The 30 straight edges an icosahedron ( constant Euclidean vertex to vertex distance, 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.