Here is a wide variety of mathematical beadwork structures by Horibe Kazunori.
Looking closely at one example, you can see how the surface curvature depends on the structure. Generally, six-sided cycles correspond to an infinite tessellation of hexagons, which makes a flat plane or can be rolled into a cylinder. But in the places where positive curvature (a spherical region) is desired, some pentagons are used instead of hexagons. And in places where negative curvature (a saddle-shaped region) is desired, some heptagons are used instead of hexagons. With this knowledge, the bead designer can control the surface outcome.
Horibe gives detailed instructions for making a beaded buckyball here. (It is in Japanese, but the pictures explain it all.)
See all of George Hart’s Math Monday columns