Kirby says: >What students should be reminded of is the rules of the game (like chess) narrow the permitted / legal moves to an exponentially tiny fragment of what's possible, but this strictness is what makes for the rigorous proofs of Euclidean geometry.
Not sure what you are getting at. Let's have the 3d necklace thingy - one can still constrain two opposite sides to be coplanar and parallel, or simply lie in parallel planes, etc. and map out the various properties of the so-and-so constrained varieties. That's a nice way to look at the whole game - as one of imposed constraints and degrees of freedom.
My point had to do with a prejudice to see (some) taxonomies as naturally forming a tree, but that is sometimes misleading.
I'm viewing your post as pointing out that there is a prejudice to seeing "stick-gons" as planar -- fair enough.