Configurations in the Plane -- Dylan Thurston
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Suppose you're given a framework with vertices in the plane connected
by rods of fixed length, with a few vertices pinned down to fixed
configurations.  What is the topology of the space of positions it can
flex to?  For instance, here's one problem that arose for designing
steam locomotives: Can you arrange the framework so that one vertex
moves around a circle while another moves along a straight line?  We
will see how to answer this and a large number of other questions by
picking the framework cleverly.