I took Math 389: Exploration in Mathematics at U of M in the 2021 winter semester. During the course we did two projects. One of them is Gluing the edges of a polygon, the other being Attraction.

For the first project, we looked into the various surfaces we obtain when we pair the edges of a polygon and “glue” them together. If you want to dive in on your own, here is the problem construction.

Readers familiar with the classification of surfaces theorem would quickly realize that the surfaces are just the connected sums of real projective planes and tori. We did in fact proved this result for glued polygons independently, but we did not prove that every surface can be obtained by glued polygons.

We also find a combinatorial formula for the number of pairings that results in a genus-1 surface, although it is not proved in the report.

Also, here are some results from the projects. Here is the project report.

Another project we looked into is called “attraction.” I feel that we did not make much interesting progress in the continuous part, but we derived a very interesting discretized problem based on the project. Here is the project description and our report.