Turtles, all the way down

Plotting rectangles on a sphere.

I tried drawing the rectangles as plannar rectangles, but that didn’t work very well. When two rectangles touch each other, you can clearly see the line created by the intersecting planes. To make it work, the rectangles have to curve around the sphere. Although it is relatively easy to draw geodesics connecting the four points, I couldn’t find a parametrized form of spherical polygons. I eventually found a fragment of Mathematica code that recursively approximates the surface with subdividing (plannar) triangles.