John Carlos Baez on Nostr: Every nonzero vector is basically just an arrow: sure, they can have different ...

Every nonzero vector is basically just an arrow: sure, they can have different lengths and point in different directions, but they all look like arrows.

For bivectors the classification is different. Some look like flat surfaces: these are the ones you can write as u ∧ v for two vectors u and v. Some can't be written this way, but can be written as a sum of two terms u ∧ v + w ∧ z for some vectors u,v,w,z. In 3d or 4d space that's as bad as it gets. But in higher dimensions there are bivectors that can only be written as a sum of three terms, or four, etc. When space has dimension at most 2n, you need at most n terms.

For trivectors the classification gets a lot more interesting. You can write every trivector as u ∧ v ∧ w when the dimension of space is low enough, namely 5-dimensional or lower. When you hit 6 dimensions you also get trivectors that you can only write as a sum of two terms, like u ∧ v ∧ w + a ∧ b ∧ c. The same holds in dimensions 7 and 8.

But in 9 dimensional space, ALL HELL BREAKS LOOSE! There are *infinitely* many kinds of trivectors.

To be honest: they can all be written as a sum of three terms, like u ∧ v ∧ w + a ∧ b ∧ c + d ∧ e ∧ f. So you could say there's just one new kind. But that's not how mathematicians think about it. I really need to say more precisely what I mean by a 'kind' of trivector.

[Typical transition for a math professor who is getting ready to talk about what they really REALLY wanted to talk about.]

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