The four bars problem

A four-bar linkage is a mechanism consisting of four rigid bars which are joined by their endpoints in a polygonal chain and which can rotate freely at the joints (or vertices). We assume that the linkage lies in the 2-dimensional plane so that one of the bars is held horizontally fixed. In this paper we consider the problem of reconfiguring a four-bar linkage using an operation called a pop. Given a four-bar linkage, a pop reflects a vertex across the line defined by its two adjacent vertices along the polygonal chain. Our main result shows that for certain conditions on the lengths of the bars, the neighborhood of any configuration that can be reached by smooth motion can also be reached by pops. The proof relies on the fact that pops are described by a map on the circle with an irrational number of rotation.