A framework (G,p) is a geometric realisation of a graph into Euclidean space Rd. (G,p) is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of Rd. The generic rigidity of frameworks in dimension 1 and 2 is well known to be characterised only by a simple counting condition on the graph. Extending this to 3 dimensions is the fundamental open problem of rigidity theory.
In this talk I will discuss a rigidity theory for bar-joint frameworks in Rd+1 whose vertices are constrained to lie on concentric d-spheres with independently variable radii. In particular I will describe a complete combinatorial characterisation of the rigidity of graphs whose vertex sets are partitioned into k colour classes, where vertices of the same colour are assigned to the same variable sphere, for the case d=1. I will also describe how far our methods can be extended to higher dimensions.

This is joint work with Bernd Schulze, Shin-Ichi Tanigawa and Walter Whiteley.