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 spherical symmetry.

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

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

• $ K=2?~$ I imagine a spiky surface somewhat like the Mathematica logo.

2. K=-1?

Thanks in advance for solution if available or other suggestions.