Saturday, September 6, 2008

The dual of buckeminsterfullerene and other thoughts about polyhedra

I've been thinking about polyhedra, and I have to thank those who wrote the Wikipedia article Polyhedron for giving me more to think about. The pentakis dodecahedron consists of 60 isosceles triangles joined at 32 vertices (12 five-folds and 20 six-folds). You can view and rotate a three-dimensional version at Technically, this is a Catalan solid. The Catalan solids are the duals of the Archimedean solids, which are semi-regular convex polyhedra composed of two or more types of regular polygons meeting in identical vertices. Catalan solids therefore have one type of polygon but two or more types of vertices. This particular polyhedron is the dual of the truncated icosahedron, or buckyball.

Left to right:
1) Buckministerfullerine (a buckyball).
The shape is known a truncated icosahedron, which is a type of Archimedean solid. This is the structure of carbon-60, which is your basic fullerene.
2) An eicosahedron nested inside of a dodecahedron. These two Platonic solids are duals. The image is from the banner of the web page for the Dept. of Chemistry and Biochemistry at the University of Maryland.
3) A soccer ball.

Buckminister Fuller at Black Mountain College. Buckminister Fuller was at Black Mountain College in the summer of 1948, when this picture was taken, and 1949, when he was director.