The Emergence of the Golden Ratio in the Spectrum of the Hessian of an H_3-Invariant Potential on S^2

We show that for a potential on the unit sphere $S^2$ invariant under the full icosahedral group $H_3$, the ratio of the eigenfrequencies of the tangential modes at the global minima equals $\varphi^2 = \varphi + 1 \approx 2.618$, where $\varphi = (1+\sqrt{5})/2$ is the golden ratio. The ratio of the Hessian eigenvalues (effective stiffnesses) is $\varphi^4$, and the square root of the frequency ratio, which is proportional to the stiffness ratio, equals $\varphi$. The potential is constructed as a linear combination of the Klein invariants $I_6$ and $I_{10}$. The minima are located at the vertices of the icosahedron. The Hessian is diagonalized in a basis distinguished by the $D_5$ symmetry of the vertex. Numerical computation confirms the result with a relative error of $<10^{-5}$. The result is independently verified on a discrete mass-spring model on the vertices of the icosahedron. The geometric origin of $\varphi$ from the structure of the $D_5 \subset H_3$ subgroup and the connection with the $H_4 \to H_3$ reduction are discussed.