I stumbled across an app that I can see using in the classroom even though it is more of an art app. This is an app that has both paid and free versions but I'll be talking about the free version. This app, GearDoodleLT, is very similar to the old spiralgraph but it is a virtual version of it.
If you look closely at the picture you will see that it is one gear inside another. The outer gear is static but the inner gear can change size and the pen changes size. You can caluclate the gear ratios to determine one set of values, you can also find the relationship between the radius of the inner circle and the outer circle and the size and type of design you get. In addition, you can add in how the size of the pen point effects the number of repetitions needed to create one full design.
There is a native way of using Archimedes's n-gon to create a circle. You start with a square and find the distance from the center to the side. Use that distance on the diagonal to make a new fold/side Do this to the four diagonals, then cut the folds to make new sides. and repeat till you get a circle. I'll try to make a video this weekend and post it.