Three Dollar Quill

Hm.

I haven't done too much more work on understanding higher-dimensional magnets. Let's just take what I was saying and try to lay it out without running the numbers yet.

I'm supposing that if I form a solenoid by coiling wire around four facets of a hypercube, then there are two kinds of facet that result: "pole" and "coil". My current intuition/belief is that I can get some idea of how various orientations of solenoids interact by enumerating the different relative orientations, given a fixed displacement, and then running the relative numbers.

I'm not sure how to look for an explanation of this stuff, so my hope is that I can find a geometric algebra expression by doing the calculations in three dimensions, seeing what kind of mathematical behavior I get in known physical arrangements, and then scaling them up. Before considering two solenoids next to each other, I should get some idea of the gradients around a single solenoid. It seems to me that a pole cell should change bivector orientation as you rotate around the face that it shares with a coil cell, but probably no change around the faces it shares with the other pole cells, but maybe my "intuitions" here are getting pushed too far?

It's possible that I'll need to take a break from this and come back to it when I'm ready to be a bit more focused and methodical, but either way, right now, I need to get to sleep.

Good night.