# 6 rings with equidistant intersection points

I should note, before I begin, I'm a programmer, not a mathematician, so if my question seems stupid, there's a reason for that. :-j

I'm trying to create a 3d model of a sphere with 6 rings around it. Where the rings intersect are important; they need to be equidistant from one another, and there can't be any more than 2 rings intersecting at any point. This is for a game, where the points of intersection are areas of maximum flexibility.

As a degenerate case, consider the case for 3 rings;

http://s15.postimg.org/e5k18koyj/ring_problem.png

As the image shows, each point where 2 rings intersect has at least 90 degrees of arc between itself and any other point of intersection. What I want is a model with 6 rings, where the distance between any arbitrary point of intersection and it's closest neighbor is maximized.

Here's the openSCAD code I used to create this image. I expect people on this forum will mostly be interested in the rotation matrix I'm using.

orbit_ring_rotations = [
[[    0,   0,   0]],
[[    0,  90,   0]],
[[    0,  90,  90]]
];
for (i = [0:1:3 -1])
rotate(orbit_ring_rotations[i])
rotate_extrude(angle = 360, \$fn=quality)
translate([3, 0, 0])
circle(r = .1);


I really don't have any idea how to attack this problem from here, except to write a program that tries all of the combinations (for integer values between 0 and 359 in the rotation matrix, anyway). But seeing as that's kinda complicated, if there's a math-y solution, I'd love to hear it.