Rigid-motion zero modes of a spring framework

For unstressed central Hookean springs, the quadratic elastic energy is $\frac{1}{2}{\sum }_{ij}{k}_{ij}[{\mathbf{n}}_{ij}\cdot ({\mathbf{u}}_i-{\mathbf{u}}_j){]}^2$, with unit edge direction ${\mathbf{n}}_{ij}$. A common translation gives zero displacement difference. An infinitesimal rotation ${\mathbf{u}}_i=\mathbf{\Omega }\times {\mathbf{r}}_i$ also gives zero extension because a vector is perpendicular to its cross product with $\mathbf{\Omega }$. Thus a noncollinear three-dimensional free framework has three translational and three rotational zero modes. Extra mechanisms can give additional zero modes; six is a lower bound until rigidity is established.