The Medieval Fourier Transform

From my (naive) point of view, the rule looked like when n=1 there are twelve evenly spaced bags around the table, when n=2 there are piles of two bags evenly spaced around the table, n=3 three bags evenly spaced, and so on but then you get to n=5. You can't put 12 bags into 5 even piles.

That's when the whole example broke down for me.

If I had to guess (keeping in mind I'm remarkably dense), I'm fixating too much on the piles. Maybe I'm supposed to be looking at superimposed circles.

It might be less confusing to think about them unrolled into a line. When n=1, I have a line twelve months long and I put the bags evenly spaced along that line. Wrapping that line into a circle shows twelve evenly spaced bags.

When n=2, I have a line 24 months long. Putting the bags evenly spaced along those 24 months, and then wrapping that line into a twice-looped circle would put the bags into piles of two.

At n=5 the line is 60 months long. The bags are evenly spaced along that line and the line is wrapped into five circles, putting the bags into a wonky sort of arrangement, but that's just how it goes.

I think that's what I'm looking at. Hard to tell without a marker and some yarn.