The moduli space M_r has dimension 2n-6, at worst isolated singularities and is always a Kahler space. If the side-lengths are integers it is a complex projective variety which coincides with the moduli space of projective equivalence classes of n ordered points on CP^1 weighted by r. If we choose two nonadjacent vertices v_i and v_j then the distance between them (the length of the "diagonal" joining them) is smooth provided v_i and v_j do not coincide. The corresponding Hamiltonian flow rotates the part of the polygon to one side of the diagonal around the diagonal at constant speed while the other part remains fixed. Such a flow will be called a "bending flow". The bending flows corresponding to two nonintersecting diagonals commute.
Now assume that we have chosen a triangulation of P as above. The triangulation is determined by n-3 nonintersecting diagonals and consequently M_r admits n-3 commuting periodic Hamiltonian flows - the bending flows associated to these diagonals. These flows are not usually everywhere defined (because a diagonal in the triangulation could have zero length) so M_r is not always toric. This brings us to the second part of the talk.
The second part of the talk starts with a construction from classical invariant theory that gives a toric degeneration of M_r,that is, a family of spaces over the line with fiber M_r and fiber over 0 a toric variety (M_r)_0. Ben Howard, Andrew Snowden, Ravi Vakil and I used this to describe a presentation for the ring of projective invariants for n ordered points on the line (now on the Archive). But the main point of my lecture is the description of (M_r)_0 as a SPACE. Indeed, (M_r)_0 has a beautiful interpretation in terms of polygonal linkages (but modulo a coarser equivalence relation than congruence). The bending flows are now everywhere defined and coincide with the action of the compact part of the complex torus that acts by virtue of the fact that (M_r)_0 is a toric variety.