Distance Configurations of Points in a Plane with a Galois group that is   not Soluble
  We have conjectured that the constraint equations defined by a generic Laman graph are not soluble by radicals when the graph is 3-connected. We prove that this conjecture follows from the following simpler conjecture: the constraint equations defined by a generic Laman graph are not soluble by radicals if the graph does not contain a proper subgraph which is itself a Laman graph. 
John C. Owen (D-Cubed Ltd, Cambridge), Stephen C. Power (Lancaster   University)