Nonconvex Cases for Carpenter\u27s Rulers

We consider the carpenter\u27s ruler folding problem in the plane, i.e., finding a minimum area shape with diameter 1 that accommodates foldings of any ruler whose longest link has length 1. An upper bound of 0.614 and a lower bound of 0.476 are known for convex cases. We generalize the problem to simple nonconvex cases: in this setting we improve the upper bound to 0.583 and establish the first lower bound of 0.073. A variation is to consider rulers with at most k links. The current best convex upper bounds are 0.486 for k = 3, 4 and 0.523 for k = 5, 6. These bounds also apply to nonconvex cases. We derive a better nonconvex upper bound of 0.296 for k = 3, 4