# Goldberg variations: New shapes for molecular cages

## Flat hexagons and pentagons come together in new twist on old polyhedra

Molecular graphics performed with the University of California, San Francisco Chimera package. Figure provided by S. Schein.

In the latest verse of a centuries-old mathematical refrain, scientists have figured a way to iron out the wrinkles in a large class of molecular cages. The cages have faces consisting of 12 regular pentagons and up to 480 irregular hexagons, which puts them into a well-known category of shapes called fullerenes. However, unlike most previously known fullerenes, the new shapes’ hundreds of faces are flat rather than warped, and the atoms in the molecule are equally spaced.

The shapes’ flat faces make them convex polyhedra, a type of highly symmetric, faceted solid first studied by the ancient Greeks. The first class to be discovered, called the Platonic solids, consists of solids with identical faces that are all regular polygons, meaning shapes with equal sides and angles. There are only five such solids, the most complicated of which is the icosahedron (familiar to game players as the shape of 20-sided dice). A less