The shape is built by cascading several transforms, like the sprocket.
1. Make a conformal cube projection using math from Lee, L. P. (1976): Conformal Projections Based on Elliptic Functions. Cartographica. Monograph 16, supplement no. 1 to Canadian Cartographer, v. 13., 1976.
2. Rearrange the net just enough to make it confusing.
3. Spin it like this; position so the three big squares meet at (0,0).
4-5. Close up the gap using z = z^(3/4).
6. Tumble the sphere so the north pole moves to the center of the 2D shape.
7. Increase lobes from 3 to 5 using z = z^(5/3).
8. Adjust the contents to show 1 sphere instead of 5/3 of a sphere. Briefly,
- stereographically map the sphere to the plane
- z = z^(3/5)
- map the plane back to the sphere
- all the while evading icky numerical trouble at the south pole by handling the north and south hemispheres separately
- fool with the azimuth so that sectors are not misplaced.