> Proving that something is impossible is a powerful act of mathematics. It shifts our perspective from that of rule follower to that of rule enforcer. And to enforce the rules, you must first understand them. You need to know not just how to apply them, but when they don’t apply. And you also need to be on the lookout for situations where rules might conflict with one another. Our octagon exploration exposes the interplay between polygons, convexity, right angles and angle sums. And it highlights how S = (n – 2) × 180º isn’t just a formula: It’s one condition in a world of competing conditions.<p>This paragraph puts to words something I’ve been mulling over for decades. Looking back to math class, or just looking at any discussion of math education, there’s always a discussion about students who just memorize and plug in the formulas, mentioned as the kinds of students who don’t understand what they’re doing. But that’s an incomplete explanation of what those students are missing that I believe the quoted paragraph completes.<p>It’s not about knowing how to apply a formula. It’s about understanding a formula as a <i>property</i> of the system you’re working with. It’s a great insight and one I’m probably going to mull over for a long while.
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.<p>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.<p>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.
One needs only a tiny bit of space curvature to get a space where a convex octagon with four right angles exists.<p>One could draw one on the surface of a sphere, for example, just as one can draw a triangle with three right angles (<a href="https://www.quora.com/How-can-I-draw-a-triangle-with-three-90-degree-angles-on-a-sphere?top_ans=2573947" rel="nofollow">https://www.quora.com/How-can-I-draw-a-triangle-with-three-9...</a>)
In the theme of this excellent article: solving proof-based problems takes skill, but designing good problems in the first place is even harder. I wonder to what extent solving such problems automatically makes you better at formulating similar problems. Reminds me of how being able to understand a spoken language doesn't help with speaking as much as I expected it would (for me, anyway).
The coin toss example given here seems tautological. Why would my unfair coin have to abide by some norm of probability? If I were designing such a coin, it would obviously include a mechanism for tracking state such that if the last toss was heads the next would always be tails. For example, it could literally redraw the upwards-facing side to the desired outcome an instant before striking the table.<p>I've never had a great head for mathematics because it feels like most impossibilities arise from arbitrary constraints and definitions. Reading such proofs feels like:
1) I've created an arbitrary universe with made up rules
2) This condition does/doesn't break one of those rules I made up
3) QED<p>For example, why on earth can't a straight line in a rectangle have a 180 degree angle at its center making that rectangle an octagon too? It's not clear why one system is epistemologically valid and the other is not. The proof of impossibility feels like a "no true scotsman"/"no true octagon" fallacy disguised by a bit of added notation.
The unfair coin that can't be more likely to give two different results than two the same assumes that each flip has the same probabilities (so k doesn't depend on how many times you flipped it or what you got). Which is fair, but maybe a coin could be built that tends to toggle between outcomes. If it was half full of honey, and you flipped it over once in your hand after catching to read it, it might be more likely to give the other result the next time.<p>Anyway, this is a bit outside the box; the point was that within a set of rules there was a problem that was impossible to solve.
I’ll share one I’ve been pondering for 30 years since my geometry teacher told me it’s impossible: It’s not possible to trisect an arbitrary angle using only a straightedge and compass.
My problem with the example is that it relies on (and illustrates) a convention that seems arbitrary.<p>Why is a polygon with three consecutive collinear vertices not a polygon? Convention. Does anyone here have insight into why it's a good convention?<p>To me it sounds something like saying "9/12 is not a valid fraction because it can be simplified".