New blog post: Salvaging the useful parts from the cult - a 2-year review of geometric algebra
In this post, I argue that a lot of the advantageous didactical techniques of geometric algebra are not exclusive to Clifford algebras, and that we are mostly better off reformulating them in classical maths and dropping the "geometric algebra" banner.
https://ash64.eu/assets/ga_review.pdf
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@vacuumbubbles great post, great explanation! I must add, the space of bivectors becomes isomorphic to 𝔰𝔬(p,q), not End(V).
Another comment: you make the argument that multivectors have more geometric intuition than k-forms. But I think this intuition cannot be stretched too far (I'm sure you know all this, but I gotta be pedantic): a (simple) bivector 𝑎∧𝑏 is NOT the parallelogram spanned by 𝑎 and 𝑏, but rather some sort of "oriented amount" of the plane spanned by the two vectors. The parallelogram is only one way to represent 𝑎∧𝑏, but there are many other parallelograms (with same signed area) that represent the same bivector. The bivector 𝑎∧𝑏 does not know anything about the angles and lengths of 𝑎,𝑏.
It becomes much more unintuitive when you ADD different bivectors - and you'll have to, because not all bivectors are simple!
Passing to 2-forms is not very hard conceptually: think of them as being "measuring devices" for bivectors in the sense that they are dual to them. In fact, in order to define a 2-form it suffices to define it on simple bivectors, so in my mind this makes them actually MORE geometrically intuitive than bivectors!
However I strongly agree with the point that we should encourage to identify bivectors, 2-forms and infinitesimal rotation once we have a metric available - this really does help! But as a Riemannian geometer I might be biased here.
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@pschwahn Those are some really good points! It completely slipped my attention that it's not End(V), but just its subspace formed by lowering an index of so(p, q) that's isomorphic to the bivectors.
Also, you're completely right that k-vectors don't encode shape information and that you can deform the parallelograms representing them arbitrarily. I glossed over that in the post, but now that I'm rereading it, it really sounds like too much simplification. Also, I kinda ignored the non-blade-k-vector thing for simplicity (there's the nice example of the spacetime bivector (\gamma_0 \wedge \gamma_1 + \gamma_2 \wedge \gamma_3) that can't be represented as a parallelogram).
And I also fully agree with your intuition for k-forms! What I've started doing is imagining 2-forms as a kind of "extruded raster" with a specific orientation, which you can intersect with a bivector, and then you look through it and count the "cells" that are "blocked" by the bivector (I've tried to depict what I mean in Blender). But that was sorta out of the scope for the post :D
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@vacuumbubbles that's a nice way of thinking about them that I hadn't considered before!
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@pschwahn @vacuumbubbles - nice conversation, guys! I mainly want to agree that it's useful to develop visual/pictorial ways of thinking about differential forms and multivectors, which also involves recognizing the limitations of any sort of picture.
I've been teaching someone about sheaves and germs, and they've been drawing pictures, and it turns out talking about the limitations of drawings is a good way to develop basic insights and intuitions into a subject!
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