How to put roughly equidistan points on the surface of a sphere, each with 4 neighbors?

How to put roughly equidistan points on the surface of a sphere, each with
4 neighbors?

Say I want to play a game like Go or Go Bang or Chess on a the surface of
a sphere (see here for a question about examples).
The grid on the sphere should have the following characterisitcs:
all nodes more or less equidistant
each node connected to four neighbors
connections don't intersect
I've tried around with constructing such a grid by adding nodes on the
edges of an octahedron, or by adding nodes to a bucky-ball like grid, both
don't work.