I have a very simple 2d graph.
6 lines separated by equal angle of 60 degrees radiate from the center of a 2d circle, intersecting with the circumference at 6 points. Suppose I know the coordinates of these 6 points, my question is how to turn this graph into a 3d spherical symmetric graph? New lines will be radiating from the center of the sphere, intersecting with the sphere at new points, a cross section of the new spherical object should give me back the original 6-point 2D circle, and how to convert the coordinates of the 6 old points to the coordinates of new vertices on the sphere?
The question seems to be trivial but didn't figure out how to do this. Need some help, thanks.
Copying my own answer from SO and ajusting it.
There is no platonic solid with six-fold symmetry
You might have to elaborate what your symmetry requirements exactly are. But the image I have in my mind when I just read “symmetric” is one where the convex hull of the points on the sphere would be a Platonic solid. And since none of these has a six-fold rotational symmetry, this would be impossible.
Six-fold symmetry with weaker symmetry concept
Since platonic solids don't allow for six-fold symmetry, you might try lower symmetry requirements. For example, six points on the equator of a sphere will satisfy the requirement: they are symmetric under rotation along the normal of the equatorial plane, under reflection in planes normal to the equatorial plane, and also under reflection in the equatorial plane itself. If you want more points, choose any number of parallels and place points at $60°$ intervals there as well. But all of this is very arbitrary, so I feel you might want to explicitely state more requirements.
Archimedean solids can have six-fold symmetry
As an intermediate route between Platonic solids and very arbitrary configurations, you may look at Archimedean solids. The cuboctahedron in particular does have six-fold symmetry: it contains four regular hexagons. It is the shape you get if you place points at the centers of all the edges of a cube, and construct their convex hull. So computing the coordinates should be simple, and the Wikipedia article does give them explicitely.