Over the holidays, I made and gave some paper constructions. This one is based on Moroccan tile designs, where lines zig-zagging at ninety degree angles are rotated around each other to make six-pointed stars.
I'd made this earlier, but I wanted to improve the design so there was less cutting and gluing. After several attempts with InkScape and some failed prototypes, I came up with this net. Printing it twice gave me the structure I needed for one sphere.
Whenever I have a design that is a flat, radially symmetric hexagon, I can fold it in to a three-dimensional radially symmetric pentagon -- and pentagons are easily turned into a dodecahedron or icasadodecahedrons. (Think variations on twelve and twenty sided dice.)
You can see how the six pointed stars remain (more-or-less) flat, while the five pointed stars are centered on corners. I suppose I could have creased the design to highlight the polyhedral nature, but I wanted this to look like a sphere.
The most tedious part was punching out the hanging chads from where the cutter-plotter didn't always cut through the card-stock. The most difficult part was gluing along a seam without pulling the previously glued parts apart.
As I got closer to finishing the sphere, I poked a pencil through the construction and used the eraser end to wield the pieces together.
For an encore, and because I had copius amounts of spare time, I designed this construction after seeing something similar on the net. "I can do that," I said. Once I figured out the equilateral triangles were being used to make the five-pointed stars, the rest was fairly easy.