value of a notation is how easy it makes it to work and think. I'm not sure if GA achieves that or not, but what's the harm in trying a new approach? AFAIK nobody is proposing to replace all of geometry with GA, only 3+1 spacetime.

Author doesn't argue against the idea of choosing a new notation, he makes very detailed arguments about why this specific new notation is clumsy to work with.