The illustration at the top where sticks are used immediately made me want to provide the "outside the box" solution of arranging the sticks in 3D. But I guess an octagon is pretty clearly defined as a planar object.
Something the article didn't touch on but that I think is fundamental when trying to construct proofs: Often, you can get further in a proof by trying to prove the opposite. Assume that what you're trying to prove is impossible and try to construct a counterexample to the original proof. You will reach a point where the construction can't go further - examine closely what invariant prevents you from completing the counterexample and you will happen across something that will advance your actual proof. Note that this need not result in a proof by contradiction - it's usually more about illuminating the way forward.
In a sense, the octagon problem is teaching exactly this step. That is, the "real" problem here (and the proof that the students should arrive at) is "prove that you cannot construct a convex octagon with 4 right angles". How do you do that? You assume that it's wrong and try to find a counterexample to the original proposition - a convex octagon with 4 right angles. The ensuing "productive struggle" leads exactly to the insights you need to prove the original proposition, as detailed in the article.
I spent most of the article thinking about non-Euclidean solutions to the octagon problem; however, non-Euclidian geometry is clearly not something one would introduce to a group of students who did not already know four right angles cannot be in a convex Euclidean octagon.
In a non-Euclidean space with sufficient negative (hyperbolic) curvature it should indeed be possible to have a convex octagon with four 90° internal angles. Just like in a space with positive curvature, such as the surface of a sphere, it is possible for a triangle to have three 90° internal angles.
If you are willing to start stretching space then we can throw all the polygon definitions out the window. With a strong enough curve in space a sphere can start having corners.
No, polygons in non-Euclidean space makes sense. Polygons and more generally, geometric simplicial complexes in other spaces is a branch of mathematics all by itself (geometric group theory), so you certainly shouldn't "throw it out the window".
Proof by contradiction was a thought that went thru my mind as well when reading this article. Not sure if high school teaches that stream of thought, can't recall using such methodology in school.
Proof by contradiction is how we were shown that sqrt(2) cannot be a rational number. I'm not from the US though (and I don't even know how this generalizes within my country).
to be fair, that would have been valid. The picture is a 2D projection of the sticks, construct it in 3D and look straight down: you have understood something much deeper than just following what seem to be the rules without understanding. Which is exactly what the article is about still, so that's pretty on point.
Something the article didn't touch on but that I think is fundamental when trying to construct proofs: Often, you can get further in a proof by trying to prove the opposite. Assume that what you're trying to prove is impossible and try to construct a counterexample to the original proof. You will reach a point where the construction can't go further - examine closely what invariant prevents you from completing the counterexample and you will happen across something that will advance your actual proof. Note that this need not result in a proof by contradiction - it's usually more about illuminating the way forward.
In a sense, the octagon problem is teaching exactly this step. That is, the "real" problem here (and the proof that the students should arrive at) is "prove that you cannot construct a convex octagon with 4 right angles". How do you do that? You assume that it's wrong and try to find a counterexample to the original proposition - a convex octagon with 4 right angles. The ensuing "productive struggle" leads exactly to the insights you need to prove the original proposition, as detailed in the article.