Excel file: 10.Spirals
This is a great file for thinking about fractions. This will be presented as a Bridges 2021 workshop paper in August, 2021. If you like math and art you should know about Bridges.
This comes right after File 1 in terms of difficulty because a simple addition to that file, the radius reduction factor r, produces spirals. Specifically, when r > 0, the radius is reduced by 1/r each time a point is created. (A static sheet, Annotated r, is included for teaching purposes. It shows these fractional reductions for r = 2, 3, 4, and 5 given a square parent polygon, n = 4.) By contrast, when r = 0, there is no radius reduction and polygons or stars result.
Unlike Files 1 through 5, these images are not closed circuits when r > 0. The starting point (at the top of the parent polygon) is not connected to the ending point which, after r jumps, is the center of the circle.
If teaching from PwP, it may be useful to cover this right after File 1. I have not placed it there because I do not want users to think that we will be doing aestheometry on spirals.