= Rigid-motion zero modes of a spring framework
For unstressed central Hookean <springs>, the quadratic elastic <energy> is $\tfrac12\sum_{ij}k_{ij}[\mathbf n_{ij}\cdot(\mathbf u_i-\mathbf u_j)]^2$, with unit edge direction $\mathbf n_{ij}$. A common <translation> gives zero <displacement> difference. An infinitesimal <rotation> $\mathbf u_i=\boldsymbol\Omega\times\mathbf r_i$ also gives zero extension because a <vector> is perpendicular to its <cross product> with $\boldsymbol\Omega$. 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.
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