just read your 3.5-agon article. I first read about such rational polygons as 7/2 some years ago in ‘regular polytopes’ by Coxeter published 1974, though like you I had also discovered them myself just by wondering what would happen if one applied the rules to a non-integer example. Do you know who first came up with the idea of rational polygons?

Love your suggestion. I’m going to try it! Thanks!

I’d follow that lesson with a follow up lesson asking them to find the sum of the measures of the angles at the star points, generalizing for each k and finally for all (n/k). With the experience of the first table they can cut their explorations in half and be able to generalize.

Where a polytope has a star-polygon for a face or the way they surround a vertex, one uses the appropriate fraction. The 3.5-gon has symbol {7/2}.

I hadn’t noticed the points to count from one vertex to another but by doing so, we can begin to see the connection to combinations and pascals triangle. You’ve given something for me to do on my trip to Denver to explore this further. What a wonderfully rich problem…and amazing discussion thread.

Keep ’em coming, Dan!

I’m new to this 3.5-gon stuff. I was just trying to make sense of the numbers and how they related to the resulting shape. I’m honored that I “discovered” something that the author of Discovering Geometry did not discover!

It is cool that Elaine Watson noticed that the denominator is also the number of cycles to complete the star polygon. I hadn’t seen that before.