In Definition I.19 Euclid defines polygons like this:
Rectilinear figures are those which are contained by straight lines.
Figures with intersecting sides like the pentagram would probably not have qualified as polygons because there are points that are not unambiguously contained in the figure or not giving rise to questions like: "What's the area of a pentagram?"
I found it quite hard to give a simple and formal definition other than "no non-adjacent sides may intersect". Furthermore, it's not easy to check if a set of points given in Cartesian coordinates does qualify.
It came as an aha-experience when I suddenly saw, how easy it is to check for polygonity when the points are given in polar coordinates (resp. complex numbers):
A polygon is given by $n$ arbitrary but pairwise different points $r_i e^{i \phi_i}$ with $0 \leq \phi_{i} \leq \phi_{i+1} \leq 2\pi$ for $i<n$.
Note the correspondence with
A line is given by two arbitrary but different points $r_1 e^{i \phi_1}$ and $r_2 e^{i \phi_2}$
So lines and polygons - when defined this way: via points and not via straight lines - are on a equal footing (both somehow first-class citizens of geometry: finite sets of points, fulfilling easy to check conditions).
I wonder if there is something deep in this observation, maybe indicating that complex numbers will be of help when understanding arithmetic and the relationship between geometry and arithmetic?
Could there have been a way for Euclid to give a definition of polygons similar-in-spirit to the one above, so the connection with complex numbers would have been discovered earlier?
Or are these considerations in fact in vain, futile, or even worse: trivial and give no deeper insight at all?