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I get that the way a vector's arrowhead points corresponds to its orientation for a given direction (line). We can also understand vectors within $V$ as isomorphic to a set of endomorphic translations.

I get that clockwise or counter-clockwise rotations represent the orientation of a bivector (with a direction given by the underlying plane). I get how to figure out the orientation of a bivector from $a \wedge b$ vs $b \wedge a$ (follow the left operand on the parallelogram, then go in the direction of the right operand). I get that the magnitude represents the absolute value of 1/2 angle of rotation (ie, just take the bivector represented as a arc of radius 1). Note that for a bivector, the origin point of that rotation isn't immediately expressed in the object's data (it's understood to be the origin of the vector space itself, which didn't matter for 1D).

However, I have 3 questions concerning trivectors (in general dimension $n$ geometric/exterior algebra, rather than just as pseudoscalars in 3D).

  1. I don't understand what orientation is for a volume. Does there exist an intuitive explanation ? My first intuition, given the analogy with curl of a vector field and rotations, would have been something like an analogy with the divergence of a vector field (ie, "flowing in or out of a volume", with the central point of divergence being the origin of the vector space ?), but that seems like it might be wrong.

  2. Additionally, I don't know if there's some standard 3D operation that would correspond to the action of a trivector (like translations are for 1D and rotations are for 2D). Maybe linear contraction/dilation of space ? Since the trivector acts as a pseudoscalar in 3D, this makes sense in that context at least. But does this extend to trivectors in $n > 3$ dimensions ? If not, is there some other operation I should know about ?

  3. Finally, I don't understand how to extend "orientation at the boundary" to "orientation of the interior" from 2D boundaries of 3D objects. I get it for 1D boundaries of 2D objects (following the vectors around the edge), but not any further than that. I'm looking at the diagrams here: https://en.wikipedia.org/wiki/Exterior_algebra but the rotating faces don't even seem coherent (ie, it seems contradictory since 3 gears in planes perpendicular to each other connected 2 by 2 are necessarily gridlocked as a whole).

Thanks for your time.

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(1) Imagine a hand in a volume. It is either a left hand or a right hand. No matter how you reorient (i.e. rotate) the volume, a left hand will remain a left hand and a right hand will remain a right hand. If you reflect the volume across a plane, though, you change the "handedness" of the volume.

(2) Not that I know of. There's probably some Clifford algebra / geometric algebra way to turn an arbitrary $n$-vector into some kind of transformation (of the original space / the algebra itself / spinors), but I feel this might not be in quite the same spirit as the 1D and 2D cases (although I'd argue the 1D and 2D cases aren't in the same spirit as each other, either), maybe closer to the 2D case though.

(3) Imagine the hand in the volume again. Assume the volume is a polyhedron. If you move your hand outward to one of the faces, then use either the right-hand rule or left-hand rule to get an orientation (clockwise or counterclockwise) on that face.

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    $\begingroup$ Thanks ! Not exactly the answer I was hoping for, but it's a helpful answer nonetheless. One thing in particular that your answers to (1) and (3) helped me understand is that I should understand the orientations on the boundary not as "gears perpendicular to each other that turn" but rather as "an orientation on the planes tangent to the volume/manifold that should be the same at each point", and this understanding seems consistent with the drawings. $\endgroup$ Nov 22, 2021 at 12:02

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