Maths always looks for studying patterns and grouping similar structures under same class. I am looking to explore the complexity classes in
2-d polygons as school classwork project.
The simplest polygon is the triangle. Then we will have squares, rectangles and then regular polygons and so on. And when all convex polygons are exhausted, we have concave polygons. I am also taking the liberty of excluding self-intersecting polygons as they get complicated, i think. Now have mathematicians classified complexity of nonconvex polygons at all? For example, say such group of polygons is the simplest nonconvex polygon of class 1 complexity, and then class 2 complexity, and so on.
I want to know how mathematicians go about such questions. For example if a person claims that he has a computer program to do some operations on very complex polygons. Then others will want to start giving him very simple polygons and then move on to very complex polygons to benchmark his software. How would such a spectrum of complexity in polygon geometry be designed?
Holes in polygons are allowed!
Also complexity can be defined more concretely. I just cannot figure how. A naive definition of complexity: large number of edges etc.