The 30 straight edges an icosahedron ( constant euclidean vertex to vertex di stance, constant sphere center to vertex distance ) have normal curvatures kn=0 in radial planes). They span and tessellate 20 equilateral triangles of Gauss curvature $K=0$. We try to find parametrization of the surface in other cases of icosahedral symmetry.

When $K=1$, we have a sphere with all $\kappa_n=1$.

How should the normal curvatures change so that the surface has:

• $ K=2?~$ I imagine spiky surfaces somewhat like the Mathematica logo and / bulgy ovaloids .
• $K=-1~? $

Thanks in advance for a solution or other suggestions.