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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.

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.

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.

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.

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 $\kappa_n=1$.

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 a solution or other suggestions.

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.