Whell, I don't know if this work but...
1 - imagine a sphere surface with center C and radius R
2 - imagine 3 lines L1, L2 and L3 intersecting in C
3 - each line define a point when intersecting the surface, that is all lines have length R from a given point C
4 - the angle between L1 and L2 should be equal to the angle between L2 and L3 and between L1 and L3
5 - the angle should be such that (360/angle)=integer
6 - the 3 points in the end of the lines define a planar triangular surface, which is a "segment" of a sphere
7 - do circular patherns of this surface around C
Can this work? I will give it a try!
Good Luck