Folding rulers inside triangles

Abstract

An l-ruler is a chain of n links, each of length l. The links, which are allowed to cross, are modeled by line segments whose endpoints act as joints. A given configuration of an l-ruler is said to fold if it can be moved to a configuration in which all its links coincide. We show that l-rulers confined inside an equilateral triangle of side 1 exhibit the following surprising alternation property: there are three values x1 ∼ 0.483, x2. = 0.5, and x3 ∼ 0.866 such that all configurations of n-link l-rulers fold if l ∈ [0. x1,] or l ∈ (x2, x3]. but, for any l ∈ (x1,x2] and any l ∈ (x3a. l ], there are configurations of l-rulers that cannot fold. In the folding cases, linear-time algorithms are given that achieve the folding. Also, a general proof technique is given that can show that certain configurations - in the nonfolding cases - cannot fold.