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Place equal masses at the six vertices $\pm {\mathbf{e}}_i$ of a regular octahedron and identical unstressed axial springs on its twelve edges. For the invariant axial displacement family ${\mathbf{u}}_{\pm i}=\pm q_i{\mathbf{e}}_i$, the quadratic potential and kinetic energies areHence $m\ddot{\mathbf{q}}=-k(I+11^{\mathsf{T}})\mathbf{q}$. The breathing vector $1=(1,1,1)$ has squared frequency $4k/m$, while every traceless axial vector with $q_1+q_2+q_3=0$ has squared frequency $k/m$. Orthogonal projection onto these subspaces solves any initial condition in this family without an eighteen-dimensional characteristic polynomial.

An octahedron is a polyhedron with eight faces. The regular octahedron has six vertices, twelve equal edges and eight equilateral triangular faces. One realization has vertices $\pm {\mathbf{e}}_1,\pm {\mathbf{e}}_2,\pm {\mathbf{e}}_3$; two vertices are adjacent exactly when they are not an opposite pair. Its edge graph is the octahedral graph.