Post #4776257 · Elektrine

2026-09-19 09:04 UTC

A vector looks like an arrow; anyone studying math or physics in college learns about these. Bivectors are important too - though less widely taught. You can create a bivector by taking the 'wedge product' u ∧ v of two vectors u and v, and you can visualize this as the parallelogram shown at left here. But in fact, an ellipse or any other flat surface with the same area lying in the same plane is just as good a picture of this bivector. Further, you can add bivectors, so there are bivectors like u ∧ v + w ∧ z that you can't picture as just a single surface. Thus, you need to be good at the algebraic manipulation of bivectors to keep from making mistakes: unlike with vectors, the picture is just a rough indication of what's going on. Yet bivectors are extremely important. For example, while the electric field is a vector field, the magnetic field is really a bivector field. In 3d space we can convert bivectors into vectors using a trick called the 'cross product': we replace u ∧ v with a vector called u × v. So that's what they teach us in college, to avoid talking about bivectors. But this trick requires an awkward 'right-hand rule', and in higher dimensional space it doesn't work at all. You can go on and define trivectors, etc., and some of the same issues show up. So now let's talk about the classification of trivectors in 9-dimensional space. [Typical way a math professor gracefully transitions to what they really wanted to talk about.] (1/n)

Replies (1)

@johncarlosbaez@mathstodon.xyz 2026-09-19 09:44

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: these new ones 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 way math professor starts to talk about what they really REALLY wanted to talk about.] (2/n)

Open ##4776256