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mattferraro.dev
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In this post we will re-invent a form of math that is far superior to the one you learned in school.

My first thought was how hyperbolic that statement is, then I realized, yeah, geometric algebra…

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I don’t know what it is about geometric algebra that brings out the cranks. It’s very handy though that they call it geometric algebra instead of Clifford algebra.

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Over a century ago, the cranks were the ones excited about vectors.

Just remember what Lord Kelvin said:

“Symmetrical equations are good in their place, but ‘vector’ is a useless survival, or offshoot from quaternions, and has never been of the slightest use to any creature.”

Letter to G. F. FitzGerald (1896) as quoted in A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (1994) by Michael J. Crowe, p. 120

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People have taken Clifford algebras very seriously. They’ve been used to try to make geometric sense of second cohomology groups, and the equivalent of Hodge theory has been worked out. Those efforts haven’t yielded a lot back. So far the returns on considering general Clifford algebras over just the exterior algebra have been brutally diminishing. But for some reason they’ve caught the imagination of a wider public. My comments are made as someone who wants to love them, but can’t justify doing so given that simpler tools yield more powerful results.

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IMO the big cool thing about Clifford algebra is how easy it is to teach and understand. Learning Cliifford algebra for me was a lot like when I bought a decent modern MIG welder. I’m genuinely confused as to why the precursors persist in curriculum in spite of being so obviously inferior.

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Why don’t you ask an actual working mathematician or theoretical physicist this question? Like quaternions, Clifford algebras are cool and elegant, but for whatever reason they just haven’t proven very useful in applications to either physics or other mathematics. Exterior algebra and differential forms, on the other hand, have proven themselves to be extremely useful in both theoretical and applied differential geometry (the theory of connections on principal and associated fiber bundles, which is the foundation of modern gauge theories in physics, is most naturally phrased in terms of differential forms).

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Yeah, the author kind of buried the lede, never actually naming this system until the very end. I started out with my crank-radar tingling, but the position made sense, so after a while I skipped forward to find what this is called, and it seems legit.

i see some people dissing it in the comments, but as someone who sort of hit a wall in 2nd-year college physics, I can say that the math derived here seems a lot clearer than the stuff I remember bouncing off of when learning rotational mechanics or E&M.

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To be honest, I didn’t really intend to diss geometric algebra. I’m largely impartial to the topic since it’s been over 8 years since any of the code I write has been related to geometry or the physical world in any way. However, I do remember being really impressed back then with how that formulation holds together and it definitely broadens one’s horizons. I don’t think the excitement these people have for it is baseless, but they do sound like me when I’m trying to explain to my half-asleep great great aunt and her neighbor why everybody should be using Nix.

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If only the folks excited about geometric algebra put a fraction of the effort into learning exterior algebra and differential forms, they would be in a much better position to understand modern differential geometry and the physics that uses it.

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Fraction of the effort of what?

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Learning geometric algebra, obviously (I thought).

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Geometric algebra posts are so predictable that they seem like a conspiracy.

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If only this blog would have a RSS feed.

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What, a newsletter subscription isn’t good enough for you? ;)

Seriously though, I believe a big part of why RSS is not a big thing anymore is that it was hard to monetize.