Deconstructing a parametric plot

Today's exponential sum looks like three octagons, slightly rotated with respect to each other.

I thought that the graph was tracing out one octagon, then another, then another. But that's not what it's doing at all. You can see for yourself by going to the exponential sum page and clicking the animate link.

The curve is being traced out in back-and-forth strokes. It's drawing thin wedges, not octagons. You can get a better picture of this by looking at the real and imaginary parts (x and y coordinates) separately.

It would be hard to look at the first plot an imagine the second, or to look at the second plot and imagine the first.