Aside: Magnetic fields are actually 4D phenomena (x/y/z/t). We use equations to think about fields by reducing fields to 2D so that we may scribble our thoughts down onto clay tablets and papyrus. There is this new medium - called “computers” - that might let us break free of 2D-only expression of 4D phenomena, using, say, 3D (x/y/t) while expressing the 4th D (z) using a trick that humanity invented, called “perspective”. Fields might be more-than-4D phenomena but our brains can only perceive 4 dimensions at this time.

You're absolutely right. That's the whole reason I'm doing these experiments in field visualization. The video we're making is fully 3d rendered, and we were dissatisfied with all the styles of 3d field visualization that we saw out there in the wild. For instance, it's common for videos to use contour lines (in 3d, so splines, but whatever) to show a 2d slice of the field around a solenoid. That works kinda-sorta if you (A) pick the correct angle to slice and (B) use a perfectly ring-shaped solenoid. But for our video, we have a variety of weirdly-shaped coils and permanent magnets moving and interacting in nuanced ways. So we wanted to try to come up with something that would reveal more of the 3d structure of the field. My experiments are happening in 2d, yes, but that's just because it's easier for me to iterate in 2d than 3d. I'm thinking in 3d as I'm working in 2d. (Feel free to do an n+1 on all these uses of "2d" and "3d" to include time, if that's your notational preference — I just always take time as a given since I'm effectively always working in time-based media)