= Axial normal modes of an octahedral spring framework
{title2=$\omega_b^2=4k/m,\quad\omega_s^2=k/m$}
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 are
$$
V=k[(q_1+q_2)^2+(q_1+q_3)^2+(q_2+q_3)^2],\qquad
T=m\sum_i\dot q_i^2.
$$
Hence $m\ddot{\mathbf q}=-k(I+\mathbf1\mathbf1^{\mathsf T})\mathbf q$. The breathing <vector> $\mathbf1=(1,1,1)$ 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>.
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