Extending a flexible unit-bar framework to a rigid one

We prove that (1) a flexible unit-bar framework G in Rn can always be extended to a rigid unit-bar framework in Rn, and (2) G is ‘congruent’ to a subgraph of a rigid unit-bar framework in Rn if and only if the Euclidean distances between joints of G are all algebraic numbers. Meanwhile, it is proved that a previous result on a framework in R2 [for any real algebraic number r>0, there is a rigid unit-bar framework in R2 having two vertices with distance r apart] extends to any dimension.