[hansvm]

The concept of negative area still feels like it'd get messy in a hurry. For a square pillar, the side lengths should be the same, suddenly giving you imaginary lengths just for the eventual area subtraction to work out. For a negative volume though, you need cubic roots of unity for the side lengths, throwing off your area calculations. Has anyone actually put together a system where the sort of concept you're describing is cohesive?

[mtizim]

You're in for a nice trip, the concept is called Geometric Algebra:

https://youtu.be/60z_hpEAtD8?si=HHs_9m0IJ43nfI3S (~50m video)

TLDW: Yes, the concept is there, makes much more sense than a cross product (which is just an oriented area) and generalizes really nicely.

Alternatively, read: https://en.m.wikipedia.org/wiki/Bivector

[hansvm]

Gotcha; it's less that my complaint doesn't apply, but more that it isn't relevant (i.e., "squares" and "cubes" aren't especially interesting constructs which need to coexist nicely, and if you relax that constraint then directional geometry can be very interesting). Does that sound right?