My math is abandoning me. I started out with the design below, which can be used to tile a plane. My thought was that I would rotate the “blade-like” hexagonal elements, since their shapes mesh almost gear-like when rotated in opposite directions. Behind them, the pinwheels would also rotate in alternating directions.
It’s not hard to convert a hexagonal tiling into rectangle tiles that will evenly cover a plane. In this case, drawing a rectangle from the centers of the the blade-like elements at 2, 4, 8, and 10 o’clock will work. Ah, but not necessarily if the elements are rotating! The rotations won’t match up. I’d need to expand the pattern and cut a larger tile from the expanded pattern.
Of course, it also quickly became obvious that I need to consider it a triangular tiling, not the hexagonal tiling that was in my head. With a triangular tiling, I can’t have the blade-like elements rotating in opposite directions because you run into this problem:
Naturally, any odd-numbered circuit will have the same issue. But say we treat the center blade-like element from the initial design differently, and replace it with a different shape, say. The six elements around it could rotate in alternating directions.
You’ll no doubt see where this is going. With this design, I could tile the plane. But what if I wanted a variation of this where the center elements, different though they were, still rotated? I end up back with a variation on my triangle problem.
So after that, I briefly considered going back and altering the original design, but instead decided I didn’t want to animate it after all.