The Non-solvability by Radicals of Generic 3-connected Planar Laman Graphs

We show that planar embeddable 3-connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertex-edge count 2v − 3 = e together with a corresponding inequality for each subgraph. The following main theorem of the paper resolves a conjecture of Owen (1991) in the planar case. Let G be a maximally independent 3-connected planar graph, with more than 3 vertices, together with a realisable assignment of generic distances for the edges which includes a normalised unit length (base) edge. Then, for any solution configuration for these distances on a plane, with the base edge vertices placed at rational points, not all coordinates of the vertices lie in a radical extension of the distance field.