We prove a simple nonlinear identity relating the Lovasz theta number of a graph to its smallest radius hypersphere embedding where each edge has unit length. We use this identity and its generalizations to establish min-max theorems and to translate results related to one of the graph invariants above to the other. Classical concepts in tensegrity theory allow good interpretations of the dual SDP for the problem of finding an optimal hypersphere embedding as above. We generate a spectrum of structured SDPs on which extensions of such interpretations are possible. This is joint work with Levent Tuncel.