My questions is motivated by Folding the Hyperbolic Crane article which presents non-Euclidean paper for origami and the existence of a special rectangle on Euclidean paper.
Actually, there exists a unique rectangle with sides a, b (a<b) with two ways of being folded along a line through its center such that the area of overlap is minimized and each area yields a different shape – a triangle and a pentagon. It looks like a 'bistable structure' with two minimum.
I am trying to understand the existence of similar 'rectangles' in other geometries. Suppose that we can specify not just side ratio but also absolute size or corner angles of the such rectangles as well.
Do you think there is a general way to prove the existence of the special rectangles?
What would be the possible constraints for such geometries?
