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.

Arrange your shapes so that they cover the sphere and so that each shape shares a boundary(doesn't matter how long) with each other shape. Pick a the longest boundary, then introduce squiggles in the other boundaries as required, increasing the lengths to match.

Simply, split the sphere into wedged quarters, rotate one half by 90 degrees(about the equator), then, with squiggles, double the arc length of the equator.