I’m not sure yet how I’m going to organize my review. So far I’ve been dabbling widely in the materials I have available. Yet I think it would be a good idea for me to mix drills in the calculation-focused subjects I may eventually want to tutor—calculus, analytic geometry, and so on—with review of some more proofs-based math. Otherwise I’m going to feel as if I’m missing half the show.
Today I was reading Chapter Zero by Carol Schumacher, one of my old textbooks, which was written as an introduction to proofs-based math. In its discussion of definitions, it says the following:
[W]e might naively define a square as “a four-sided, equilateral polygon,” but we would quickly see that such a figure need not have right angles. If we are actually only interested in right-angled figures, then we could refine our definition to be “a four-sided, equilateral polygon with four right angles.”
That immediately had me wondering whether, to know that a four-sided, equilateral polygon is a square, one must know the measures of all of its angles. I suspect that one is sufficient. I believe there are no equilateral, four-sided polygons that have one right angle and do not have four right angles. I haven’t sat down to work out a proof yet, though.
What if the vertices of the polygon, listed in order, are (0,0), (1,0), (0,0), (0,1), and then back to (0,0)? That figure isn’t a square. It’s not convex, and it has edges lying atop one another. Is it a polygon? One of the angles at (0,0) is 90 deg, though. The other angle there is 270 deg. The angles at (1,0) and (0,1) are both 0 deg. The sum of the angles in a quadrilateral should be 360 deg, and that’s true of this figure. You might like a book that excited people when I was a student called Proofs and Refutations, by Imre Lakatos. He’s interested there in Euler’s Theorem that in a polyhedron with V vertices, E edges, and F faces, V-E+F=2. The book is a dialogue in which the theorem keeps getting broken by people pushing the definition of polyhedron just like I’m pushing the definition of polygon. “Is a star a pentagon?” for instance. At one point in the dialogue, they’ve agreed that they want edges to meet only at vertices, faces to meet only at edges, and so on. Ah, says one of the characters, so a cube completely inside another cube is a polyhedron, then. But of course for it, V-E+F=4, twice the value for a single cube. Lakatos thinks theorems come first and definitions later, and that math proceeds by people finding counterexamples and then refining definitions and making more subtle proofs, following which there’s yet another counterexample… Just something fun to think about.
Good questions. I didn’t know whether such a shape was a polygon by established definitions, nor did I have a clear instinct about whether we would want it to be. Mathworld says that there is “unfortunately substantial disagreement over the definition of a polygon” and quotes some definitions that would include your figure and at least one that would not. So I guess that if I start working on this question, I will have to be explicit about that.
I probably will get to it eventually, but I suspect it may be a complicated problem, at least if I don’t want to start with too much proven already. For instance, if I knew that a four-sided, equilateral (convex) polygon had to be a parallelogram (which I suspect is true) and was prepared to use the angle theorems I learned in 8th grade geometry, then the proof would be very simple. I’d like to start more basic than that, but maybe that would be letting myself in for trying to rediscover a couple books of Euclid.
I’m pretty sure I have a copy of Proofs and Refutations. I will check it out.