Nonconvex cases for carpenter's rulers

Abstract

We consider the carpenter's 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 π/3 − √ 3/4 = 0.614 . . . and a lower bound of 10 + 2 √ 5/8 = 0.475 . . . 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 (about) 0.486 for k = 3, 4 and π/6 = 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.