Today’s exponential sum is highly symmetric:

These sums are often symmetric, but not always. For example, here’s the sum from a couple days ago:

It’s not obvious from looking at the parameters whether a sum will be symmetric or not. Maybe someone could find a prove criteria for a sum to have certain symmetries.

For more exponential sums, see this page. There’s a link labeled details that explains what is being plotted, and each page has an animate link to show the steps in creating that image.

I feel like if you let N be larger, you’d get something that’s linearly “periodic” instead of circularly periodic.

That’s true of the 7/7/19 image. It will keep reproducing itself moving from right to left. But the 7/9/19 image will only retrace itself if you increase N.