Blowing Up Polygonal Linkages

{Consider a planar linkage, consisting of disjoint polygonal arcsand cycles of rigid bars joined at incident endpoints (polygonal chains), withthe property that no cycle surrounds another arc or cycle. We prove that thelinkage can be continuously moved so that the arcs become straight, the cyclesbecome convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease thedistance between any pair of vertices, and preserves any symmetry present inthe initial configuration.In particular, this result settles the well-studied carpentersrule conjecture.