# Classifying planar polygons based on complexity [closed]

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?

PS:

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.

## closed as unclear what you're asking by Noah Schweber, астон вілла олоф мэллбэрг, ncmathsadist, zhoraster, LeucippusFeb 3 '17 at 4:56

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• I don't see what you mean about "complex polygons". Do you mean very intricated non convex ones ? This is not a problem for computing areas (see the shoelace formula (en.wikipedia.org/wiki/Shoelace_formula)) – Jean Marie Feb 2 '17 at 21:50
• i reworded my question differently to avoid confusion. I meant that any computational in general meant for polygons. By complex polygons i mean something like this: imgur.com/a/99TrQ – user_1_1_1 Feb 2 '17 at 22:21
• One extension: If you consider polygons with holes, with computation of a Euler number, it can become tricky (imagine a polygon as an island, with holes (lakes), then little islands in lakes, then again, little lakes in these islands etc.) – Jean Marie Feb 2 '17 at 22:30
• Exactly! i allow for holes also. So how to categorize the involved complexities upto a finite level of complexity? – user_1_1_1 Feb 2 '17 at 22:46
• You write "When all convex polygons have been exhausted, we have concave polygons." Really? There are extremely simple concave polygons (e.g. chevrons) and extremely complicated convex polygons (e.g. place lots of points around the unit circle at a complicated series of distances); and honestly, I find such simple concave polygons vastly simpler than, say, a regular $p$-gon for a large prime $p$. Before this question can be answered, you need to make precise what you mean by "complexity". – Noah Schweber Feb 2 '17 at 22:56