The 30 straight edges of an icosahedron (with constant Euclidean vertex to vertex distance, and constant sphere center to vertex distance) have normal curvatures $\kappa_n=0$ 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/or bulgy ovaloids.
• $K=-1$?

Thanks in advance for a solution or other suggestions.