# Background

A linkage is a collection of line segments, or links, in the plane which are joined at their ends (joints) to form a graph. Linkages can be reconfigured by a continuous motion of the joints that preserves the length of each link. Here, we enfore that links cannot intersect each other during such motions.

Examples of linkages abound: think of a robotic arm, or the many linkages used in mechanical devices (for instance, to convert a linear motion into a circular motion in an engine.) Or consider a carpenter's ruler which can be folded up, or even the folding of a cardboard box...

Examples of linkages: a robot arm, and a carpenter's ruler.

The most fundamental question about linkages remained open until recently: Can every chain be reconfigured into any other chain with the same sequence of link lengths? A chain is just a linkage whose underlying graph is a path. We can reformulate this as: Can every chain be straightened? If such is the case, we will be able to move between any two configurations with the same link sequence by first straightening, and then "un-straightenening" the chain, since the motions are reversible.