Thursday, March 17, 2011

Ih-Symmetric C380: A layer-by-layer inspection

Previously, Bih-Yaw posted a beaded version of this molecule. It is one particular example of a "High-genus fullerene", where the phrase high-genus literally means "many holes", and fullerene is a nickname of a family of cage-like molecules that consist of exclusively graphitic carbon. You can view these molecular structures as two-layered figures. And it is exactly the holes that connect the two layers.

Here we shall inspect the molecule, which is composed of 380 carbon atoms, layer by layer. Readers can then follow this route to construct their own beaded version of the structure. However, please bear in mind that beads should be placed at the edges of the figure. Bead represents chemical bond instead of atom despite of its round shape.

The inner layer is composed of 20 nine-member rings (nonagon):

They sit at the threefold rotational axis of icosahedral symmetry, similar to the hexagons in the pattern of a soccer ball. We notice that there are twelve fivefold symmetric holes each surrounded by five nonagons. It would not surprise a mathematician that these holes are located at the fivefold rotational axis of icosahedral symmetry, and again, similar to the pentagons of a soccer ball.

Next, each of the holes is crowned with five extra heptagons. Simple arithmetic tells us that there are total sixty of them.

Finally, the molecule is completed by filling out the space between the holes with hexagons. There are ninety of them in total.

I've mentioned that this molecule has 380 atoms. (So it is abbr. as C380 by chemists.) Since beads stand for bonds instead of atoms, 380*3/2=570 beads are required to build this molecule.

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