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Poor Foundations in Geometric Algebra

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Re: Poor Foundations in Geometric Algebra

#71

Post by runevault »

forrestthewoods wrote: post #70
runevault wrote: post #68

Earlier quoted context omitted.

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.

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

Freya's content around math is always incredible, and yeah that's exactly what I meant.

Re: Poor Foundations in Geometric Algebra

#72

Post by stainforth »

forrestthewoods wrote: post #67

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 w…

"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."

Re: Poor Foundations in Geometric Algebra

#73

Post by ajkjk »

hkmaxpro wrote: post #64
kevinventullo wrote: post #60

Earlier quoted context omitted.

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

The author responded to this exact statement at this Reddit comment: https://old.reddit.com/r/math/comments/1b5s32x/the_case_agai... You may also read other discussions by the author and others at the Reddit post: https://old.reddit.com/r/math/comments/1b5s32x/the_case_agai... A few Reddit comments appear to agree at that non-mathematicians (e.g. amateurs and game programmers) use geometric algebra non-rigorously. I…

Well the real issue is that the GA proponents do this weird bait-and-switch where they talk about caring a lot about being intuitive and simple, and then they do a bunch of nonsense with the geometric product that makes no sense. It's all at least possibly rigorous (depends on who you're reading), but the thing I want people to question is: is it good? Is the GA way better?

When GA has clean formulas for calculating stuff, sure. But should you write all your linear algebra formulas in terms of the geometric product? heck no. For most people GA is their first exposure to wedge products, though, and those actually are great, so they think that's GA. No, that's just ordinary well-known material that should be in linear algebra classes already. What GA adds on top of that is mostly really weird, although there is a kernel of quality inside it somewhere.

Re: Poor Foundations in Geometric Algebra

#74

Post by ajkjk »

mgarym wrote: post #55
rhelz wrote: post #22

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 circl…

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.

Re: Poor Foundations in Geometric Algebra

#75

Post by ajkjk »

gugagore wrote: post #63
prof-dr-ir wrote: post #51

Earlier quoted context omitted.

> "linear algebra with a uselessly nonstandard notation" Let me chime in that as a physicist (who does use the "pseudo" stuff occasionally) I very much share this opinion. The notation may be really cool and compact, but I just do not see the benefit - for example, d*F = j and dF = 0 is compact enough for me. It is all fine if people use this language to learn linear algebra or differential geometry. And maybe it has…

What other way is there around avoiding polar vs axial vectors looking the same but behaving differently? Or normal vs tangent vectors transforming differently?

Those concepts are from exterior algebra which GA is built on. You do not need GA (the geometric product, etc) for them.

Re: Poor Foundations in Geometric Algebra

#76

Post by nyssos »

jmount wrote: post #62

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 disting…

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

Re: Poor Foundations in Geometric Algebra

#77

Post by nyssos »

gowld wrote: post #29

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.

"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.

Re: Poor Foundations in Geometric Algebra

#78

Post by zarzavat »

forrestthewoods wrote: post #70
runevault wrote: post #68

Earlier quoted context omitted.

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.

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

It’s surprising to me that someone could know what a for loop is but not summation notation. Is summation notation not taught in school?

You have to learn about summation before you learn calculus, because any definition of integration requires series.

Re: Poor Foundations in Geometric Algebra

#79

Post by forrestthewoods »

zarzavat wrote: post #78
forrestthewoods wrote: post #70

Earlier quoted context omitted.

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

It’s surprising to me that someone could know what a for loop is but not summation notation. Is summation notation not taught in school? You have to learn about summation before you learn calculus, because any definition of integration requires series.

Imagine the idea being used for more complex math operations. Summation/Product notation is pretty straight forward (if you were ever taught it). The vast majority of math notation is not. Or rather most people are not taught most math notation. And even if you were taught it it's really hard to remember!

Re: Poor Foundations in Geometric Algebra

#80

Post by ganzuul »

pvg wrote: post #46
ganzuul wrote: post #44

Earlier quoted context omitted.

You can't be serious. Reconsider your assumptions about me. What you are saying in no way supports your perspective.

I don't have any assumptions about you, it's just stuff you should avoid in comments because it tends to trash the conversation and, like the guidelines point out, isn't interesting.

This is obviously not so.
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