The building block of type-II high-genus fullerenes can be chosen to be an arbitrary Goldberg polyhedron. Puncturing three holes along three carefully chosen pentagons can create a basic unit with three coordination (or a trivalent unit).
I use C60 and its Schlegel diagram to illustrate how to puncture a hole on an arbitrary pentagon.
1. Schlegel diagram of C60
a fused C120 with dumbbel-shape.
3. Of course, if we like, we can puncture two holes on a C60. There are three possible ways. Here I only show the situation with two pentagons separated by two hexagons. The resulting structure will contain two holes connected (or separated) by two heptagons.
4. Punctured C60 with three holes:
5. Here are two possible weaving path. I usually used the first path though.
a. non-spiral path
6. I am working on a project with teachers and students of the Taipei First Girls High School (北一女). We are going to make a giant buckyball consisting of sixty units of punctured C60s. Here are a few basic units I made: