The continuum limit of icosahedral Goldberg–Coxeter fullerenes: the flat icosahedron with twelve cone points

Let G_ν = GC_{ν,0}(D) be the Goldberg–Coxeter fullerene of the dodecahedron D, with 20ν² vertices, and let L_ν = 3I − A_ν be its graph Laplacian. Omori, Naito and Tate showed that the bottom o(ν²) eigenvalues tend to 0. We identify the limit of the rescaled bottom spectrum ν²λ_k(L_ν): it is E²μ_k/4, where μ_k are the eigenvalues of the Laplace–Beltrami operator on the surface of the regular icosahedron of edge E. That surface is flat except at its twelve vertices, which are cone points of angle 5π/3; it is not the round sphere. Two independent computations reach the same limits to about 10⁻⁹: exact lattice spectra up to ν = 96 (184,320 vertices), and P1/P2 finite elements on the flat icosahedron, validated against the exactly solvable flat tetrahedron. Both show the convergence rate h^(12/5) forced by the cone angle. The departure from the round sphere is quantified: the first nonzero eigenvalue is 0.999024744 times that of the sphere of equal area, the ratio of the second to the first eigenvalue shell is 2.990861832 rather than 3, and the ℓ = 3 shell splits into T2u ⊕ Gu with relative splitting 0.131080131. In the normalisation of Greif, Kaplan, Strichartz and Wiese, our first eigenvalue agrees with theirs to every printed digit, and our higher eigenvalues refine theirs by about four further digits. Scripts and data reproducing every number are included.