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Poor Foundations in Geometric Algebra
91–100 of 131 posts
Re: Poor Foundations in Geometric Algebra
#92Earlier quoted context omitted.
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.
That's fine. That's still a step beyond just "linear algebra".
Re: Poor Foundations in Geometric Algebra
#93I'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.
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.
Re: Poor Foundations in Geometric Algebra
#94[dead]
Re: Poor Foundations in Geometric Algebra
#95Note 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 ht…
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
Re: Poor Foundations in Geometric Algebra
#96Earlier quoted context omitted.
What about Clifford algebras over the imaginaries? Is that included in your idea of reals?
Imaginary numbers aren't a field, so there's no such thing. Clifford algebras over the complex numbers work fine, but it's usually not what the people talking about "geometric algebra" are doing.
How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)
Re: Poor Foundations in Geometric Algebra
#97I 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.
Re: Poor Foundations in Geometric Algebra
#98Earlier quoted context omitted.
Imaginary numbers aren't a field, so there's no such thing. Clifford algebras over the complex numbers work fine, but it's usually not what the people talking about "geometric algebra" are doing.
How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)
You seem to be calling complex numbers imaginary numbers, but they're not the same thing. Imaginary numbers are a subset of the complex numbers consisting of the imaginary axis without 0, e.g. i, 2i, -3.1i. Complex numbers also include the real numbers and all combinations of real and imaginary.
Re: Poor Foundations in Geometric Algebra
#99I'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.
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
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.
Re: Poor Foundations in Geometric Algebra
#100As 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 notatio…
Modern vector notation (i, j ,k) came to be thanks to an electrical engineer bored of dealing with quaternions lol (Oliver Heaviside)