Geometry question, I guess

This isn't homework, just something I'm working on to pass the time, and I've hit a dead end.  What I'm trying to think of is 4 two-dimensional shapes on the surface of a sphere that each share the exact same amount of their perimeter with EVERY other shape.  Eg, if shape #1 shares 5 units along it's borders with shape #2, it also shares 5 units with #3 and #4, #2 shares 5 units with #3 and #4, and so on.  I just can't seem to get it.  Was about to give up, and then I remembered /sci/ and I thought anon might have some insight somewhere in his collective.

Map your sphere to a plane(stereographic projection) for easier visualization if that's a problem and go from there. Nothing immediately comes to mind, but I'll think about it for a bit.

Are the shapes the same?