Here's an origami-related math question that I thought might be of interest to others in the wider origami community.
If you take a (finite) sheet of paper of any shape and fold it just once, the resulting model will have an area that is greater than or equal to half the area of the original shape, with equality if and only if you fold along a line of bilateral symmetry. The area of the folded model is also less than that of the original sheet. So now let's define R to be the minimum, over all folds, of the ratio of the area of the folded model to that of the original sheet, we know ½ <= R < 1. So for any given shape of paper, we get a specific value R (which doesn't depend on scale). How close to 1 can R get? In fact, you can get arbitrarily close to 1. Now here's the question: What is the least upper bound on R over triangularly shaped sheets? (And what is the shape of the "worst" triangle, or, what is a sequence of triangles whose R's approach this least upper bound?) It seems like a high-school geometry problem, and every now and then, I try to determine what it is...it is some intrinsic constant associated with triangles. Yet, the computations always bog down and I haven't been able to figure it out. Can anyone figure it out?
