Grid on a sphere

Question

For which n can we divide the surface of a sphere into n equal parts, such that each part has k neighboring parts, and the total grid has k-fold rotational symmetry around an axis from any of the parts center to the center of the sphere? And no part can be distinguished from another other then fixing a coordinate system.

Is it the same as for the plane ?

Answer 1

Your choices are either $n\ge 2$ identical wedges each stretching from north pole to south pole (with $k=2$), or one of the five Platonic solids projected radially on their circumscribed sphere:

• $k=3, n=4$ (tetrahedron)
• $k=3, n=8$ (octahedron)
• $k=3, n=20$ (icosahedron)
• $k=4, n=6$ (cube)
• $k=5, n=12$ (dodecahedron)

There is no solution where the parts have smaller angular diameter than for the icosahedron.