Folding rulers inside triangles

Anl-ruler is a chain ofn links, each of lengthl. The links, which are allowed to cross, are modeled by line segments whose endpoints act as joints. A given configuration of anl-ruler is said to fold if it can be moved to a configuration in which all its links coincide. We show thatl-rulers confined inside an equilateral triangle of side 1 exhibit the following surprising alternation property: there are three valuesx1≈0.483,x2=0.5, andx3≈0.866 such that all configurations ofn-linkl-rulers fold ifl∈[0,x1] orl∈(x2,x3], but, for anyl∈(x1,x2] and anyl∈(x3, 1], there are configurations ofl-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.